# Math¶

 trueq.math.decomposition.decompose_unitary Decomposes a target multi-qubit Gate into alternating cycles of local operations and the given multi-qubit Gate. trueq.math.decomposition.dekron Finds two unitary matrices whose tensor product is closest to the specified matrix. trueq.math.decomposition.QubitMode Enumerates possible qubit gate decomposition modes. trueq.math.decomposition.split_cycle Exhaustively attempts to split all Gates in this Cycle into Gates that act on as few subsystems as possible. trueq.math.frame.Frame A frame (that is, a spanning set) of a vector space. trueq.math.general.embed_unitary Returns a new unitary where extra dimensions have been added to every subsystem and the given unitary has been embedded into the lowest energy levels. trueq.math.general.int_base Returns the integer solution to base ** n == x. trueq.math.general.int_log Returns the integer logarithm of x in the given base. trueq.math.general.kron Returns the Kronecker product of all supplied arrays (two or more). trueq.math.general.make_confusion_matrix Constructs a confusion matrix from a single error probability or a vector of error probabilities. trueq.math.general.prime_base If the given number is a power of a prime number less than 20, returns the prime. trueq.math.general.proc_infidelity Calculates the process infidelity between two unitary matrices. trueq.math.general.reshuffle Reshuffles the subsystems of a multi-partite matrix. trueq.math.kak_utils.find_kak_equivalent Finds a gate inside of a Config that is equivalent to the target gate up to single-qubit operations. trueq.math.kak_utils.kak Decomposes an arbitrary two-qubit unitary matrix using the KAK decomposition. trueq.math.random.random_bcsz Returns a random CPTP superoperator matrix drawn from the BCSZ distribution (). trueq.math.random.random_density Returns a random density matrix drawn from the trace-normalized Ginibre ensemble of the given rank. trueq.math.random.random_unitary Returns a Haar-random unitary matrix of size (dim, dim). trueq.math.rotation.FixedRotation A fixed unitary rotation. trueq.math.rotation.Layer Parent class for Rotation and FixedRotation. trueq.math.rotation.Rotation A representation of a rotation of the form $exp^{-i \cdot G \cdot \text{param}}$ for some Hermitian matrix $G$. trueq.math.superop.Superop Represents a quantum superoperator. trueq.math.tensor.OperatorTensor Represents an operator (e.g., unitary) or superoperator (e.g., CPTP channel) on a multipartite qubit system. trueq.math.tensor.StateTensor Represents a pure or mixed state on a multipartite qubit system. trueq.math.tensor.Tensor Represents a multipartite tensor on an arbitrary number of subsystems. trueq.math.tensor.to_rowstack_super Returns the superoperator of the operation $X\mapsto AXB^T$ in the subsystem-wise row stacking basis. trueq.math.weyl.WeylBase Base class for symplectic representations of collections and mappings of qudit Weyl operators. trueq.math.weyl.Weyls Represents a list of multi-qudit projective Weyl operators. trueq.math.weyl.Clifford Represents a multi-qudit Clifford operator via its symplectic representation. trueq.math.weyl.Stabilizers Class that represents a list of multi-qudit Weyl operators with associated phases that are constrained to certain discrete values on the unit circle. trueq.math.weyl.StabilizerGroup A class which stores the symplectic representation of a group of multi-qudit stabilizer operators.

## Decomposition tools¶

trueq.math.decomposition.decompose_unitary(given_gate, target_gate, max_depth=3, tol=1e-10, dim=None, max_resets=20)

Decomposes a target multi-qubit Gate into alternating cycles of local operations and the given multi-qubit Gate.

import trueq as tq

circuit = tq.math.decompose_unitary(tq.Gate.cx, tq.Gate.swap)
circuit.draw()

Parameters
• given_gate (Gate) – The Gate to use for the decomposition.

• target_gate (Gate) – The Gate to decompose.

• max_depth (int) – The maximum number of given_gates to use in the decompostion.

• tol (float) – The tolerance for convergence as defined by process infidelity.

• dim (int | NoneType) – The dimension of the subsystems. The default value of None will result in the dimension inferred from given_gate.

• max_resets (int) – The number of times to attempt to reset the algorithm.

Returns

A circuit that implements the target gate up to a global phase.

Return type

Circuit

trueq.math.decomposition.dekron(A, dim=None, force=False)

Finds two unitary matrices whose tensor product is closest to the specified matrix.

import numpy as np
import trueq as tq

U = np.diag([1, 1, -1, -1])

print(tq.math.dekron(U, dim=2, force=True))

(array([[ 1.+0.j,  0.+0.j],
[ 0.+0.j, -1.+0.j]]), array([[1.+0.j, 0.+0.j],
[0.+0.j, 1.+0.j]]))

Parameters
• A (numpy.array) – A matrix to decompose into a tensor product of two unitary matrices.

• dim (int | NoneType) – Dimension of the first square submatrix a. The default value of None will result in the dimension inferred from a.

• force (bool) – Whether to force a decomposition to the nearest tensor factors (True) or raise an error if no exact decomposition is found (False).

Returns

The closest unitary tensor factors of the input matrix.

Return type

tuple

Raises

DecompError – if force=False and no valid decomposition is found.

class trueq.math.decomposition.QubitMode(value)

Enumerates possible qubit gate decomposition modes. A decomposition mode of ZXZ, for example, results in decompositions of the form [("Z", angle1), ("X", angle2), ("Z", angle3)], where entries appear in chronological order, and angles are specified in degrees

Some decomposition modes contain 5 rotation axes, such as ZXZXZ. These modes rely on identities of the form $R_X(\theta)=R_Z(-\pi/2)R_X(\pi/2)R_Z(\pi-\theta)R_X(\pi/2)R_Z(-\pi/2)$. Therefore, these modes guarantee that the second and fourth rotation are always +90 degrees. This is useful when virtual gates are used for one rotation axis. If virtual Z gates are used, and the mode is ZXZXZ, then gates will be decomposed in the form [("Z", angle1), ("X", 90), ("Z", angle2), ("X", 90), ("Z", angle3)] and only one pulse shape is required to create any single qubit unitary.

decompose(gate)

Decomposes the given gate into this mode, where the output format is a list of tuples of the form (pauli, angle).

import trueq as tq

tq.math.QubitMode.ZXZ.decompose(tq.Gate.y)

[('Z', 0.0), ('X', 180.0), ('Z', 180.0)]

Returns

A list of axes and angles to generate the gate from this mode.

Return type

list

trueq.math.decomposition.split_cycle(cycle, max_size=6)

Exhaustively attempts to split all Gates in this Cycle into Gates that act on as few subsystems as possible.

import trueq as tq

# Generate a three-qubit gate from a product of seperate single-qubit gates.
# This is done using the & shorthand for Kronecker product.
gate = tq.Gate.s & tq.Gate.h & tq.Gate.y
cycle = tq.Cycle({(0, 1, 2): gate})
tq.math.split_cycle(cycle)

True-Q formatting will not be loaded without trusting this notebook or rerunning the affected cells. Notebooks can be marked as trusted by clicking "File -> Trust Notebook".
 Marker 0 Compilation tools may only recompile cycles with equal markers. (0): Gate.s Name: Gate.s Aliases: Gate.s Gate.sz Gate.cliff9 Generators: 'Z': 90.0 Matrix: 1.00j -1.00 (1): Gate.h Name: Gate.h Aliases: Gate.h Gate.cliff12 Generators: 'X': 127.279 'Z': 127.279 Matrix: 0.71 0.71 0.71 -0.71 (2): Gate.y Name: Gate.y Aliases: Gate.y Gate.cliff2 Generators: 'Y': 180.0 Matrix: -1.00 1.00

Note

The cost of this function scales exponentially with the number of qubits. Setting max_size to be greater than 6 for qubits could result in extremely long delays.

Parameters
• cycle (Cycle) – The cycle containing the gates to be split.

• max_size (int) – The maximum subsystem size to attempt to split. Any gate acting on a larger subsystem will be skipped.

Returns

A Cycle containing gates that act on as few subsystems as possible.

Return type

Cycle

## Frame¶

class trueq.math.frame.Frame(frame)

A frame (that is, a spanning set) of a vector space.

import trueq as tq

# Make a frame with elements "U", "D" where U=[1, 0] and D=[0, 1]
b = tq.math.Frame({"U": [1, 0], "D": [0, 1]})

Parameters

frame (dict) – A dictionary of frame elements.

shape(n_sys)

Returns the shape of the vector space spanned by tensor products of the frame operators.

Parameters

n_sys (int) – The number of tensor factors.

Return type

tuple

labels(n_sys)

Returns an iterator of basis element labels for multiple systems. The order is the same as elements.

import trueq as tq

b = tq.math.Frame({"U": [1, 0], "D": [0, 1]})
list(b.labels(2))

['UU', 'UD', 'DU', 'DD']

Parameters

n_sys (int) – a positive integer specifying the number of systems under consideration.

Return type

map

get_element(label)

Returns the array corresponding to the given label or the Kronecker product of the arrays for an iterable of labels.

import trueq as tq

b = tq.math.Frame({"U": [1, 0], "D": [0, 1]})
print(b.get_element("D"))
print(b.get_element("DU"))

[0.+0.j 1.+0.j]
[0.+0.j 0.+0.j 1.+0.j 0.+0.j]

Parameters

label (Iterable) – Which element to fetch.

Returns

The array for a given basis element.

Return type

numpy.ndarray

elements(n_sys=1)

Returns an iterator of frame elements for multiple systems. The order is the same as labels.

import trueq as tq

b = tq.math.Frame({"U": [1, 0], "D": [0, 1]})
list(b.elements(2))

[array([1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j]),
array([0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]),
array([0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j]),
array([0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j])]

Parameters

n_sys (int) – A positive integer specifying the number of systems under consideration.

Return type

map

n_sys(arr)

Determines how many systems an array acts on.

Parameters

arr (array_like) – An array with shape equal to the shape of this basis.

Returns

The number of systems an array acts on.

Return type

int

to_coeffs(arr)

Expands the given array in this frame, returning a 1D vector in the same order as labels. This function is the inverse of from_coeffs().

Parameters

arr (array_like) – An array with shape consistent with the shape of this frame for some positive integer.

Returns

A vector of expansion coefficients.

Return type

numpy.ndarray

from_coeffs(coeffs)

Takes a 1D array, interprets it as a list of coefficients with respect to this basis, and returns the array resulting from summing up these coefficients times their corresponding basis element. This function is the inverse of to_coeffs().

Parameters

coeffs (array_like) – A 1D array of coefficients.

Returns

The matrix specified by a vector of expansion coefficients.

Return type

numpy.ndarray

to_expansion(arr)

Expands the given array in this basis, returning a dictionary mapping labels to coefficients.

Note

Only non-zero coefficients are normally included. However, in the case where arr is the zero array, instead of returning an empty dictionary, a dictionary with the first member labels mapped to $0$ is returned.

Parameters

arr (array_like) – An array with shape consistent with the shape of this basis for some positive integer.

Returns

A sparse representation of arr with respect to this frame.

Return type

dict

from_expansion(expansion)

Takes a dictionary mapping basis elements to coefficients and returns the array resulting from summing up these coefficients times their corresponding frame element. This function is the inverse of to_expansion().

Parameters

expansion (dict) – A dictionary mapping basis elements to coefficients.

Returns

A dense matrix from a sparse set of expansion coefficients.

Return type

numpy.ndarray

plot_arr(arr, ax=None)

Plots the given array as the coefficients in this basis/frame.

Parameters
• arr (array_like) – An array with shape consistent with the shape of this basis for some positive integer or a list thereof.

• ax (matplotlib.Axis) – An existing axis to plot on. If not given, a new figure will be created.

plot_superoperator(arr, ax=None)

Plots the superoperator that acts by conjugation by an array or a list of arrays on a state expanded in this frame.

Parameters
• arr (array_like) – An array with shape consistent with the shape of this basis for some positive integer or a list thereof.

• ax (matplotlib.Axis) – An existing axis to plot on. If not given, a new figure will be created.

superoperator(arr)

Returns the superoperator that acts by conjugation by an array or a list of arrays on a state expanded in this frame.

Parameters

arr (array_like) – An array with shape consistent with the shape of this basis for some positive integer or a list thereof.

Returns

A matrix representation of a superoperator with respect to this frame.

Return type

numpy.ndarray

frame.pauli_basis = <trueq.math.frame.Frame object>

## General tools¶

Accepts a matrix representing an operator acting on labels, which is then reshuffled based on the ordering of all_labels. For each subsystem in all_labels not contained in labels, an identity matrix is inserted in the returned matrix, respecting the order of all_labels.

For example, given the matrix representation of $X \otimes Z \otimes Y$, labels of (0, 2, 3), and a target all_labels of (2, 1, 0, 3), this method would return the matrix reperesentation of $Z \otimes I \otimes X \otimes Y$.

Parameters
• mat (ndarray) – The matrix which is being manipulated.

• labels (tuple) – The current labels associated with the provided matrix.

• all_labels (tuple) – The target label ordering, including any new labels.

Return type

ndarray

Raises

ValueError – If labels is not a strict subset of all_labels, or if there are repeat labels in either tuple, or if mat has an incompatible shape with labels.

trueq.math.general.embed_unitary(u, n_sys, extra_dims)

Returns a new unitary where extra dimensions have been added to every subsystem and the given unitary has been embedded into the lowest energy levels. The returned unitary acts trivially on the extra levels.

import trueq as tq

# inject the CNOT gate into two qutrits
cnot3 = tq.math.embed_unitary(tq.Gate.cnot.mat, 2, 1)
tq.visualization.plot_mat(cnot3, xlabels=3, ylabels=3) Parameters
• u (numpy.ndarray-like) – The unitary matrix to embed.

• n_sys (int) – The number of subsystems the unitary acts on.

• extra_dims – The number of extra dimensions to add to each subsystem.

Returns

An embedding of u into a larger space.

Return type

int

trueq.math.general.int_base(x, n)

Returns the integer solution to base ** n == x. Returns None when no exact solution exists to numerical precision.

import trueq as tq

print(tq.math.int_base(16, 4))
print(tq.math.int_base(9, 2))
print(tq.math.int_base(12, 2))

2
3
None

Parameters
• x (int) – The positive integer to find the base of.

• n – A positive integer; the power to which the base should be raised to get x.

Returns

The integer solution to base ** n == x.

Return type

int | NoneType

trueq.math.general.int_log(x, base)

Returns the integer logarithm of x in the given base. Returns None when no exact solution exists to numerical precision.

import trueq as tq

print(tq.math.int_log(16, 2))
print(tq.math.int_log(16, 4))
print(tq.math.int_log(16, 3))

4
2
None

Parameters
• x (int) – The positive integer to compute the logarithm of.

• base (int) – The base of the logarithm.

Returns

The integer logarithm in a specified base.

Return type

int | NoneType

trueq.math.general.kron(*arrs)

Returns the Kronecker product of all supplied arrays (two or more). Note that this function does not work with numpy.complex128 data type.

Parameters

*arrs – One or more numpy.ndarrays.

Returns

The Kronecker product of the specified arrays.

Return type

numpy.ndarray

trueq.math.general.make_confusion_matrix(error, n_sys=1, dim=None, n_outcomes=None)

Constructs a confusion matrix from a single error probability or a vector of error probabilities. If a matrix is given, it is checked to have columns that sum to unity before being returned.

If $p$ is the single probability error or a vector of error probabilities, it is turned into confusion matrix by setting the diagonal elements to $1-p$ and the off-diagonal elements of the first m = \text{min}(n_o, d)^n columns to $p/(n_o - 1)$, and the elements of the remaining columns to $p/n_o$, where $n$ is n_sys, $d$ is dim, and $n_o$ is n_outcomes. Typically, one will want $d=n_o$, which is the default behaviour.

import trueq.math as tqm

print(tqm.make_confusion_matrix(0.01))
print(tqm.make_confusion_matrix([0.1, 0.2, 0.3]))

[[0.99 0.01]
[0.01 0.99]]
[[0.9  0.1  0.15]
[0.05 0.8  0.15]
[0.05 0.1  0.7 ]]

Parameters
• error (float | array_like) – An error probability, a vector of error probabilities, or a matrix whose columns sum to unity.

• n_sys (int) – The number of subsystems.

• dim (int) – The dimension of each subsystem.

• dim – The number of outcomes.

Return type

numpy.ndarray

Raises
• ValueError – If the columns of the output do not sum to unity or if any entry is negative.

• ValueError – If any provided dimensions are not consistent with the value of error.

trueq.math.general.prime_base(n)

If the given number is a power of a prime number less than 20, returns the prime. Otherwise, returns None.

Parameters

n (int) – The number to check.

Return type

int | NoneType

trueq.math.general.proc_infidelity(A, B=None)

Calculates the process infidelity between two unitary matrices.

Parameters
• A (numpy.ndarray) – A unitary matrix of shape (d, d).

• B (numpy.ndarray | NoneType) – A unitary matrix of shape (d, d). If B is None, then its assumed to be an identity and a faster calculation takes place.

Returns

The process infidelity between the two unitary matrices.

Return type

float

trueq.math.general.reshuffle(mat, perm)

Reshuffles the subsystems of a multi-partite matrix.

import numpy as np
import trueq as tq

# Cnot on qubits (0, 1)
mat = np.array([[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 0, 1],
[0, 0, 1, 0]])

# We can reverse the direction of the CNOT matrix so that the control qubit
# becomes 1 instead of 0:
tq.visualization.plot_mat(tq.math.reshuffle(mat, [1, 0]), xlabels=2, ylabels=2) Parameters
• mat (numpy.ndarray) – A numpy.ndarray object to be reshuffled, which must have an (n, n) shape where n = dim ** len(perm).

• perm (Iterable) – A list of indices to permute which must contain all integers from 0 to the number of systems in the mat.

Returns

The reshuffled matrix.

Return type

numpy.ndarray

## KAK tools¶

trueq.math.kak_utils.kak(X)

Decomposes an arbitrary two-qubit unitary matrix using the KAK decomposition.

The decomposition is of the form np.kron(a0, a1) @ U @ np.kron(b0, b1) where U is a unitary matrix with hamiltonian only containing "II", "XX", "YY", and "ZZ" terms.

This returns (a0, a1), (b0, b1), and the generators for a Gate "U".

The original unitary can be reconstructed as follows.

import trueq as tq

A, B, k = tq.math.kak(tq.Gate.cnot)
gen = {p: v for p, v in zip(['XX', 'YY', 'ZZ'], k)}
tq.Gate(np.kron(*A) @ tq.Gate.from_generators(gen).mat @ np.kron(*B))

True-Q formatting will not be loaded without trusting this notebook or rerunning the affected cells. Notebooks can be marked as trusted by clicking "File -> Trust Notebook".
Name:
• Gate.cx
Aliases:
• Gate.cx
• Gate.cnot
Likeness:
• CNOT
Generators:
• 'IX': 90.0
• 'ZI': 90.0
• 'ZX': -90.0
Matrix:
Parameters

X (Gate) – A two-qubit gate.

Returns

A decomposition of the input.

Return type

tuple

trueq.math.kak_utils.find_kak_equivalent(conf, gate, param_scaling=180, tol=0.01)

Finds a gate inside of a Config that is equivalent to the target gate up to single-qubit operations.

Parameters
• conf (Config) – A Config object containing multi-qubit gates.

• gate (Gate) – The target gate to find inside the config.

• param_scaling (float) – Upper bound on the parameter range that is searched to match parameter values for parametrized gates.

• tol (float) – Tolerance to decide if convergence was successful, success is defined as the L1 distance between target KAK and found KAK is less than the tol.

Returns

a list of Gates that are equivalent to the target.

Return type

list

## Random Generators¶

trueq.math.random.random_bcsz(dim, rank=None)

Returns a random CPTP superoperator matrix drawn from the BCSZ distribution (). The superoperator is in the global row stacking basis.

See also random_bcsz() which invokes this method to create a new Superop() object.

Parameters
• dim (int) – The dimension the superoperator will act on.

• rank (int | NoneTye) – A positive integer specifying the output Choi rank. If None, then the value dim ** 2 is used. A value of 1 results in a random unitary superoperator.

Returns

A matrix of shape (dim**2, dim**2).

Return type

numpy.ndarray

trueq.math.random.random_density(dim, rank=1)

Returns a random density matrix drawn from the trace-normalized Ginibre ensemble of the given rank. By default, the rank is one so that this function returns random pure states.

Parameters
• dim (int) – The dimension of the density matrix.

• rank (int) – A positive integer specifying the rank of the output density matrix.

Returns

A random density matrix of a specified matrix rank.

Return type

numpy.ndarray

trueq.math.random.random_unitary(dim)

Returns a Haar-random unitary matrix of size (dim, dim).

See also random_unitary() which invokes this method to create a new Superop() object.

Parameters

dim (int) – The size of the unitary matrix.

Returns

A Haar-random unitary matrix.

Return type

numpy.ndarray

## Rotations¶

class trueq.math.rotation.FixedRotation(mat)

A fixed unitary rotation.

import trueq as tq
import numpy as np
from trueq.math import FixedRotation

g = FixedRotation(tq.Gate.x.mat)
assert g.pauli_angle == ("X", 180)

Parameters

mat (numpy.array) – A Unitary matrix.

property pauli_angle

If this layer can be represented by a fixed rotation along a Pauli axis such as $exp^{-i \cdot \frac{\pi}{360} \text{Pauli} \cdot \text{angle}}$, this will be a tuple (Pauli, angle), otherwise this is (None, None)

import trueq as tq
from trueq.math import FixedRotation

g = FixedRotation.from_pauli("XY", 90)
assert g.pauli_angle == ("XY", 90)

g = FixedRotation.from_pauli("Y", 11)
assert g.pauli_angle == ("Y", 11)

Type

tuple

static from_pauli(pauli, angle)

A convenience method to create a FixedRotation from a Pauli string.

This is the equivalent to $exp^{-i \frac{\pi}{360} \text{pauli} \cdot \text{angle}}$.

import trueq as tq
from trueq.math import FixedRotation

gen = FixedRotation.from_pauli("XY", 90)
assert gen.pauli_angle == ("XY", 90)

gen = FixedRotation.from_pauli("Y", 11)
assert gen.pauli_angle == ("Y", 11)

Parameters
• pauli (str) – A Pauli which describes the axis of the rotation.

• angle (float) – The angle of rotation as defined above.

Return type

FixedRotation

property width

The width of the matrix returned by this layer.

Type

int

class trueq.math.rotation.Layer

Parent class for Rotation and FixedRotation.

property pauli_angle

If this layer can be represented by a fixed rotation along a Pauli axis such as $exp^{-i \cdot \frac{\pi}{360} \text{Pauli} \cdot \text{angle}}$, this will be a tuple (Pauli, angle), otherwise this is (None, None)

import trueq as tq
from trueq.math import FixedRotation

g = FixedRotation.from_pauli("XY", 90)
assert g.pauli_angle == ("XY", 90)

g = FixedRotation.from_pauli("Y", 11)
assert g.pauli_angle == ("Y", 11)

Type

tuple

property pauli_const

If this layer can be represented by a rotation along a Pauli axis such as: $exp^{-i \frac{\pi}{360} \text{Pauli} \cdot \text{constant} \cdot \text{param}}$, this will be a tuple (Pauli, constant), otherwise this is (None, None).

import trueq as tq
import numpy as np
from trueq.math import Rotation

g = Rotation(tq.Gate.y.mat * np.pi / 360, "theta")
assert g.pauli_const == ("Y", 1)

g = Rotation.from_pauli("X", "phi", 5)
assert g.pauli_const == ("X", 5)

gen = tq.Gate.random(2).mat
g = Rotation(gen + gen.conj().T, "theta")
assert g.pauli_const == (None, None)

Type

tuple

abstract property width

The width of the matrix returned by this layer.

Type

int

class trueq.math.rotation.Rotation(gen, param_name)

A representation of a rotation of the form $exp^{-i \cdot G \cdot \text{param}}$ for some Hermitian matrix $G$.

import trueq as tq
import numpy as np
from trueq.math import Rotation

g = Rotation(tq.Gate.x.mat / 2, "theta")
assert tq.Gate(g(np.pi / 2)) == tq.Gate.cliff5
pauli, const = g.pauli_const
assert pauli == "X"
assert abs(const - 180 / np.pi) < 1e-12

Parameters
• gen (numpy.array) – A Hermitian matrix to generate this rotation.

• param_name (str) – The name associated with the param, e.g. theta, phi, t, etc.

property pauli_const

If this layer can be represented by a rotation along a Pauli axis such as: $exp^{-i \frac{\pi}{360} \text{Pauli} \cdot \text{constant} \cdot \text{param}}$, this will be a tuple (Pauli, constant), otherwise this is (None, None).

import trueq as tq
import numpy as np
from trueq.math import Rotation

g = Rotation(tq.Gate.y.mat * np.pi / 360, "theta")
assert g.pauli_const == ("Y", 1)

g = Rotation.from_pauli("X", "phi", 5)
assert g.pauli_const == ("X", 5)

gen = tq.Gate.random(2).mat
g = Rotation(gen + gen.conj().T, "theta")
assert g.pauli_const == (None, None)

Type

tuple

static from_pauli(pauli, param_name, constant=1)

A convenience method to create a Rotation from a Pauli string.

This is the equivalent to $exp^{-i \frac{\pi}{360} \text{pauli} \cdot \text{constant} \cdot \text{param}}$.

import trueq as tq
from trueq.math import Rotation

gen = Rotation.from_pauli("XY", "theta")
assert gen.pauli_const == "XY"

gen = Rotation.from_pauli("XY", "theta", constant=10)
assert gen.pauli_const == ("XY", 10)

Parameters
• pauli (str) – A Pauli which describes the axis of the rotation.

• param_name (str) – The name of the associated free parameter.

• constant (float) – A constant scaling term as defined above.

Return type

Rotation

property width

The width of the matrix returned by this layer.

Type

int

property id_scale

An estimate of a non-zero value where this rotation is identity.

If the rotation is “well behaved”, such as a rotation around a Pauli axis, this value will typically be the smallest non-zero value where the rotation is the identity.

If no reasonable estimate can be calculated, then this value is None.

import trueq as tq
import numpy as np

gen = np.flipud(np.eye(5))
rot = tq.math.Rotation(gen, "phi")
assert rot.id_scale is not None
assert tq.math.proc_infidelity(rot(rot.id_scale)) < 1e-12

# D matrix where the terms are irrational fractions of one another
rot2 = tq.math.Rotation(np.diag([0, 1, np.sqrt(2), np.pi]), "phi")
assert rot2.id_scale is None

Type

float | NoneType

find_closest(mat, tol=1e-12)

Attempts to find the parameter value where this rotation creates a matrix which is equal to the target matrix up to a global phase.

If no suitable parameter value is found, then this returns None

import trueq as tq
import numpy as np

gen = np.flipud(np.eye(5))
rot = tq.math.Rotation(gen, "phi")
target = rot(0.5)
assert rot.find_closest(target) == 0.5
assert rot.find_closest(tq.math.random_density(5)) is None

Parameters
• mat (numpy.array) – The target matrix.

• tol (float) – Allowable process infidelity to be considered a match.

Return type

NoneType | float

## Superoperators¶

class trueq.math.superop.Superop(mat, dim=None)

Represents a quantum superoperator. A superoperator is a linear operation that maps density matrices to density matrices. This class contains utilities to compute various superoperator properties and metrics, methods for composing and combining superoperators, and methods to convert between the various superoperator representations.

Generally, instances of this class should be created by calling one of the static methods starting with from_. For example:

import numpy as np
import trueq as tq

k1 = np.sqrt(0.95) * np.eye(2)
k2 = np.sqrt(0.05) * tq.math.random_unitary(2)
s = tq.math.Superop.from_kraus([k1, k2])

# whether s is cp and tp, and print the choi rank
print(s.is_cp, s.is_tp, s.choi_rank)

# compute some metrics of u
print(s.infidelity, s.coherent_infidelity)

# apply s to a random density matrix
print(s.apply(tq.math.random_density(2)))

# compute the 2-norm distance to the identity channel
i = tq.math.Superop.from_unitary(np.eye(2))
print((s - i).norm(2))

# plot the pauli transfer matrix of s tensored with i
(s & i).plot_ptm()

True True 2
0.044683411820047536 0.001292797902951115
[[0.4749691 +3.05350944e-19j 0.42056211-1.90734890e-01j]
[0.42056211+1.90734890e-01j 0.5250309 +1.48696354e-19j]]
0.13369130386086825 Parameters
• mat (numpy.ndarray-like) – A superoperator matrix in the row-stacking basis.

• dim (int | NoneType) – The dimension of each subsystem this superoperator acts on, e.g. 2 for qubits. If None, the constructor will try to automatically infer the subsystem dimension by choosing the smallest compatible prime.

property dim

The dimension of each subsystem. For example, this is equal to 2 for a superoperator that acts on two qubits.

Type

int

property n_sys

The number of subsystems that this superoperator acts on.

Type

int

property total_dim

The total Hilbert space dimension that this superoperator acts on.

Type

int

property choi_rank

The rank of the Choi matrix of this superoperator, that is, the minimal number of Kraus operators required to describe this superoperator.

Type

int

property is_cp

Whether this superoperator is completely positive (CP), i.e. whether this superoperator always outputs positive states when given positive states (even when this superoperator is tensored with an ancilla space).

Type

bool

property is_cptp

Whether this superoperator is completely positive and trace-preserving (CPTP). See also is_cp and is_tp.

Type

bool

property is_tp

Whether this superoperator is trace preserving (TP), i.e. whether the trace of output density matrices is always equal to the trace of input density matrices.

Type

bool

property is_unital

Whether this superoperator is unital, i.e. whether it maps the completely mixed state to the completely mixed state.

Type

bool

property is_unitary

Whether this superoperator is a unitary chanel, i.e. whether there existis a unitary $U$ such this superoperator acts as $U\rho U^\dagger$ for any input $\rho$. If so, it can be extracted as the sole Kraus operate using kraus.

Type

bool

A new Superop instance equal to the adjoint of this superoperator.

Type

Superop

property avg_gate_fidelity

The average gate fidelity of this superoperator, defined by

$\overline{F} = \int d\psi \langle\psi|\mathcal{E}(|\psi\rangle\langle\psi|)|\psi\rangle$

for the superoperator $\mathcal{E}$. This is related to the process fidelity $F_E$ (see fidelity) via $\overline{F}=(d^2F_E+d)/(d(d+1))$ where $d$ is the dimension of the underlying Hilbert space.

import trueq.math as tqm

s = tqm.Superop(tqm.random_bcsz(3))
u = tqm.Superop.from_unitary(tqm.random_unitary(3))

# find the average gate fidelity between these superoperators

0.3783764257861908

Type

float

property avg_gate_infidelity

The average gate infidelity of this superoperator, often denoted as $r$, equal to one minus avg_gate_fidelity; see the definition there.

import trueq.math as tqm

s = tqm.Superop(tqm.random_bcsz(3))
u = tqm.Superop.from_unitary(tqm.random_unitary(3))

# find the average gate infidelity between these superoperators

0.6777292229090173

Type

float

property coherent_infidelity

The coherent infidelity of this superoperator, defined by

$e_U = e_F - e_S$

for the superoperator $\mathcal{E}$, where $e_F$ is the process infidelity and $e_S$ is the stochastic_infidelity. This quantity ranges between 0 and the process infidelity of this superoperator, where a larger value indicates more coherent (i.e. unitary or calibration) noise. This quantity is reported by the XRB protocol when SRB results are also present to estimate $e_F$.

Type

float

property dnorm

The diamond norm of this superoperator.

import trueq as tq

# ideal gate
s1 = tq.math.Superop.from_unitary(tq.Gate.cnot.mat)

# gate simulated with depolarizing
s2 = sim.operator(tq.Cycle({(0,1): tq.Gate.cnot})).mat()
s2 = tq.math.Superop.from_rowstack(s2)

# compute the diamond norm of the difference (see the note below for why
# this is commented out)
# (s1 - s2).dnorm


Note

This is computed via Watrous’ semidefinite program, see .

Note

This function requires cvxpy and cvxopt to be properly installed. In particular, their dependency scs must also be installed, which also requires blas and lapack libraries to be installed. It may take some effort to get this all set up in your environment, hence the example above has dnorm commented out so that the documentation will build.

Type

float

property fidelity

The process fidelity of this superoperator, which can be defined as

$F_E = \operatorname{Tr}(\mathcal{E})/d^2$

for the superoperator $\mathcal{E}$ where $d$ is the dimension of the underlying Hilbert space. This quantity is related to the average gate fidelity $\overline{F}$ (see avg_gate_fidelity) via $\overline{F}=(d^2F_E+d)/(d(d+1))$.

import trueq.math as tqm

s = tqm.Superop(tqm.random_bcsz(3))
u = tqm.Superop.from_unitary(tqm.random_unitary(3))

# find the process fidelity between these superoperators

0.1497325327437782

Type

float

property infidelity

The process infidelity of this superoperator, defined as $e_F=1-F_E$ where $F_E$ is the process fidelity. This quantity is estimated by protocols like SRB and CB.

import trueq.math as tqm

s = tqm.Superop(tqm.random_bcsz(3))
u = tqm.Superop.from_unitary(tqm.random_unitary(3))

# find the process infidelity between these superoperators

0.9212084013268468

Type

float

norm(p=2)

Returns the Schatten p-norm of this superoperator, equal to

$\|\mathcal{E}\|_p = ( \operatorname{Tr}[(\mathcal{E}^\dagger\mathcal{E})^{p/2}])^{1/p} )$

for the superoperator $\mathcal{E}$. These norms are unitarily invariant, so the basis (row-stacking, column-stacking, normalized Weyl, etc.) is irrelevant. Certain values of $p$ have special names:

• $p=1$ is the trace norm.

• $p=2$ is the Frobenius norm and is the fastest to compute.

• $p=\infty$ is the spectral (or operator) norm.

Parameters

p (float) – Any real number greater than or equal to one.

Return type

float

Raises

ValueError – If $p<1$.

property stochastic_infidelity

The stochastic infidelity of this superoperator, defined by

$e_S = 1 - \sqrt{\operatorname{Tr}(\mathcal{E}\mathcal{E}^\dagger)} / d$

for the superoperator $\mathcal{E}$, where $d$ is the dimension of Hilbert space, see . This quantity ranges between 0 and the process infidelity of this superoperator, where a larger value indicates more stochastic noise. The stochastic infidelity is reported by the XRB protocol. For unital TP maps, the stochastic fidelity is related to the unitarity by the formula

$e_S = 1 -\sqrt{u(1-1/d^2)+1/d^2}$
Type

float

property unitary_fraction

The unitary fraction of this superoperator, defined as

$f_u = 1 - (1-1/d^2)\frac{1 - \sqrt{u}}{e_F}$

where $e_F$ is the process infidelity and $d$ is the dimension of Hilbert space. A unitary fraction equal to 0 represents the least possible amount of unitarity a superoperator can have, given its fidelity, and a value of 1 represents a unitary superoperator.

Type

float

property unitarity

The unitarity of this superoperator, defined by

$u = \frac{d}{d-1}\int d\psi \operatorname{Tr}[ \mathcal{E}'(|\psi\rangle\langle\psi|) ]$

for the superoperator $\mathcal{E}$, where

$\mathcal{E}'(A) = \mathcal{E}(A) - \operatorname{Tr}[\mathcal{E}(A)]\frac{I}{\sqrt{d}}$

is the traceless projection of $\mathcal{E}$, and where $d$ is the dimension of the underlying Hilbert space . In other words, the unitarity is the purity of output states averaged over all pure input states, normalized by possible trace-decreasing behaviour. This quantity is estimated by the XRB protocol.

Type

float

static eye(total_dim, dim=None)

Constructs a new identity superoperator.

Parameters
• total_dim (int) – The total dimension the superoperator acts on, e.g. 16 for four qubits.

• dim (int | NoneType) – The dimension of each subsystem this superoperator acts on, e.g. 2 for qubits. If None, the constructor will try to automatically infer the subsystem dimension by choosing the smallest compatible prime.

Return type

Superop

static zero(total_dim, dim=None)

Constructs a new superoperator that sends all inputs to the zero matrix.

Parameters
• total_dim (int) – The total dimension the superoperator acts on, e.g. 16 for four qubits.

• dim (int | NoneType) – The dimension of each subsystem this superoperator acts on, e.g. 2 for qubits. If None, the constructor will try to automatically infer the subsystem dimension by choosing the smallest compatible prime.

Return type

Superop

static random_bcsz(total_dim, rank=None, dim=None)

Constructs a new random superoperator drawn from the BCSZ distribution ().

import trueq.math as tqm

s = tqm.Superop.random_bcsz(9, 5)
assert s.is_cptp
assert s.choi_rank == 5

Parameters
• total_dim (int) – The total dimension the superoperator acts on, e.g. 16 for four qubits.

• dim (int | NoneType) – The dimension of each subsystem this superoperator acts on, e.g. 2 for qubits. If None, the constructor will try to automatically infer the subsystem dimension by choosing the smallest compatible prime.

• rank (int | NoneTye) – A positive integer specifying the output Choi rank. If None, then the value total_dim ** 2 is used. A value of 1 results in a random unitary superoperator.

Return type

Superop

static random_constrained_cptp(total_dim, infidelity, coherent_fraction, equibility=1, rank=None, dim=None)

Constructs a random CPTP (see is_cptp) channel whose process infidelity and level of unitary are (approximately) given. This is done by composing a random unitary channel of fidelity coherent_fraction (see random_constrained_unitary()) with a random decoherent channel (see random_decoherent()) of infidelity (1 - coherent_fraction) * infidelity. The infidelity and stochastic infidelity of the resulting composition is accurate up to $O(r^2)$ with $r$ the infidelity, see .

import trueq.math as tqm

# random 2-qutrit gate
s = tqm.Superop.random_constrained_cptp(9, 0.01, 0.4)
assert s.is_cptp
print(s.infidelity, s.stochastic_infidelity)

0.009976800614242998 0.005996535845802731

Parameters
• total_dim (int) – The total dimension the superoperator acts on, e.g. 16 for four qubits.

• infidelity (float) – The desired process infidelity; a number in $[0,1]$.

• coherent_fraction (float) – The desired fraction of the process infidelity to be unitary or coherent in nature; a number in $[0,1]$.

• equibility – A positive number indicating the equibility of the resulting channel in arbitrary units. A large value (say, 100) results in very uniform singular values of the leading kraus term, whereas they are more varied for a small value like 1.

• rank (int | NoneTye) – A positive integer specifying the output Choi rank. If None, then the value total_dim ** 2 is used. A value of 1 results in a random unitary superoperator.

• dim (int | NoneType) – The dimension of each subsystem this superoperator acts on, e.g. 2 for qubits. If None, the constructor will try to automatically infer the subsystem dimension by choosing the smallest compatible prime.

Return type

Superop

static random_constrained_unitary(total_dim, infidelity, dim=None)

Constructs a random unitary channel constrained to have the given process infidelity, see infidelity.

import trueq.math as tqm

s = tqm.Superop.random_constrained_unitary(9, 0.1)
assert s.is_unitary
assert abs(s.infidelity - 0.1) < 1e-10

Parameters
• total_dim (int) – The total dimension the superoperator acts on, e.g. 16 for four qubits.

• infidelity (float) – The desired process infidelity; a number in $[0,1]$.

• dim (int | NoneType) – The dimension of each subsystem this superoperator acts on, e.g. 2 for qubits. If None, the constructor will try to automatically infer the subsystem dimension by choosing the smallest compatible prime.

Return type

Superop

static random_decoherent(total_dim, infidelity, equibility=1, rank=None, dim=None)

Constructs a random decoherent channel with a given process infidelity (approximately), see infidelity. A decoherent channel is one whose polar decomposition of the leading Kraus term (see lkpd()) returns a unitary portion equal to the identity, i.e. the main unitary action of the channel is trivial. The infidelity of the resulting composition is accurate up to $O(r^2)$ with $r$ the infidelity, see .

import trueq.math as tqm

s = tqm.Superop.random_decoherent(9, 0.1)
assert s.is_cptp
print(s.infidelity, s.stochastic_infidelity)

0.0999978550076519 0.09788150593289602

Parameters
• total_dim (int) – The total dimension the superoperator acts on, e.g. 16 for four qubits.

• infidelity (float) – The desired process infidelity; a number in $[0,1]$.

• equibility (float) – A positive number indicating the equibility of the resulting channel in arbitrary units. A large value (say, 100) results in very uniform singular values of the leading kraus term, whereas they are more varied for a small value like 1.

• rank (int | NoneTye) – A positive integer specifying the output Choi rank. If None, then the value total_dim ** 2 is used. A value of 1 results in a random unitary superoperator.

• dim (int | NoneType) – The dimension of each subsystem this superoperator acts on, e.g. 2 for qubits. If None, the constructor will try to automatically infer the subsystem dimension by choosing the smallest compatible prime.

Return type

Superop

Raises

ValueError – If the rank and infidelity are inconsistent—the rank can only be 1 when the infidelity is 0.0, and vise versa.

static random_stochastic(total_dim, infidelity, rank=None, dim=None)

Constructs a random stochastic channel with a given process infidelity, see infidelity. For the purposes of this function, a stochastic channel is one that can be written as convex mixture of orthogonal unitary channels, one of which is the identity, and the rest of which are traceless with evenly distributed eigenvalues. Such a channel is CPTP (see is_cptp), and the process fidelity is given by the weight of the identity mixture component.

The main difference between this method and random_decoherent() is that this method does not allow non-unital (e.g. $T_1$ noise is non-unital because the completely mixed state reverts to something with polarization) behaviour or state-dependent leakage.

import trueq.math as tqm

s = tqm.Superop.random_stochastic(9, 0.1)
assert s.is_cptp
print(s.infidelity)

0.09999999999999998

Parameters
• total_dim (int) – The total dimension the superoperator acts on, e.g. 16 for four qubits.

• infidelity (float) – The desired process infidelity; a number in $[0,1]$.

• rank (int | NoneTye) – A positive integer specifying the output Choi rank. If None, then the value total_dim ** 2 is used. A value of 1 results in a random unitary superoperator.

• dim (int | NoneType) – The dimension of each subsystem this superoperator acts on, e.g. 2 for qubits. If None, the constructor will try to automatically infer the subsystem dimension by choosing the smallest compatible prime.

Return type

Superop

Raises

ValueError – If the rank and infidelity are inconsistent—the rank can only be 1 when the infidelity is 0.0, and vise versa.

static random_unitary(total_dim, dim=None)

Constructs a Haar-random unitary channel.

import trueq.math as tqm

s = tqm.Superop.random_unitary(9)
assert s.is_unitary

Parameters
• total_dim (int) – The total dimension the superoperator acts on, e.g. 16 for four qubits.

• dim (int | NoneType) – The dimension of each subsystem this superoperator acts on, e.g. 2 for qubits. If None, the constructor will try to automatically infer the subsystem dimension by choosing the smallest compatible prime.

Return type

Superop

static from_kraus(*ops, dim=None)

Instantiates a new Superop from a given set of Kraus operators. A superoperator in the Kraus representation is a set of $m$ operators ${K_0,\ldots,K_{m-1}}$ that act on a given density matrix by the formula $\rho\mapsto\sum_{i=0}^{m-1}K_i \rho K^\dagger_i$. A superoperator admits a Kraus representation if and only if it is completely positive (see is_cp), but it is only trace preserving (see is_tp) if $\sum_{k=0}^{m-1}K^\dagger_i K_i = \mathbb{I}$.

import numpy as np
import trueq as tq

# instantiate a depolarizing channel
p = 0.05
superop = tq.math.Superop.from_kraus(
np.sqrt(1-p) * tq.Gate.id.mat,
np.sqrt(p / 3) * tq.Gate.x.mat,
np.sqrt(p / 3) * tq.Gate.y.mat,
np.sqrt(p / 3) * tq.Gate.z.mat
)

Parameters
• *ops – One or more array_like Kraus operators.

• dim (int | NoneType) – The dimension of each subsystem this superoperator acts on, e.g. 2 for qubits. If None, the constructor will try to automatically infer the subsystem dimension by choosing the smallest compatible prime.

Returns

The superoperator corresponding to the given set of Kraus operators.

Return type

Superop

property kraus

The Kraus representation of this superoperator; see from_kraus() for details about the representation. Note that Kraus representations are not unique, and this decomposition is only possible if the superoperator is CP (see is_cp). The format is a 3-D array ops of shape (n_ops, dim, dim) where ops[i,:,:] is the $i^{th}$ Kraus operator.

Type

numpy.ndarray

plot_kraus(n_columns=4, abs_max=None, axes=None)

Plots a Kraus representation of this superoperator; see from_kraus() for details about the representation. Note that Kraus representations are not unique.

Parameters
• n_columns (int) – If new axes are made, how many columns of subplots to use.

• abs_max (NoneType | float) – The value to scale absolute values of the matrix by; the value at which plotted colors become fully opaque. By default, this is the largest absolute magnitude of the input matrix.

• axes (Iterable | NoneType) – An existing list of axes to plot on. They are created otherwise.

static from_unitary(u, dim=None)

Instantiates a new Superop from a given unitary matrix. A unitary matrix $U$ acts on a given matrix by the formula $\rho\mapsto U\rho U^\dagger$.

import trueq as tq

# instantiate a CNOT as a superoperator
superop = tq.math.Superop.from_unitary(tq.Gate.cnot.mat)

Parameters
• u (numpy.ndarray) – A unitary matrix.

• dim (int | NoneType) – The dimension of each subsystem this superoperator acts on, e.g. 2 for qubits. If None, the constructor will try to automatically infer the subsystem dimension by choosing the smallest compatible prime.

Returns

The superoperator that conjugates input states by the given unitary matrix.

Return type

Superop

static from_function(fcn, total_dim=None, dim=None)

Instantiates a new Superop from a python function.

import trueq as tq

def depolarizing(rho, p=0.05):
d = rho.shape
return (1 - p) * rho + p * np.eye(d) * np.trace(rho) / d

# instantiate a qubit depolarizing channel
superop = tq.math.Superop.from_function(depolarizing, 2)

# instantiate a qutrit depolarizing channel
superop = tq.math.Superop.from_function(depolarizing, 3)


Note

The function is assumed to be a linear function of a single density matrix. Passing a nonlinear function will result in a linear approximation that reproduces the action of the function on a basis of the operator space.

Parameters
• fcn (function) – A function that takes a density matrix and returns a density matrix of the same shape.

• total_dim (int | NoneType) – The total Hilbert space dimension that this superoperator acts on. This will control the size of density matrix that is input into fcn. The default value of None will result in the default dimension obtained from get_dim().

• dim (int | NoneType) – The dimension of each subsystem this superoperator acts on, e.g. 2 for qubits. If None, the constructor will try to automatically infer the subsystem dimension by choosing the smallest compatible prime.

Returns

The superoperator that maps a density matrix to the state returned by the given function.

Return type

Superop

Raises

ValueError – If dim does not divide total_dim.

from_qobj()

Instantiates a new Superop from a QuTiP operator object.

import qutip as qt
import trueq as tq

# make some QuTiP operator
cx = qt.sigmax()

# convert to a Superop and plot
s = tq.math.Superop.from_qobj(cx)
s.plot_ptm() Parameters

qobj (qutip.Qobj) – A QuTiP superoperator object.

Returns

The superoperator for a given qobj.

Return type

Superop

Raises
• RuntimeError – :If QuTiP is not imported.

• ValueError – If the given qobj has non-uniform subsystem dimensions.

property qobj

This Superop as a QuTiP superoperator object.

import qutip as qt
import trueq as tq

# construct a Superop from a simulation
s = tq.math.Superop(sim.operator(tq.Cycle({(0,1): tq.Gate.cx})).mat())

# convert to a Qobj and plot
o = s.qobj
qt.visualization.matrix_histogram_complex(o)

(<Figure size 432x288 with 2 Axes>, <Axes3D:>) QuTiP is an optional dependency and must be installed for this function to work.

Type

qutip.Qobj

Raises

RuntimeError – If QuTiP is not imported.

static from_ptm(ptm)

Instantiates a new Superop from a Pauli transfer matrix (PTM). A PTM represents how density matrices are transformed when expanded in the Pauli basis. In particular, suppose that for $n$ qubits the Pauli matrices are denoted by $\{P_i\}_{i=0}^{4^n-1}$, and sorted lexicographically. For example with $n=2$, we have the ordering

$\{P_i\}_{i=0}^{15} =\{II, IX, IY, IZ, XI, XX, XY, XZ, YI, YX, YY, YZ, ZI, ZX, ZY, ZZ\}.$

Given any density matrix $\rho$, we can expand it in the Pauli basis as $\rho=\sum_{i=4^n-1} c_i P_i$. A PTM $A$ performs the transformation $c\mapsto A c$ where $c$ is the vector $c=(c_0,c_1,...,c_{4^n-1})$.

import numpy as np
import trueq as tq

# instantiate a qubit depolarizing channel
superop = tq.math.Superop.from_ptm(np.diag([1, 0.99, 0.99, 0.99]))


The PTM can be generalized to subsystems with $d>2$ using the Weyl operators, which are a set of $d^2$ mutually orthogonal unitary matrices. However, unlike the Paulis, they are not generally Hermitian which would lead to non-real elements of the PTM for CP (see is_cp) maps. Therefore, we use a symmetrized version of the Weyl matrices, see Weyls and herm_mat for details. The expected order corresponds to lex=False in all.

Parameters

ptm (numpy.ndarray) – A square matrix.

Returns

The superoperator for a given PTM.

Return type

Superop

Raises

ValueError – If the PTM does not have have a width that is an even power of a small prime number.

property ptm

The Pauli transfer matrix representation of this superoperator; see from_ptm() for details on this representation.

Type

numpy.ndarray

plot_ptm(abs_max=None, ax=None)

Plots the Pauli transfer matrix representation of this superoperator; see from_ptm() for details on this representation.

Parameters
• abs_max (NoneType | float) – The value to scale absolute values of the matrix by; the value at which plotted colors become fully opaque. By default, this is the largest absolute magnitude of the input matrix.

• ax (matplotlib.Axis | NoneType) – An existing axis to plot on. One is created otherwise.

static from_choi(choi, dim=None)

Instantiates a new Superop from a Choi matrix. If a channel $\Lambda$ transforms a given density matrix as $\rho\mapsto \Lambda(\rho)$, then the Choi matrix (in the row convention) is equal to $\Lambda_r=\sum_{i,j=0}^{d-1}\Lambda(E_{i,j})\otimes E_{i,j}$ where $d$ is the dimension, and $E_{i,j}$ is the $d\times d$ matrix of zeros with a 1 in the $(i,j)^\text{th}$ entry.

The Choi matrix is positive semidefinite if and only if the superoperator it represents is completely positive (see is_cp). Therefore, there is a correspondence between completely positive superoperators and density matrices (which must also be positive semidefinite) that is often called the Choi-Jamiolkowski isomorphism.

import numpy as np
import trueq as tq

# instantiate a random CP qubit channel
superop = tq.math.Superop.from_choi(2 * tq.math.random_density(4))

Parameters
• choi (numpy.ndarray) – A Choi matrix.

• dim (int | NoneType) – The dimension of each subsystem this superoperator acts on, e.g. 2 for qubits. If None, the constructor will try to automatically infer the subsystem dimension by choosing the smallest compatible prime.

Returns

The superoperator for a given Choi matrix.

Return type

Superop

property choi

The Choi matrix representation of this superoperator; see from_choi() for details on this representation.

Type

numpy.ndarray

plot_choi(abs_max=None, ax=None)

Plots the Choi representation of this superoperator; see from_choi() for details on this representation.

Parameters
• abs_max (NoneType | float) – The value to scale absolute values of the matrix by; the value at which plotted colors become fully opaque. By default, this is the largest absolute magnitude of the input matrix.

• ax (matplotlib.Axis | NoneType) – An existing axis to plot on. If omitted, a new one is created.

static from_rowstack(mat, dim=None)

Instantiates a new Superop from a Liouville superoperator matrix in the row-stacking basis. Note that since this is the native basis of the Superop class, this method is equivalent to the constructor.

This is the best representation for superopertors for simulation if density matrices are C-ordered (i.e. row-major, so that entire rows are contiguous in memory). This is because if our row-stacking superoperator matrix is $M$ and our density matrix $\rho$, then if we take the rows of $\rho$ and stack them together into one column vector, denoted $\operatorname{vec}(\rho)$, then the output density matrix is given by regular matrix multiplication, $\operatorname{vec}^{-1}(M\operatorname{vec}(\rho))$, where $\operatorname{vec}^{-1}$ is the unstacking operator.

import numpy as np
import trueq as tq

# instantiate the identity channel on a qutrit
superop = tq.math.Superop.from_rowstack(np.eye(9))

Parameters
• mat (numpy.ndarray) – A superoperator in the row-stacking basis.

• dim (int | NoneType) – The dimension of each subsystem this superoperator acts on, e.g. 2 for qubits. If None, the constructor will try to automatically infer the subsystem dimension by choosing the smallest compatible prime.

Return type

Superop

property rowstack

The matrix representation of this superoperator in the row-stacking basis; see from_rowstack() for details on this representation.

Type

numpy.ndarray

plot_rowstack(abs_max=None, ax=None)

Plots the matrix representation of this superoperator in the row-stacking basis; see from_rowstack() for details on this representation.

Parameters
• abs_max (NoneType | float) – The value to scale absolute values of the matrix by; the value at which plotted colors become fully opaque. By default, this is the largest absolute magnitude of the input matrix.

• ax (matplotlib.Axis | NoneType) – An existing axis to plot on. If omitted, a new one is created.

static from_rowstack_subsys(mat, dim=None)

Instantiates a new Superop from a Liouville superoperator matrix in the row-stacking subsystem-wise basis. If a unitary $U=U_1\otimes U_2$ acts on some bipartite Hilbert space, then rowstack (which is also the native representation used by this class) is equal to

$U\otimes \overline{U} =U_1 \otimes U_2 \otimes \overline{U}_1 \otimes \overline{U}_2$

which we call the global rowstacking convention. It is sometimes more useful to use the subsystem-wise rowstacking convention where the wires of each subsystem are grouped together, corresponding to the unitary

$U\otimes \overline{U} =U_1 \otimes \overline{U}_1 \otimes U_2 \otimes \overline{U}_2$

Indeed, this is the convention used by OperatorTensor and hence also by the Simulator. This idea generalizes to n-partite superoperators.

import numpy as np
import trueq as tq

# instantiate the identity channel on two qubits
superop = tq.math.Superop.from_rowstack_subsys(np.eye(16))

Parameters
• mat (numpy.ndarray) – A superoperator in the row-stacking subsystem-wise basis.

• dim (int | NoneType) – The dimension of each subsystem this superoperator acts on, e.g. 2 for qubits. If None, the constructor will try to automatically infer the subsystem dimension by choosing the smallest compatible prime.

Return type

Superop

property rowstack_subsys

The matrix representation of this superoperator in the row-stacking basis; see from_rowstack_subsys() for details on this representation.

Type

numpy.ndarray

plot_rowstack_subsys(abs_max=None, ax=None)

Plots the matrix representation of this superoperator in the row-stacking subsystem-wise basis; see from_rowstack_subsys() for details on this representation.

Parameters
• dim (int) – The dimension of each Hilbert space subsystem, e.g. 2 for qubits.

• abs_max (NoneType | float) – The value to scale absolute values of the matrix by; the value at which plotted colors become fully opaque. By default, this is the largest absolute magnitude of the input matrix.

• ax (matplotlib.Axis | NoneType) – An existing axis to plot on. If omitted, a new one is created.

static from_colstack(mat, dim=None)

Instantiates a new Superop from a superoperator in the column-stacking basis.

This is the best representation for superopertors for simulation if density matrices are Fortran-ordered (i.e. column-major, so that entire columns are contiguous in memory). This is because if our column-stacking superoperator matrix is $M$ and our density matrix $\rho$, then if we take the columns of $\rho$ and stack them together into one column vector, denoted $\operatorname{vec}(\rho)$, then the output density matrix is given by regular matrix multiplication, $\operatorname{vec}^{-1}(M\operatorname{vec}(\rho))$, where $\operatorname{vec}^{-1}$ is the unstacking operator.

import numpy as np
import trueq as tq

# instantiate the identity channel on a qutrit
superop = tq.math.Superop.from_colstack(np.eye(9))

Parameters
• mat (numpy.ndarray) – A superoperator in the column-stacking basis.

• dim (int | NoneType) – The dimension of each subsystem this superoperator acts on, e.g. 2 for qubits. If None, the constructor will try to automatically infer the subsystem dimension by choosing the smallest compatible prime.

Return type

Superop

property colstack

The matrix representation of this superoperator in the column-stacking basis; see from_colstack() for details on this representation.

Type

numpy.ndarray

plot_colstack(abs_max=None, ax=None)

Plots the matrix representation of this superoperator in the col-stacking basis; see from_colstack() for details on this representation.

Parameters
• abs_max (NoneType | float) – The value to scale absolute values of the matrix by; the value at which plotted colors become fully opaque. By default, this is the largest absolute magnitude of the input matrix.

• ax (matplotlib.Axis | NoneType) – An existing axis to plot on. If omitted, a new one is created.

property lkpd

The leading Kraus polar decomposition (LKPD) of this superoperator. This is a tuple u, p where u is unitary and p is positive semi-definite, and the product u @ p is equal to the leading Kraus term.

See  for a comprehensive overview of this form. In short, when eigenvectors of a positive semi-definite Choi matrix are scaled by the square-roots of their eigenvalues and devectorized into square matrices, they form a set of mutually orthogonal Kraus operators for the channel. The Kraus operator corresponding to the largest eigenvalue forms a convenient approximation to the channel when the channel is nearly unitary. The QR decomposition of this leading Kraus term separates leading dynamics of the channel into unitary and stochastic behaviour.

Note

This is a partial representation because it does not contain all information about the superoperator; non-leading terms of the Kraus representation are dropped.

Type

tuple

plot_lkpd(axes=None, abs_max=None)

Plots both matrices of the polar decomposition of the leading Kraus term of this superoperator; see lkpd() for details on this (partial) representation.

Parameters
• abs_max (NoneType | float) – The value to scale absolute values of the matrix by; the value at which plotted colors become fully opaque. By default, this is the largest absolute magnitude of the input matrix.

• axes (Iterable | NoneType) – An existing list of axes to plot on. They are created otherwise.

apply(state)

Applies this superoperator to a given density matrix or pure state. The output is always a density matrix of shape (dim, dim).

import numpy as np
import trueq as tq

# make a superoperator acting on a qutrit
s = tq.math.Superop(np.random.randn(9, 9))

# apply to a pure state
s.apply([1, 0, 0])

array([[ 1.94082752+0.j,  0.27099433+0.j,  0.17139886+0.j],
[-0.17845968+0.j,  0.76931995+0.j, -2.35326482+0.j],
[-1.36828211+0.j, -0.58994251+0.j,  1.32023084+0.j]])

Parameters

state (numpy.ndarray-like) – An input density matrix or pure state of the correct shape.

Returns

The image of state under this superoperator.

Return type

numpy.ndarray

## Tensors¶

class trueq.math.tensor.Tensor(output_shape, input_shape=None, spawn=None, value=None, dtype=None)

Represents a multipartite tensor on an arbitrary number of subsystems. Each subsystem owns a number of ordered wires (or indices), which can each be input or output wires (e.g., lower and upper indices in Einstein’s notation). One important restriction is that all subsystems have the same wire description: the same number, dimensions, and order for input and output wires.

Here are some motivating cases for various output and input shapes output, input:

• (d,), (): Pure states on a multipartite qubit system. Each subsystem has a single output wire of dimension $d$ and no input wires (i.e., a column vector). See StateTensor.

• (d,), (): Probability distributions over Cartesian products of dits. Each subsystem has a single output wire of dimension $d$ and no input wires (i.e., a column vector).

• (d,d), (): Vectorized mixed states on a multipartite qubit system. Each subsystem has two output wires each with dimension $d$ and no input wires. The first output wire represents the output wire of the unvectorized state, and the second output wire the input wire of the unvectorized state (this is the row-stacking convention). See StateTensor.

• (d,), (d,): Unitaries acting on a multipartite qubit system. Each subsystem has a single output wire of dimension $d$ and a single input wire of dimension $d$ (i.e., a square matrix). See OperatorTensor.

• (d,d), (d,d): Superoperators acting on a multipartite qubit system. Each subsystem has two input wires and two output wires, all of dimension $d$. The two input wires act on a vectorized density matrix (or contract against another superoperator in composition). See OperatorTensor.

A main feature of this class is the ability to efficiently left-multiply square matrices onto specific subsystems (see apply_matrix()), which can be strictly smaller than all of the subsystems present in the tensor. Therefore this class can be used to maintain the state of a simulation, and is general enough to track pure states, mixed states, unitaries, and superoperators.

import trueq as tq
import numpy as np

t = tq.math.Tensor(2, 2, spawn=np.eye(2))
t.apply_matrix((0, 2), tq.Gate.cnot.mat)
t.apply_matrix((1,), tq.Gate.h.mat)

# display the matrix corresponding to the circuit {(0, 2): cnot, (1,): h}
tq.visualization.plot_mat(t.mat().reshape(8, 8)) The internal storage format of this class is a dictionary mapping labels to NumPy arrays, such as the following:

{
(0,): [[0, 1], [1, 0]],
(2, 3): [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0]]
}

{(0,): [[0, 1], [1, 0]],
(2, 3): [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0]]}


In the above example, the label (0,) stores an X gate, and the labels (2, 3) store a CNOT gate.

Parameters
• output_shape (tuple | int) – The output wire dimension, or a tuple of output wire dimensions. Set to an empty tuple for no output wires.

• input_shape (tuple | int | NoneType) – The input wire dimension, or a tuple of input wire dimensions. Set to an empty tuple or None for no input wires.

• spawn (numpy.ndarray-like) – The default value to assign to new subsystems. The size must match the provided input and output shapes. By default, this parameter is set to the tensor of all zeros, except the first element, which is 1.

• value (dict) – The initial value of the tensor, set as a dictionary mapping tuples of subsystem indexes to tensors of the correct shape.

• dtype – The data type of the arrays, numpy.float64 by default.

property dtype

The data type of this tensor.

Type

type

property labels

Which subsystem labels of this tensor have a value.

Type

tuple

property n_sys

The number of subsystem labels with a value

Type

int

property shape

The output and input shape of each subsystem, e.g., ((2, 2), (2, 2)). See the arguments input_shape and output_shape of the constructor of Tensor.

Type

tuple

property total_dim

The total dimension of this tensor; the total subsystem dimension to the power of the number of subsystems.

Type

int

property spawn

The array that is assigned to new subsystems by default.

Type

numpy.ndarray

conj(copy=True)

Takes the complex conjugate of each array in this tensor.

Parameters

copy (bool) – Whether to mutate the given tensor, or to create a copy and leave the original intact.

Returns

This instance or a copy.

Return type

Tensor

Takes the complex conjugate and transpose of each array in this tensor. Here, transpose refers to swapping the input shape with the output shape.

Parameters

copy (bool) – Whether to mutate the given tensor, or to create a copy and leave the original intact.

Returns

This instance or a copy.

Return type

Tensor

transpose(output_perm, input_perm, copy=True)

Permutes wire dimensions of this tensor, possibly switching input wires with output wires.

from trueq.math import Tensor

# turn a column vector into a row vector
t = Tensor((2,))
print(t.shape, t.transpose([], ).shape)

# vectorize a matrix with row-stacking
t = Tensor((2,), (2,))
print(t.shape, t.transpose([0, 1], []).shape)

# vectorize a matrix with column-stacking
t = Tensor((2,), (3,))
print(t.shape, t.transpose([1, 0], []).shape)

# shuffle some wires
t = Tensor((2, 3), (5,))
print(t.shape, t.transpose([2, 1], ).shape)

((2,), ()) ((), (2,))
((2,), (2,)) ((2, 2), ())
((2,), (3,)) ((3, 2), ())
((2, 3), (5,)) ((5, 3), (2,))

Parameters
• output_perm (Iterable) – A list of indices referencing positions in the joint tuple shape = output_shape + input_shape. Therefore output_perm + input_perm must be a permutation of range(len(shape)).

• input_perm (Iterable) – A list of indices referencing positions in the joint tuple output_shape + input_shape.

• copy (bool) – Whether to mutate the given tensor, or to create a copy and leave the original intact.

Returns

This instance or a copy.

Return type

Tensor

value(order='subsystem')

Returns the value of this tensor: a dictionary mapping tuples of subsystem labels to array values.

The argument of this method changes the order of indexes of the arrays. For simulation performance reasons, the internal storage arranges them subsystem-wise, so that all input wires for a given subsystem are contiguous, and likewise all output wires for a given subsystem are contiguous, and this is the default order of this method. This can optionally be changed so that indices are riffled over subsystems.

As an example, suppose the input shape is (2, 4) and the output shape is (5,) and that there are 3 subsystems in an array. Then the array will have shape (2, 4, 2, 4, 2, 4, 5, 5, 5) if the order is "subsystem" (default), or the array will have shape (2, 2, 2, 4, 4, 4, 5, 5, 5) if the order is "composite". As a convenience, an order of "composite-group" flattens each group of subsystems, giving a shape (8, 64, 125) in this example, and an order of "composite-flat" flattens all inputs and all outputs separately, giving a shape (512, 125) in this example.

Parameters

order (str) – Which order to arrange the wires, either "subsystem", "composite", "composite-group", or "composite-flat". Or, one of "s", "c", "cg", or "cf" for short.

Return type

dict

mat(order='subsystem', labels=None)

Returns the full matrix representation of this tensor, with subsystems ordered by sorted labels.

The argument of this method changes the order of indexes of the arrays. For simulation performance reasons, the internal storage arranges them subsystem-wise, so that all input wires for a given subsystem are contiguous, and likewise all output wires for a given subsystem are contiguous, and this is the default order of this method. This can optionally be changed so that indices are riffled over subsystems.

As an example, suppose the input shape is (2, 4) and the output shape is (5,) and that there are 3 subsystems in an array. Then the array will have shape (2, 4, 2, 4, 2, 4, 5, 5, 5) if the order is "subsystem" (default), or the array will have shape (2, 2, 2, 4, 4, 4, 5, 5, 5) if the order is "composite". As a convenience, an order of "composite-group" flattens each group of subsystems, giving a shape (8, 64, 125) in this example, and an order of "composite-flat" flattens all inputs and all outputs separately, giving a shape (512, 125) in this example.

Note

Note that this property will create an array that is exponentially large in the number of subsystems, and causes any distinct subsystems found in value() to be permanently merged; calling this attribute mutates the tensor’s internal representation.

Parameters
• order (str) – Which order to arrange the wires, either "subsystem", "composite", "composite-group", or "composite-flat". Or, one of “s”, "c", "cg", or "cf" for short.

• labels (Iterable) – The desired subsystem label order, which must be a permutation of labels. By default, labels are ordered from smallest to biggest.

Return type

numpy.ndarray

Updates the number of subsystems to include labels provided, if they do not already exist. If the labels are already present, this call doesn’t change anything.

Parameters

labels (Iterable) – A list of labels to be added to this tensor.

Returns

This instance.

Return type

Tensor

Does nothing. This is to be used by subclasses that require a mutation of shape on a call to marginalize(). See, for instance, StateTensor.upgrade() and OperatorTensor.upgrade().

Returns

This instance.

Return type

Tensor

marginalize(labels, assume_unital=True, copy=True)

Keeps the provided labels of this tensor by marginalizing the remaining labels. Here, marginalization is defined as the sum over all tensor indices of the remaining labels.

import trueq as tq
import numpy as np

# Make a random probability distribution on length-4 bitstrings
S = np.random.rand(16)
p = tq.math.Tensor(2, value={(0, 1, 2, 3): S / S.sum()})

# Keep two of the bitstrings and marginalize the rest
p.marginalize((1, 2))

Tensor(<[(2,), ()] on labels [(1, 2)]>)

Parameters
• labels (tuple) – Which labels to keep. The order does not matter.

• assume_unital (bool) – Assumes that each factor in the tensor has a sum of 1. Otherwise, in the case where entire entries in value() need to be removed, sums are still computed and multiplied onto some other entry so that the overall sum of the tensor is preserved. In the case where all labels in the tensor are marginalized, label (0,) is re-added so that the total sum of the tensor can be preserved.

• copy (bool) – Whether to make a copy of this tensor, or mutate it in place.

Returns

This instance.

Return type

Tensor

update(other)

Updates this tensor with some new values. If values are placed on labels that do not exist, creates and sets them. If values are placed on labels that already exist, they are first removed by calling marginalize().

Parameters

other (dict | Tensor) – A dictionary or another Tensor. In either case, the size of subsystems in other needs to match this tensor.

Returns

This instance.

Return type

Tensor

apply_matrix(labels, matrix)

Mutates this tensor by applying the given square matrix to the given labels.

The size of the matrix must be compatible with the output shape of this tensor; just as with regular matrix multiplication, $AB$, the number of columns of $A$ must match the number of rows of $B$. The rule enforced here is that the number of rows of the given matrix, when reshaped into a square, must be equal to the product of this tensor’s output wire dimensions. Square matrices are enforced to avoid changing the shape of the tensor.

import trueq as tq
import numpy as np

t = tq.math.Tensor((2, 3), (2,))

# we need a 6x6 since output shape is (2, 3)
t.apply_matrix((0,), np.random.randn(6, 6))

# we need a 36x36 since we are applying to 2 systems
t.apply_matrix((0, 3), np.random.randn(36, 36))

(0, 3)

Parameters
• labels (Iterable) – Which subsystems to apply the matrix to.

• matrix (numpy.ndarray-like) – A square matrix.

Returns

The labels to which the matrix was applied, including labels that needed to be merged together.

Return type

tuple

apply_to_all(mat, copy=False)

Applies a matrix to every subsystem of this tensor.

Parameters
• mat (numpy.ndarray) – A matrix to apply to this system, which must be square if copy == False.

• copy (bool) – Whether to make a copy of this tensor, or mutate it in place.

Returns

This instance, or the copied instance.

Return type

Tensor

dot(rhs, dtype=None, overwrite_rhs=False)

Contracts this tensor with another tensor, returning a new tensor. The only shape restriction is that this tensor’s input shape must match the output tensor’s output shape. The returned tensor has input-output shapes (this.output_shape, rhs.input_shape); the inner shapes have been contracted against each other.

import trueq.math as tqm
import numpy as np

lhs = tqm.Tensor(5, (2, 3), value={(0, 1): np.random.randn(25, 36)})
rhs = tqm.Tensor((2, 3), 4, value={(1, 5): np.random.randn(36, 16)})
product = lhs.dot(rhs)

print(product.shape)
print(product.labels)

((5,), (4,))
(0, 1, 5)

Parameters
• rhs (Tensor) – Another tensor with a compatible shape.

• dtype (numpy.dtype) – The data type of the output tensor. If either of the inputs are complex, and the output is real, the real part is taken without warning.

• overwrite_rhs (bool) – Whether to mutate rhs so that its final value is the product. Otherwise, a copy of rhs is made.

Return type

Tensor

sample(n_shots=1, labels=None)

Assumes that this tensor represents a categorical probability distribution over ditstrings and samples n_shots from it, or a marginalized version of it if labels are provided.

import trueq.math as tqm

probs = tqm.Tensor(2, value={(0,): [0, 1], (1, 2): [0, 0.5, 0, 0.5]})

probs.sample(100)
# Results({"101": 52, "111": 48})

probs.sample(100, labels=(0, 2))
# Results({"11": 100})

Results({'11': 100})

Parameters
• n_shots (int) – The number of samples to draw from the distribution.

• labels (Iterable) – A list of labels to sample from.

Return type

Results

to_results(labels=None, clip=True)

Constructs a new Results object whose ditstring values are equal to the corresponding matrix entries of this tensor.

import trueq.math as tqm

probs = tqm.Tensor(2, value={(1,): [0, 1], (0, 2): [0, 0.5, 0, 0.5]})
probs.to_results()

Results({'011': 0.5, '111': 0.5})

Parameters
• labels (Iterable) – A list of labels to slice, in case a subset of all labels present in this tensor is desired. If None, all labels present in the tensor are used (in sorted order).

• clip (bool) – Whether or not to clip the values to the interval $[0, 1]$.

Return type

Results

class trueq.math.tensor.StateTensor(dim=None, spawn=None, value=None)

Represents a pure or mixed state on a multipartite qubit system. This is one of the main output formats of the Simulator. It differs from just a pure matrix, such as a numpy.ndarray, in that it also stores qubit label information, and attempts to store a sparse representation when the state is not entangled between various subsystems.

Depending on the types of noise in your simulator, this state could be pure or mixed. This will affect the dimension. You can check which case you have with is_mixed. You can get the matrix representation with mat().

import trueq as tq

# construct a circuit that makes a 3-qubit cat state
circuit = tq.Circuit()
circuit.append({(0,): tq.Gate.h})
circuit.append({(0, 1): tq.Gate.cnot})
circuit.append({(1, 2): tq.Gate.cnot})

# use a simulator to fetch the state of this circuit as a StateTensor
state = tq.Simulator().state(circuit)

# print the 1D pure state vector
print(state.mat())

# get the density matrix representation
tq.visualization.plot_mat(state.mat())

[0.70710678+0.j 0.        +0.j 0.        +0.j 0.        +0.j
0.        +0.j 0.        +0.j 0.        +0.j 0.70710678+0.j] What follows is information intended for users that want to do a bit more than get the matrix representation of a state. This class has a few main distinctions from its parent class Tensor:

1. The datatype is always numpy.complex128.

2. The mat() and value() methods return a more convenient tensor order and shape by default.

3. The upgrade() method, which transforms this state from a pure state into a mixed state by taking appropriate outer products and changing the output shape from (d,) to (d,d). Certain methods automatically call upgrade(), for example, when a superoperator-shaped matrix is given to apply_matrix().

4. Certain functions like marginalize have been special-cased to quantum states.

import trueq as tq
import numpy as np

# make a pure state on a tensor product of qubits
psi = tq.math.StateTensor(2)

# we can manually upgrade it to a mixed state

# or we can automatically upgrade it to a mixed state by doing something that
# requires a mixed state. here, we apply something of superoperator size, rather
# than unitary size
psi = tq.math.StateTensor(2)
psi.apply_matrix((0,), np.eye(4))

# we can also create a mixed state on instatiation based on spawn/value shapes
rho = tq.math.StateTensor(2, spawn=[[1, 0], [0, 0]])


Note

This class does not assume that the represented state is positive or that it has unit trace, or that applied operators are unitary, CP , TP, unital, etc.: it just stores and manipulates complex numerical arrays.

Parameters
• dim (int | NoneType) – The Hilbert space dimension of a single subsystem. The default value of None will result in the default dimension obtained from get_dim().

• spawn (numpy.ndarray-like) – The default state to set a subsystem to when its label is refered to, but it does not exist yet. This spawn can be pure or mixed. See also the constructor of Tensor.

• value (dict) – The initial value of this state, which can either be pure or mixed. See also the constructor of Tensor.

property is_mixed

Whether this state currently is a mixed state (as opposed to a pure state). Equivalent to is_upgraded.

Type

bool

Whether this state currently is a mixed state (as opposed to a pure state). Equivalent to is_mixed.

Type

bool

property dim

The Hilbert space dimension of the subsystems of this state.

Type

int

If this state represents a pure state, upgrades it to a mixed state by taking outer products. If this state is already a mixed state, does nothing.

Returns

This instance.

Return type

StateTensor

value(order='composite-group')

Returns a dictionary mapping tuples of subsystem labels to states on those subsystems. If this state is currently pure, then states will be 1D vectors, and if mixed, then states will be 2D density matrices.

Parameters

order (str) – Advanced users see Tensor.value().

Return type

dict

mat(order='composite-group', labels=None)

If this state is currently pure, returns the pure state vector, otherwise returns the density matrix.

Note

Note that this property will create an array that is exponentially large in the number of subsystems, and causes any distinct subsystems found in value() to be permanently merged; calling this attribute mutates the tensor’s internal representation.

Parameters
• order (str) – Advanced users see Tensor.value().

• labels (Iterable) – The desired subsystem label order, which must be a permutation of labels. By default, labels are ordered from smallest to biggest.

Return type

numpy.ndarray

apply_matrix(labels, matrix)

Mutates this state by applying the given square matrix to the given labels.

If acting on $n$ subsystems, this matrix can be a $d^n \times d^n$ unitary, or a $d^{2n}\times d^{2n}$ superoparator. If this state is currently pure and a superoperator is applied, upgrade() is called. If this state is currently mixed and a unitary is applied, the unitary is made into a superoperator before applying.

import trueq as tq
import numpy as np

t = tq.math.StateTensor(2)

# apply a 2-qubit unitary
t.apply_matrix((0, 1), tq.Gate.cnot.mat)

# apply a 1% bitflip superoperator to qubit 1. This upgrades to mixed state
s = 0.99 * np.eye(4) + 0.01 * np.kron(tq.Gate.x.mat, tq.Gate.x.mat)
t.apply_matrix((1,), s)

# display the final mixed state
tq.visualization.plot_mat(t.mat()) The vectorization convention of superoperators is important. Mixed states are stored in a subsystem-wise row-stacked vectorization convention, and therefore superoperators need to use this convention too. See to_rowstack_super().

Parameters
• labels (Iterable) – Which subsystems to apply the matrix to.

• matrix (numpy.ndarray-like) – A square matrix.

marginalize(labels, assume_unital=False, copy=False)

Keeps the provided labels of this tensor by taking the partial trace of the remaining labels. This will usually upgrade() a pure state to a mixed state.

The special case of no upgrade happens when this state is pure and all the labels being traced-out are known to be unentangled with other labels (this is known when every label tuple in the keys of value() is either a strict subset of the given labels, or does not intersect at all with the given labels).

Parameters
• labels (tuple) – Which labels to keep. The order does not matter.

• assume_unital (bool) – Assumes that each factor in the tensor has a trace of 1. Otherwise, in the case where entire entries in value() need to be removed, traces are still computed and multiplied onto some other entry so that the overall trace of the tensor is preserved. In the case where all labels in the tensor are traced-out, label (0,) is re-added so that the total trace of the tensor can be preserved.

• copy (bool) – Whether to make a copy of this tensor, or mutate it in place.

Returns

This instance.

Return type

StateTensor

probabilities(povm=None)

Computes the probabilities of all possible measurement outcomes. A Positive Operator Valued Measurement (POVM), if provided, defines the measurement outcomes. Otherwise, the outcomes are computational basis measurements.

import trueq.math as tqm
import numpy as np

# compute probability distribution of a qubit superposition
psi = tqm.StateTensor(2, value={(0,): np.array([1, -1j]) / np.sqrt(2)})
print(psi.probabilities().value())

# define a POVM corresponding to readout error that has a 1% chance of
# flipping a |0> measurement and a 10% chance of flipping a |1> measurement
p00 = 0.99
p11 = 0.9
povm = [[[p00, 0], [0, 1 - p11]], [[1 - p00, 0], [0, p11]]]
povm = tqm.Tensor((2,), (2, 2), spawn=povm)
print(psi.probabilities(povm).value())

# a more efficient implementation of the above, using a stochastic confusion
# matrix instead of quantum measurement operators
povm = tqm.Tensor(2, 2, spawn=[[p00, 1 - p11], [1 - p00, p11]])
print(psi.probabilities(povm).value())

{(0,): array([0.5, 0.5])}
{(0,): array([0.545, 0.455])}
{(0,): array([0.545, 0.455])}

Parameters

povm (Tensor) – A Tensor with shape ((k,), (d, d)) or ((k,), (d,)) where d is the Hilbert space dimension, and (k,) is the number of possible outcomes per subsystem. In the former case, each slice [i,:,:] should be a positive matrix representing a measurement operator, and the sum over i should be the identity matrix to be probability preserving. In the latter case, the tensor represents a confusion matrix with which to modify the computational basis probabilities, and the sum of each row should be one to be probability preserving.

Return type

dict

class trueq.math.tensor.OperatorTensor(dim=None, spawn=None, value=None, dtype=None)

Represents an operator (e.g., unitary) or superoperator (e.g., CPTP channel) on a multipartite qubit system. This is one of the main output formats of the Simulator. It differs from just a pure matrix, such as a numpy.ndarray, in that it also stores qubit label information, and attempts to store a sparse representation when the operator is not entangling between various subsystems.

Depending on the types of noise in your simulator, this operator could be a unitary or a superoperator. This will affect the dimension. You can check which case you have with is_superop. You can get the matrix representation with mat().

import trueq as tq

# construct a circuit that makes a 3-qubit cat state
circuit = tq.Circuit()
circuit.append({(0,): tq.Gate.h})
circuit.append({(0, 1): tq.Gate.cnot})
circuit.append({(1, 2): tq.Gate.cnot})

# use a simulator to fetch the unitary of this circuit as an OperatorTensor
op = tq.Simulator().operator(circuit)

# display the unitary matrix itself
tq.visualization.plot_mat(op.mat()) What follows is information intended for users that want to do a bit more than get the matrix representation of an operator. This class has a few distinctions from its parent class Tensor:

1. The datatype is numpy.complex128 by default.

2. The mat() and value() methods return a more convenient tensor order and shape by default.

3. The upgrade() method, which transforms this operator into a superoperator mixed state by taking appropriate kroneckers and changing the input and output shapes from (d,) to (d,d). Certain methods automatically call upgrade(), for example, when a superoperator shaped matrix is given to apply_matrix().

4. Certain functions like marginalize have been special-cased to quantum operators.

import trueq as tq
import numpy as np

# make a unitary channel for a tensor product of qubits
psi = tq.math.OperatorTensor(2)

# we can manually upgrade it to a superoperator

# or we can automatically upgrade it to a superoperator by doing something that
# requires a superoperator. here, we apply something of superoperator size,
# rather than unitary size
psi = tq.math.OperatorTensor(2)
psi.apply_matrix((0,), np.eye(4))

# we can also create a superoperator on instatiation based on spawn/value shapes
rho = tq.math.OperatorTensor(2, spawn=np.eye(4))


Note

This class does not assume that the represented channel or anything applied to it is unitary, CP, TP, unital, etc.: it is just stores and manipulates numerical arrays.

Parameters
• dim (int | NoneType) – The Hilbert space dimension of a single subsystem. The default value of None will result in the default dimension obtained from get_dim().

• spawn (numpy.ndarray-like) – The default operator to set a subsystem to when its label is referred to, but it does not exist yet. This spawn can be operator or superoperator. See also the constructor Tensor.

• value (dict) – The initial value of this state, which can either be an operator or a superoperator. See also the constructor of Tensor.

• dtype (type) – The datatype of this operator.

property is_superop

Whether this tensor represents a superoperator (as opposed to a unitary).

Type

bool

Whether this tensor represents a superoperator (as opposed to a unitary). Equivalent to is_superop.

Type

bool

property dim

The Hilbert space dimension that subsystems of this state act on.

Type

int

marginalize(labels, assume_unital=False, copy=False)

Keeps the provided labels of this tensor by taking the partial trace of the remaining labels. This will usually upgrade() a unitary tensor into a superoperator tensor.

The special case of no upgrade happens when this state is pure and all the labels being traced-out are known to be unentangled with other labels (this is known when every label tuple in the keys of value() is either a strict subset of the given labels, or does not intersect at all with the given labels).

Parameters
• labels (tuple) – Which labels to keep. The order does not matter.

• assume_unital (bool) – Assumes that each factor in the tensor has a trace of 1. Otherwise, in the case where entire entries in value() need to be removed, traces are still computed and multiplied onto some other entry so that the overall trace of the tensor is preserved. In the case where all labels in the tensor are traced-out, label (0,) is re-added so that the total trace of the tensor can be preserved.

• copy (bool) – Whether to make a copy of this tensor, or mutate it in place.

Returns

This instance.

Return type

StateTensor

If this operator represents a unitary, upgrades it to a tensor representing a superoperator. If this tensor already represents a superoperator, does nothing.

Returns

This instance.

Return type

OperatorTensor

value(order='composite-flat')

Returns a dictionary mapping tuples of subsystem labels to operators on those subsystems. Operators acting on n subsystems will have shape (d**n,d**n) or (d**(2*n),d**(2*n)) depending on the current value of is_superop.

Parameters

order (str) – Advanced users see Tensor.value().

Return type

dict

mat(order='composite-flat', labels=None)

Returns the matrix form of this operator. The tensor order is the sorted list of all present labels. If n subsystems are present, this matrix will have shape (d**n,d**n) or (d**(2*n),d**(2*n)) depending on the current value of is_superop.

If this is a superoperator, then the basis used is the composite row-stacking basis.

Note

Calling this method will create an array that is exponentially large in the number of subsystems, and causes any distinct subsystems found in value to be permanently merged; calling this method mutates the tensor’s internal representation.

Parameters
• order (str) – Advanced users see Tensor.value().

• labels (Iterable) – The desired subsystem label order, which must be a permutation of labels. By default, labels are ordered from smallest to biggest.

Return type

numpy.ndarray

apply_matrix(labels, matrix)

Mutates this operator by applying the given square matrix to the given labels.

If acting on $n$ subsystems, this matrix can be a $d^n \times d^n$ unitary, or a $d^{2n}\times d^{2n}$ superoparator. If this operator is currently a unitary and a superoperator is applied, upgrade() is called. If this state is currently a superoperator and a unitary is applied, the unitary is made into a superoperator before applying.

import trueq as tq
import numpy as np

t = tq.math.OperatorTensor(2)

# apply a unitary to qubits 0 and 1
t.apply_matrix((0, 1), tq.Gate.cnot.mat)

# apply a 1% bitflip superoperator. this upgrades to a superoperator
s = 0.99 * np.eye(4) + 0.01 * np.kron(tq.Gate.x.mat, tq.Gate.x.mat)
t.apply_matrix((1,), s)

# show the full superoperator
tq.visualization.plot_mat(t.mat()) The vectorization convention of superoperators is important. Mixed states are stored in a subsystem-wise row-stacked vectorization convention, and therefore superoperators need to use this convention too. See to_rowstack_super().

Parameters
• labels (Iterable) – Which subsystems to apply the matrix to.

• matrix (numpy.ndarray-like) – A square matrix.

diagonal(mat=None, force_real=True)

Returns a new 1D Tensor whose values are the diagonal elements of this operator conjugated by the given matrix.

For example, if the matrix is the change of basis matrix from the vectorized Pauli basis to the row-stacking basis, then the result is proportional to the Pauli transfer matrix diagonal:

import trueq as tq

circuit = tq.Circuit([{(0, 1): tq.Gate.cnot}])

# the PTM off by a scale of 2 on every subsystem
basis = [tq.math.pauli_basis.get_element(p).ravel() for p in "IXYZ"]
s.diagonal(np.array(basis).T).mat("f")

array([ 4.00000000e+00,  1.98000000e+00,  1.98000000e+00,  2.22044605e-16,
0.00000000e+00,  0.00000000e+00,  0.00000000e+00,  0.00000000e+00,
0.00000000e+00,  0.00000000e+00,  0.00000000e+00,  0.00000000e+00,
3.96000000e+00,  1.96020000e+00,  1.96020000e+00, -1.11022302e-16])

Parameters
• mat (numpy.ndarray) – A dimension-compatible square matrix to change the basis by.

• force_real (bool) – Forces the dtype of the returned tensor to be real.

Return type

Tensor

trueq.math.tensor.to_rowstack_super(A, B, d, n)

Returns the superoperator of the operation $X\mapsto AXB^T$ in the subsystem-wise row stacking basis.

See https://arxiv.org/abs/1111.6950 Section V for details on subsystem-wise vs. composite stacking vectorization bases (though beware that this reference uses the column stacking convention).

Parameters
• A – The left-multiplication operator, a square matrix.

• B (numpy.ndarray) – The right-multiplication operator, a matrix with the same shape as A.

• d (int) – The subsystem dimension.

• n (int) – The number of subsystems.

Returns

The superoperator corresponding to conjugating an input by the given matrices.

Return type

numpy.ndarray

## Weyl Operators¶

class trueq.math.weyl.WeylBase(powers, dim=None)

Base class for symplectic representations of collections and mappings of qudit Weyl operators.

Common to all subclasses, and present in this base class, is a matrix of integer powers which represent the desired operator(s). Each row of the powers matrix corresponds to a tensor product of single-qudit Weyls, $W_{x,z}=\otimes_{j=1,\ldots, n} W_{x_j,z_j}$ where $x,z\in\mathbb{Z}_d^n$ and $d\geq 2$ is the subsystem dimension. A single-qudit Weyl is the product of a power of the shift operator, $X=\sum_{i\in\mathbb{Z}_d}|i+1\rangle\langle i|$, and the clock operator, $Z=\sum_{i\in\mathbb{Z}_d}e^{2\pi i/d}|i\rangle\langle i|$, given as $W_{x,z}=X^x Z^z$ for $x,z\in\mathbb{Z}_d$. For $d=2$, these correspond to the Pauli operators up to a phase for $Y$.

In a given row of the powers matrix, the first half of a row, $x$, stores the shift operator powers, and the second half of a row, $z$, stores the clock operator powers; each row of the powers matrix represents a multi-qudit Weyl operator. Some subclasses of this base class may contain a vector of phases with one phase for each row of the matrix.

For convenience, the constructor allows the powers matrix to be specified as a string. For $d=2$, the strings may be upper case Pauli strings, where multiple multi-qubit Paulis are separated by the / character:

from trueq.math.weyl.base import WeylBase

p = WeylBase("XXYZ/IXYZ")
print(p)
p.powers

XXYZ/IXYZ

array([[1, 1, 1, 0, 0, 0, 1, 1],
[0, 1, 1, 0, 0, 0, 1, 1]], dtype=uint64)


For $d\geq 2$, single-qudit Weyl operators can be entered in the form, e.g., W21 which represents $W_{2,1}$. These are concatenated together, and multiple multi-qudit Weyl operators are separate by the / character:

from trueq.math.weyl.base import WeylBase

p = WeylBase("W20W11W01/W21W00W02", dim=3)
print(p)
p.powers

W20W11W01/W21W00W02

array([[2, 1, 0, 0, 1, 1],
[2, 0, 0, 1, 0, 2]], dtype=uint64)

Parameters
• powers (array_like | string | WeylBase) – An array of powers or a string representation, both of which are described above. All numbers are taken the modulo of the dimension at construction. Alternatively, this can also be another WeylBase (or child class) instance, from which the powers and dimension will be taken.

• dim (int | NoneType) – The qudit dimension, which must be one of SMALL_PRIMES. The default value of None will result in the default dimension obtained from get_dim(). This value is unused if powers is given as another WeylBase instance.

Raises
• ValueError – If powers argument is empty.

• ValueError – If a string specification is ragged.

• ValueError – If a Pauli string is given but $d\neq 2$.

• ValueError – If an array is given with an odd number of columns.

DTYPE

alias of numpy.uint64

ROW_CHAR = '/'

The row separation character for string representations.

WEYL_CHAR = 'W'

The Weyl prefix character for string representations.

property dim

The dimension of each subsystem.

Type

int

property powers

The array storing the symplectic representation via powers of the shift and clock operators. See also x (shift powers) and z (clock powers).

Type

numpy.ndarray

property n_sys

The number of qudits in each multi-qudit Weyl.

Type

int

property x

The 2D array of shift powers; the left half of powers.

Type

numpy.ndarray

property z

The 2D array of clock powers; the right half of powers.

Type

numpy.ndarray

classmethod eye(n_sys, dim=None)

Returns an identity instance of this class.

Parameters
• n_sys (int) – The number of subsystems.

• dim (int | NoneType) – The subsystem dimension.

Return type

WeylBase

classmethod random(n_sys, n_weyls=1, dim=None, include_id=True)

Returns a random instance of this class.

Parameters
• n_sys (int) – The number of subsystems.

• n_weyls (int) – The number of multi-qudit Weyls.

• dim (int | NoneType) – The qudit dimension, which must be one of SMALL_PRIMES. The default value of None will result in the default dimension obtained from get_dim().

• include_id (bool) – Whether to include any identites in the output.

Return type

WeylBase

classmethod from_dict(dic)

Converts a dictionary representation into a Weyl instance, see also to_dict().

Parameters

dic (dict) – The dictionary representation of the Weyls.

Return type

WeylBase

to_dict()

Converts this instance into a dictionary representation, see also from_dict().

Return type

dict

class trueq.math.weyl.Weyls(powers, dim=None)

Represents a list of multi-qudit projective Weyl operators. If the dimension is 2, then this is equivalent to a list of multi-qubit Pauli operators. See WeylBase for information about the storage format.

import trueq.math as tqm

# create a list of two three-qubit Paulis, XYZ and ZIY
tqm.Weyls("XYZ/ZIY")

# create a list of two three-qutrit Weyls
tqm.Weyls("W00W02W11/W11W01W20", 3)

Weyls('W00W02W11/W11W01W20', dim=3)


Here are examples exploring some of the main feature of this class:

import trueq.math as tqm

# two equivalent ways of creating W_{1, 0}, equal to Pauli X
assert tqm.Weyls("X") == tqm.Weyls([1, 0])

# two ways of creating W_{0, 1} W_{2, 3} in dimension 5
assert tqm.Weyls("W01W23", 5) == tqm.Weyls([[0, 2, 1, 3]], 5)

# two equivalent ways to list all single qubit Weyls, I, X, Y, Z
assert tqm.Weyls("I/X/Y/Z") == tqm.Weyls([[0, 0], [1, 0], [1, 1], [0, 1]])

# get the first three qubit Pauli from a list
assert tqm.Weyls("XYZ/ZIY")[0, :] == tqm.Weyls("XYZ")

# get a list of the Weyl operators acting on only the second qutrit
assert tqm.Weyls("W00W02W11/W11W01W20", 3)[:, 1] == tqm.Weyls("W02/W01", 3)

# Weyls can be multiplied, up to a phase
assert tqm.Weyls("X") @ tqm.Weyls("Y") == tqm.Weyls("Z")

# mutliplication distributes across broadcastable shapes
assert tqm.Weyls("X") @ tqm.Weyls("X/Y/Z/I") == tqm.Weyls("I/Z/Y/X")

Parameters
property iter_sys

An iterator yielding each subsystem of this instance as a new single-qudit Weyls.

Type

generator

property n_weyls

The number of multi-qudit Weyls stored in this instance.

Type

int

property gate

Constructs and returns the unitary matrix representation of this Weyl operator in the canonical basis as a Gate. If this instance contains multiple Weyl operators, that is, if n_weyls is greater than 1, an error will be raised. In this case, consider gates instead.

For example,

import trueq as tq
import trueq.math as tqm

assert tqm.Weyls("XX").gate == tq.Gate.x & tq.Gate.x

# note that there is a -I phase difference here which Gate equality ignores
assert tqm.Weyls("Y").gate == tq.Gate.y

Type

Gate

Raises

ValueError – If this instance contains more than one multi-qudit Weyl.

property gates

Constructs and returns a list containing the unitary matrix representations of each Weyl operator in this instance, in the canonical basis as a Gate.

For example,

import trueq as tq
import trueq.math as tqm

assert tqm.Weyls("Y").gates == [tq.Gate.y]

assert tqm.Weyls("XX").gates == [tq.Gate.x & tq.Gate.x]

assert tqm.Weyls("I/Z").gates == [tq.Gate.id, tq.Gate.z]

Type

list

property herm_mat

A Hermitian version of mat such that:

• Each distinct multi-qudit Weyl operator results in a fixed, unique Hermitian matrix.

• Matrices on $n$ $d$-dimensional subsystems are normalized to $\sqrt{d^n}$.

• Unequal multi-qudit Weyl operators result in orthogonal matrices.

• When $d=2$ the usual multi-qubit Pauli matrices are used, where, in particular, $Y=\left(\begin{smallmatrix}0&-i\\i&0\end{smallmatrix}\right)$ rather than $XZ=iY$, as used by mat.

• The span of any set of multi-qudit Weyl operators along with their adjoints results in the corresponding span of these Hermitian matrices.

The above properties are obtained with the following construction, where, unlike other methods and attributes of this class, we use the phase convention $W_{x,z}=e^{i \pi xz}X^xZ^z$ for a pair of powers $x,z\in\mathbb{Z}_d$. This is done to ensure the orthogonality criterion. Given some $W_{x,z}$, its adjoint is $W_{x,z}^\dagger=W_{-x,-z}=e^{i \pi xz}X^{-x}Z^{-z}$, and a Hermitian matrix is constructed from it by normalizing either $W_{x,z}+W_{x,z}^\dagger$ or $iW_{x,z}-iW_{x,z}^\dagger$. The former choice is made whenever $(x,z)>=(d-x,d-z)\mathrm{mod}d$ lexicographically, and the latter choice is made otherwise.

Like mat, the return shape is 3D if n_weyls is greater than one, where the output’s first index is over multi-qudit Weyls, otherwise the return shape is 2D.

Type

ndarray

property mat

The matrix representation, or an array of matrix representations, of the Weyl(s) contained in this instance, in the canonical basis. The return shape is 3D if n_weyls is greater than one where the first index is over multi-qudit Weyls, otherwise the return shape is 2D.

Type

ndarray

classmethod all(n_sys, dim=None, include_id=True, lex=True)

Returns a new Weyls instance where every possible projective multi-qudit Weyl operator appears as a row.

If lex is True, then the order is primarily lexicographic in the subsystems with the right-most subsystem rastering fastest, and secondarily lexicographic in the integer pair (x, z) for each subsystem. Otherwise, the secondary ordering is chosen so that operators that are adjoints of each other are adjacent, and also so that when $d=2$, the subsystem order is $I, X, Y, Z$.

Parameters
• n_sys (int) – The number of subsystems.

• dim (int | NoneType) – The subsystem dimension.

• include_id (bool) – Whether to include any identites in the output.

• lex (bool) – Whether to order subsystems lexicographically in $(x,z)$, as described above.

Return type

WeylBase

class trueq.math.weyl.WeylDecomp(powers, coeffs, dim=None)

Represents the decomposition of a multi-qudit operator in the Weyl operator basis. Each operator is decomposed into a vector of coefficients that correspond to each of the Weyl operators. See WeylBase for information about the storage format of the Weyl operators.

import trueq as tq
import trueq.math as tqm
import numpy as np

# construct a Weyl Decomposition with a list of Weyls and vector of coefficients
decomp1 = tqm.WeylDecomp("X/Z", [0.5, 3 - 2j])

# assert that the resulting matrix is as expected
mat = 0.5 * tq.Gate.x.mat + (3 - 2j) * tq.Gate.z.mat
assert np.allclose(decomp1.mat, mat)

decomp2 = tqm.WeylDecomp.from_mat(mat)
assert np.allclose(decomp2.powers, tqm.Weyls("X/Z").powers)
assert np.allclose(decomp2.coeffs, [0.5, 3 - 2j])
assert decomp1 == decomp2

Parameters
• powers (array_like | string | WeylBase) – A powers matrix specification, as described by WeylBase.

• coeffs (array_like) – A vector of complex floats corresponding to the coefficients of each Weyl operator specified by powers.

• dim (int | NoneType) – The qudit dimension, which must be one of SMALL_PRIMES. The default value of None will result in the default dimension obtained from get_dim().

Raises

ValueError – If either the powers matrix or coeffs vector have an invalid shape.

property coeffs

Returns the vector of coefficients that corresponds to each Weyl operator for this instance.

Type

ndarray

classmethod eye(n_sys, dim=None)

Returns an identity instance of this class.

Parameters
• n_sys (int) – The number of subsystems.

• dim (int | NoneType) – The subsystem dimension.

Return type

WeylBase

classmethod from_mat(mat)

Returns a new instance of this class, as generated by the given operator.

Parameters
• mat (array_like) – A mulit-qudit operator, in matrix form.

• dim (int | NoneType) – The qudit dimension, which must be one of SMALL_PRIMES. The default value of None will result in the default dimension obtained from get_dim().

Raises
• ValueError – If mat is not square.

• ValueError – If mat size is greater than 50.

• ValueError – If mat size is not a power of dim.

property mat

Returns the matrix representation of the decomposition stored by this instance. Calculates the summation of the product of each Weyl operator and its corresponding coefficient.

Type

ndarray

classmethod random(n_sys, dim=None)

Returns a random instance of this class. All Weyl operators will be used as a basis for powers, while coeffs will be randomly distributed.

Parameters
• n_sys (int) – The number of subsystems.

• n_weyls (int) – The number of multi-qudit Weyls.

• dim (int | NoneType) – The qudit dimension, which must be one of SMALL_PRIMES. The default value of None will result in the default dimension obtained from get_dim().

Return type

WeylBase

class trueq.math.weyl.Stabilizers(powers, phase_ints, dim=None)

Class that represents a list of multi-qudit Weyl operators with associated phases that are constrained to certain discrete values on the unit circle. Specifically, each multi-qudit Weyl operator in an instance of this class must leave a nontrivial subspace invariant and so cannot generate a group that contains nontrivial multiples of the identity. Common parent class to Clifford and StabilizerGroup.

An instance of this class contains a representation in terms of two integer arrays. The first is an array of powers, where the left half contains powers of the single-qudit $X$ operator, and the right half contains powers of the single-qudit $Z$ operator, so that each row represents a multi-qudit operator. The second is a vector of phase powers, one for each multi-qudit operator. For a particular multi-qudit operator with length-$n$ power vectors $x$ and $z$, a phase integer of value $j$ results in a phase of $exp(\pi i (2j + (x.z % 2)) / d)$ for $d=2$, and a phase of $exp(2\pi ij/ d)$ for $d>2$.

import trueq.math as tqm

# define stabilizers using two Weyls and a list of phases
stab_pauli = tqm.Stabilizers(tqm.Weyls("W00W11/W00W22"), [0, 1])

#get the list of phases in complex form
stab_pauli.phases

array([ 6.123234e-17+1.0000000e+00j, -1.000000e+00+1.2246468e-16j])

Parameters
• powers (array_like | string | WeylBase) – A powers matrix specification, as described by WeylBase.

• phase_ints (array_like) – A vector of integers corresponding to the phases of each Weyl operator specified by powers.

• dim (int | NoneType) – The qudit dimension, which must be one of SMALL_PRIMES. The default value of None will result in the default dimension obtained from get_dim().

Raises

ValueError – If either the powers matrix or phase_ints vector have an invalid shape.

property phase_ints

The array corresponding to the phases of each Weyl generator in the stabilizer group, represented as integers modulo dim.

Type

numpy.ndarray

property phases

The array corresponding to the phases of each Weyl generator in the stabilizer group, evaluated to their complex values.

Type

numpy.ndarray

classmethod eye(n_sys, dim=None)

Returns an identity instance of this class.

Parameters
• n_sys (int) – The number of subsystems.

• dim (int | NoneType) – The subsystem dimension.

Return type

WeylBase

classmethod random(n_sys, n_weyls=1, dim=None)

Returns a random instance of this class.

Parameters
• n_sys (int) – The number of subsystems.

• n_weyls (int) – The number of multi-qudit Weyls.

• dim (int | NoneType) – The qudit dimension, which must be one of SMALL_PRIMES. The default value of None will result in the default dimension obtained from get_dim().

Return type

WeylBase

class trueq.math.weyl.Clifford(powers, phase_ints, dim=None)

Represents a multi-qudit Clifford operator via its symplectic representation. See WeylBase and Stabilizers for more information.

An $n$-qudit Clifford is fully defined by its action on a set of generators of the Weyl group on $n$-qudits. For consistency, we fix this set of generators to be the Weyl $X$ and $Z$ Weyl operators on each qudit. These generators are ordered as $X_1$ through $X_n$ followed by $Z_1$ through $Z_n$, where $n$ is the number of qudits in the system.

Below is an example in which we define a two-qubit Clifford that maps $XI \rightarrow IX, IX \rightarrow XI, ZI \rightarrow IZ, IZ \rightarrow ZI$, and a single-qutrit Clifford which maps $X \rightarrow Z, Z\rightarrow X$.

import trueq.math as tqm
import numpy as np

# define a set of 4 two-qubit Paulis
paulis = tqm.Weyls("IX/XI/IZ/ZI")

# define a Clifford with the generating set mapping to the above ordered Paulis
cliff = tqm.Clifford(paulis, [0, 0, 0, 0])

IX = tq.Gate.id & tq.Gate.x
XI = tq.Gate.x & tq.Gate.id
assert np.array_equal(cliff.mat @ IX @ cliff.mat.conj(), XI)

Parameters
• powers (array_like | string | WeylBase) – A powers matrix specification, as described by WeylBase. Any provided input will be turned into a sorted list of generators which produce the same stabilizer group.

• phase_ints (array_like) – A vector of integers corresponding to the phases of each Weyl operator specified by powers.

• dim (int | NoneType) – The qudit dimension, which must be one of SMALL_PRIMES. The default value of None will result in the default dimension obtained from get_dim().

Raises
• ValueError – If either the powers matrix or phase_ints vector have an invalid shape.

• ValueError – If this Clifford does not preserve commutation relations

static random(n_sys, dim=None)

Construct a random Clifford on n_sys qudit subsystems of dimension dim.

Parameters
• n_sys (int) – The number of subsystems.

• dim (int | NoneType) – The qudit dimension, which must be one of SMALL_PRIMES. The default value of None will result in the default dimension obtained from get_dim().

Return type

Clifford

property mat

Returns the unitary representation of a Clifford operation.

Return type

ndarray

classmethod eye(n_sys, dim=None)

Returns an identity instance of this class.

Parameters
• n_sys (int) – The number of subsystems.

• dim (int | NoneType) – The subsystem dimension.

Return type

WeylBase

class trueq.math.weyl.StabilizerGroup(powers, phase_ints, dim=None)

A class which stores the symplectic representation of a group of multi-qudit stabilizer operators. See WeylBase and Stabilizers for more information.

For a given set of generators, this class will automatically check if the set is reducible and find a minimal set of generators for the group. Once constructed, class property state can be used to obtain a pure state that is invariant under all elements of this group.

import trueq.math as tqm
import numpy as np

# define a stabilizer group using two two-qudit Paulis and a list of phases
weyl = tqm.Weyls("YI/IY")
stab = tqm.StabilizerGroup(weyl, [0, 1])

# observe that the pure state is invariant under the elements in this group
state = stab.state
assert np.allclose((stab.phases * weyl.gates) @ state, state)
assert np.allclose((stab.phases * weyl.gates) @ state, state)

Parameters
• powers (array_like | string | WeylBase) – A powers matrix specification, as described by WeylBase. Any provided input will be turned into a sorted list of generators which produce the same stabilizer group.

• phase_ints (array_like) – A vector of integers corresponding to the phases of each Weyl operator specified by powers.

• dim (int | NoneType) – The qudit dimension, which must be one of SMALL_PRIMES. The default value of None will result in the default dimension obtained from get_dim().

Raises
• ValueError – If either the powers matrix or phase_ints vector have an invalid shape.

• ValueError – If some of the elements of the group do not commute.

classmethod random(n_sys, k=0, dim=None)

Construct a random stabilizer group over n_sys qudit subsystems of dimension dim. This stabilizer group will act on k logical qudits and thus will have n_sys minus k Weyl generators.

Parameters
• n_sys (int) – The number of subsystems.

• k (int) – The number of logical qudits. The default value of 0 will result in a pure state specified by n_sys generators.

• dim (int | NoneType) – The qudit dimension, which must be one of SMALL_PRIMES. The default value of None will result in the default dimension obtained from get_dim().

Return type

StabilizerGroup

property state

Returns a pure state in the simultaneous +1 eigenspace of all elements in the stabilizer group.

Return type

ndarray