Math¶

 trueq.math.decomposition.decompose_su4 Decompose a target two-qubit Gate into alternating cycles of local operations and the given two-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_gate Exhaustively attempts to reduce the given Gate or Cycle into a cycle containing gates that act on as few systems 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.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.kak Decompose an arbitrary two-qubit unitary matrix using the KAK decomposition. 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.random.random_bcsz Returns a random CPTP superoperator matrix drawn from the BCSZ distribution ([1]). 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.

Decomposition tools¶

trueq.math.decomposition.decompose_su4(given_gate, target_gate, max_depth=3, tol=1e-08)

Decompose a target two-qubit Gate into alternating cycles of local operations and the given two-qubit Gate.

import trueq as tq

circuit = tq.math.decompose_su4(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.

Returns

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

Return type

Circuit

trueq.math.decomposition.dekron(A, dim=2, force=False)

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

This can also be used to find the closest unitary to a matrix in terms of the Frobenius norm.

import numpy as np
import trueq as tq

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

print(tq.math.dekron(U, dim=2, force=True))
print(tq.math.dekron(U, dim=1, 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]]))
(array([[-1.+0.j]]), array([[-1.+0.j,  0.+0.j,  0.+0.j,  0.+0.j],
[ 0.+0.j, -1.+0.j,  0.+0.j,  0.+0.j],
[ 0.+0.j,  0.+0.j,  1.+0.j,  0.+0.j],
[ 0.+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) – Dimension of the first square submatrix 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 nagles are specified in degrees

Some decoposition 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_gate(gate, labels=None, max_size=6)

Exhaustively attempts to reduce the given Gate or Cycle into a cycle containing gates that act on as few systems as possible.

import trueq as tq

# Generate a 3 qubit gate object that is the product of seperate single
# qubit gate operations. This is using the & shorthand for kronecker product.
gate = tq.Gate.x & tq.Gate.y & tq.Gate.z

print(tq.math.split_gate(gate=gate, labels=(0, 1, 2)))

# Make a 90 degree rotation about XX on qubits (0, 2) and a T gate on qubit 1
gate = tq.Gate.from_generators('XIX', 90) @ tq.Gate.from_generators('IZI', 45)

print(tq.math.split_gate(gate, labels=(0, 1, 2)))

Cycle((0,): Gate.x, (1,): Gate.y, (2,): Gate.z)
Cycle((0, 2): Gate(XX), (1,): Gate.t)


Note

The cost of this function scales exponentially with the number of qubits. Below are best- and worst-case runtimes on a typical desktop computer for a single gate. Setting max_size to be greater than 6 for qubits could result in extremely long delays.

# of Qubits

Best

Worst

2 Qubits

0.5 ms

0.5 ms

3 Qubits

0.9 ms

1.2 ms

4 Qubits

1.4 ms

10 ms

5 Qubits

5.3 ms

250 ms

6 Qubits

0.03 s

22 sec

Parameters
• gate (Gate | Cycle) – The gate or cycle to be split into pieces if possible.

• labels (tuple | list) – The given labels associated with the provided gate. Must be omitted if the input is a cycle.

• max_size (int) – Spliting will only be attempted if the number of subsystems is less than this value. If the number of subsystems exceed this, the original gate is returned unsplit.

Returns

A Cycle containing the factored gates and associated labels.

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)

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

Parameters

n_sys (int) – the number of tensor factors.

Returns

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

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.

Returns

An iterator over basis element labels for multiple systems.

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)

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.

Returns

An iterator of the frame elements for multiple systems.

Return type

map

n_sys(arr)

Determine 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 in 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 in 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

sparse_superoperator(arr)

Returns a sparse representation of 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 sparse representation of a superoperator with respect to this frame.

Return type

dict

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

General tools¶

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.

print(int_base(16, 4))

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

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

Returns

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

Return type

int | None

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 | None

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 (tuple) – A tuple of numpy.ndarrays.

Returns

The Kronecker product of the specified arrays.

Return type

numpy.ndarray

trueq.math.general.proc_infidelity(A, B)

Calculates the process infidelity between two unitary matrices.

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

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

Returns

The process infidelity between the two unitary matrices.

Return type

float

trueq.math.general.reshuffle(mat, perm, dim=2)

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 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.

• dim (int) – The dimension of the systems, 2 for qubits, 3 for qutrits, etc.

Returns

The reshuffled matrix.

Return type

numpy.ndarray

KAK tools¶

trueq.math.kak_utils.kak(X)

Decompose 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 ([1]). The superoperator is in the global row stacking basis.

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

• rank (int) – A positive integer specifying the output Choi rank. If None, then the value dim ** 2 is used.

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).

Parameters

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

Returns

A Haar-random unitary matrix.

Return type

numpy.ndarray

Rotations¶

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.

trueq.math.rotation.Layer()

Parent class for Rotation and FixedRotation.

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
assert g.pauli_const == ("X", 180 / np.pi)

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.

Superoperators¶

class trueq.math.superop.Superop(mat)

Represents a quantum superoperator. A superoperator is a linear operation that maps density matrices to density matrices. The main utility of this class is converting between the various superoperator representations such as Kraus operators, Choi matrices, and Pauli transfer matrices.

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

superop = tq.math.Superop.from_kraus([np.eye(2)])


Note

This class is in support of future features, and is not currently connected to other parts of the library. However, it is still fully functional as a stand-alone utility, and can be used, for instance, to generate inputs to the simulator method add_kraus.

Parameters

mat (numpy.ndarray-like) – A superoperator matrix in the row-stacking basis.

property dim

The Hilbert space dimension that this superoperator acts on. For example, this is equal to 4 if this superoperator acts on $4 \times 4$ density matrices.

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_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_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 adj

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

0.31874888354513703

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

0.6504239085011243

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

Computes 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 [2].

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

0.13133752036677926

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

0.8974901366868904

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 [3]. 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 [4]. 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 from_kraus(*ops)

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 (Iterable) – An iterable of Kraus operators.

Returns

The superoperator corresponding to the given set of Kraus operators.

Return type

Superop

property kraus

Returns a 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).

Returns

A 3-D array ops of shape (n_ops, dim, dim) where ops[i,:,:] is the $i^{th}$ Kraus operator.

Return 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 (None | 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 | None) – An existing list of axes to plot on. They are created otherwise.

static from_unitary(u)

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.

Returns

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

Return type

Superop

static from_function(fcn, dim)

Instantiates a new Superop from a python function.

import trueq as tq

def depolarizing(rho, p=0.05):
d = rho.shape[0]
return (1 - p) * rho + p * np.eye(d) * np.trace(rho) / d

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

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

• dim (int) – The dimension of Hilbert space.

Returns

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

Return type

Superop

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.

Raises

RuntimeError if QuTiP is not imported.

Return type

Superop

property qobj

Returns 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>,
<mpl_toolkits.mplot3d.axes3d.Axes3D at 0x7fe9728d9be0>)


QuTiP is an optional dependency and must be installed for this function to work.

Raises

RuntimeError – If QuTiP is not imported.

Type

qutip.Qobj

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 for $n$ qubits, the Pauli matrices are denoted by $\{P_i\}_{i=0}^{4^n-1}$, and sorted lexicographically. For example, for $n=2$, then 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([0.97, 0.01, 0.01, 0.01]))

Parameters

ptm (numpy.ndarray) – A PTM. This must have a dimension that is a power of 4.

Returns

The superoperator for a given PTM.

Return type

Superop

property ptm

Returns the Pauli transfer matrix representation of this superoperator; see from_ptm() for details on this representation. Note that the dim of this superoperator must be a power of 4 for this to be valid.

Returns

The Pauli transfer matrix.

Return 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 (None | 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 | None) – An existing axis to plot on. One is created otherwise.

static from_choi(choi)

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.

Returns

The superoperator for a given Choi matrix.

Return type

Superop

property choi

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

Returns

The Choi matrix.

Return 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 (None | 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 | None) – An existing axis to plot on. One is created elsewise.

static from_rowstack(mat)

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.

Return type

Superop

property rowstack

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

Returns

The superoperator matrix.

Return 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 (None | 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 | None) – An existing axis to plot on. One is created elsewise.

static from_colstack(mat)

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.

Return type

Superop

property colstack

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

Returns

The superoperator matrix.

Return 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 (None | 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 | None) – An existing axis to plot on. One is created elsewise.

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([[-1.21588151+0.j,  1.48638485+0.j, -0.16432768+0.j],
[ 0.89769996+0.j,  0.02039868+0.j, -3.09048936+0.j],
[ 1.54505627+0.j, -0.39947328+0.j, -0.60418306+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 | None) – 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 datatype 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

adj(copy=True)

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([], [0]).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], [0]).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

add_labels(labels)

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

upgrade()

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 eachother.

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=2, 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) – The Hilbert space dimension of a single subsystem.

• 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).

Type

bool

property dim

The Hilbert space dimension of the subsystems of this state.

Type

int

upgrade()

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. If a Positve Operator Valued Measurement (POVM) is not provided, then these are the probabilities of 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())

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

Parameters

povm (Tensor) – A Tensor with shape ((k,), (d, d)) where d is the Hilbert space dimension, and (k,) is the number of possible outcomes per subsystem. Each slice [i,:,:] should be a positive matrix, and the sum over i should be the identity matrix, to be probability preserving.

Return type

dict

class trueq.math.tensor.OperatorTensor(dim=2, 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) – The Hilbert space dimension of a single subsystem.

• 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

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

upgrade()

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, -1.11022302e-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,  2.22044605e-16])

Parameters

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

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