Operations
Parent class of all primitive quantum operations. 

Represents a unitary quantum gate. 

A measurement of a single qubit in the computational basis. 

A subclass of 

A preparation of a single qubit into the ground state of the computational basis. 
Operation
 class trueq.Operation
Parent class of all primitive quantum operations.
Gate
 class trueq.Gate(u)
Represents a unitary quantum gate.
import trueq as tq tq.Gate([[0, 1], [1, 0]])
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 Gate.x
 Aliases:

 Gate.x
 Gate.cliff1
 Generators:

 'X': 180.0
 Matrix:

Many standard gates are predefined as attributes:
import trueq as tq print("All available predefined gate names:") print(list(tq.Gate.ALIASES)) # Accessing a predefined gate: tq.Gate.cx
All available predefined gate names: ['id', 'x', 'y', 'z', 'cx', 'cy', 'cz', 'swap', 'iswap', 'h', 's', 'sx', 'sy', 't', 'ch', 'i', 'sz', 'f', 'cnot', 'cliff0', 'cliff1', 'cliff2', 'cliff3', 'cliff4', 'cliff5', 'cliff6', 'cliff7', 'cliff8', 'cliff9', 'cliff10', 'cliff11', 'cliff12', 'cliff13', 'cliff14', 'cliff15', 'cliff16', 'cliff17', 'cliff18', 'cliff19', 'cliff20', 'cliff21', 'cliff22', 'cliff23', 'id3', 'x3', 'y3', 'z3', 'cx3', 'cz3', 'f3', 'x3pow2', 'y3pow2', 'x3pow1', 'y3pow1', 'cnot3', 'id5', 'x5', 'y5', 'z5', 'cx5', 'cz5', 'f5', 'x5pow2', 'x5pow3', 'x5pow4', 'y5pow2', 'y5pow3', 'y5pow4', 'x5pow1', 'y5pow1', 'cnot5', 'id7', 'x7', 'y7', 'z7', 'cx7', 'cz7', 'f7', 'x7pow2', 'x7pow3', 'x7pow4', 'x7pow5', 'x7pow6', 'y7pow2', 'y7pow3', 'y7pow4', 'y7pow5', 'y7pow6', 'x7pow1', 'y7pow1', 'cnot7']
TrueQ 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:

 'ZX': 90.0
 'IX': 90.0
 'ZI': 90.0
 Matrix:

The static method
from_generators()
can be used to construct gates from Pauli rotations.Note
Gates are not checked to be unitary on instantiation; call
is_unitary
manually to check. Parameters
u (
numpy.ndarray
like dict
) – A unitary matrix in the computational basis.
 ALIASES = {'ch': Gate.ch, 'cliff0': Gate.id, 'cliff1': Gate.x, 'cliff10': Gate.cliff10, 'cliff11': Gate.cliff11, 'cliff12': Gate.h, 'cliff13': Gate.cliff13, 'cliff14': Gate.cliff14, 'cliff15': Gate.cliff15, 'cliff16': Gate.cliff16, 'cliff17': Gate.cliff17, 'cliff18': Gate.cliff18, 'cliff19': Gate.cliff19, 'cliff2': Gate.y, 'cliff20': Gate.cliff20, 'cliff21': Gate.cliff21, 'cliff22': Gate.cliff22, 'cliff23': Gate.cliff23, 'cliff3': Gate.z, 'cliff4': Gate.cliff4, 'cliff5': Gate.sx, 'cliff6': Gate.cliff6, 'cliff7': Gate.sy, 'cliff8': Gate.cliff8, 'cliff9': Gate.s, 'cnot': Gate.cx, 'cnot3': Gate.cx3, 'cnot5': Gate.cx5, 'cnot7': Gate.cx7, 'cx': Gate.cx, 'cx3': Gate.cx3, 'cx5': Gate.cx5, 'cx7': Gate.cx7, 'cy': Gate.cy, 'cz': Gate.cz, 'cz3': Gate.cz3, 'cz5': Gate.cz5, 'cz7': Gate.cz7, 'f': Gate.h, 'f3': Gate.f3, 'f5': Gate.f5, 'f7': Gate.f7, 'h': Gate.h, 'i': Gate.id, 'id': Gate.id, 'id3': Gate.id3, 'id5': Gate.id5, 'id7': Gate.id7, 'iswap': Gate.iswap, 's': Gate.s, 'swap': Gate.swap, 'sx': Gate.sx, 'sy': Gate.sy, 'sz': Gate.s, 't': Gate.t, 'x': Gate.x, 'x3': Gate.x3, 'x3pow1': Gate.x3, 'x3pow2': Gate.x3pow2, 'x5': Gate.x5, 'x5pow1': Gate.x5, 'x5pow2': Gate.x5pow2, 'x5pow3': Gate.x5pow3, 'x5pow4': Gate.x5pow4, 'x7': Gate.x7, 'x7pow1': Gate.x7, 'x7pow2': Gate.x7pow2, 'x7pow3': Gate.x7pow3, 'x7pow4': Gate.x7pow4, 'x7pow5': Gate.x7pow5, 'x7pow6': Gate.x7pow6, 'y': Gate.y, 'y3': Gate.y3, 'y3pow1': Gate.y3, 'y3pow2': Gate.y3pow2, 'y5': Gate.y5, 'y5pow1': Gate.y5, 'y5pow2': Gate.y5pow2, 'y5pow3': Gate.y5pow3, 'y5pow4': Gate.y5pow4, 'y7': Gate.y7, 'y7pow1': Gate.y7, 'y7pow2': Gate.y7pow2, 'y7pow3': Gate.y7pow3, 'y7pow4': Gate.y7pow4, 'y7pow5': Gate.y7pow5, 'y7pow6': Gate.y7pow6, 'z': Gate.z, 'z3': Gate.z3, 'z5': Gate.z5, 'z7': Gate.z7}
A dictionary containing predefined qudit gates for dimensions 2, 3, 5, and 7, where the keys are the gate names, for example
CNOT
, orH
. Gates in this dictionary can be accessed usingtrueq.Gate.<gate_name>
, for example:import trueq as tq tq.Gate.cnot
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 Gate.cx
 Aliases:

 Gate.cx
 Gate.cnot
 Likeness:

 CNOT
 Generators:

 'ZX': 90.0
 'IX': 90.0
 'ZI': 90.0
 Matrix:

 QUDIT_DIM2_ALIASES = {'ch': Gate.ch, 'cliff0': Gate.id, 'cliff1': Gate.x, 'cliff10': Gate.cliff10, 'cliff11': Gate.cliff11, 'cliff12': Gate.h, 'cliff13': Gate.cliff13, 'cliff14': Gate.cliff14, 'cliff15': Gate.cliff15, 'cliff16': Gate.cliff16, 'cliff17': Gate.cliff17, 'cliff18': Gate.cliff18, 'cliff19': Gate.cliff19, 'cliff2': Gate.y, 'cliff20': Gate.cliff20, 'cliff21': Gate.cliff21, 'cliff22': Gate.cliff22, 'cliff23': Gate.cliff23, 'cliff3': Gate.z, 'cliff4': Gate.cliff4, 'cliff5': Gate.sx, 'cliff6': Gate.cliff6, 'cliff7': Gate.sy, 'cliff8': Gate.cliff8, 'cliff9': Gate.s, 'cnot': Gate.cx, 'cx': Gate.cx, 'cy': Gate.cy, 'cz': Gate.cz, 'f': Gate.h, 'h': Gate.h, 'i': Gate.id, 'id': Gate.id, 'iswap': Gate.iswap, 's': Gate.s, 'swap': Gate.swap, 'sx': Gate.sx, 'sy': Gate.sy, 'sz': Gate.s, 't': Gate.t, 'x': Gate.x, 'y': Gate.y, 'z': Gate.z}
A dictionary containing predefined qubit gates where the keys are the gate names, for example
CNOT
, orH
. Gates in this dictionary can be accessed usingtrueq.Gate.<gate_name>
, for example:import trueq as tq tq.Gate.cx
TrueQ 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:

 'ZX': 90.0
 'IX': 90.0
 'ZI': 90.0
 Matrix:

 QUDIT_DIM3_ALIASES = {'cnot3': Gate.cx3, 'cx3': Gate.cx3, 'cz3': Gate.cz3, 'f3': Gate.f3, 'id3': Gate.id3, 'x3': Gate.x3, 'x3pow1': Gate.x3, 'x3pow2': Gate.x3pow2, 'y3': Gate.y3, 'y3pow1': Gate.y3, 'y3pow2': Gate.y3pow2, 'z3': Gate.z3}
A dictionary containing predefined gates for qudits of dimension 3, where the keys are the gate names, for example
CNOT3
, orX3
. Gates in this dictionary can be accessed usingtrueq.Gate.<gate_name>
, for example:import trueq as tq tq.Gate.cnot3
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 Gate.cx3
 Aliases:

 Gate.cx3
 Gate.cnot3
 Matrix:

 QUDIT_DIM5_ALIASES = {'cnot5': Gate.cx5, 'cx5': Gate.cx5, 'cz5': Gate.cz5, 'f5': Gate.f5, 'id5': Gate.id5, 'x5': Gate.x5, 'x5pow1': Gate.x5, 'x5pow2': Gate.x5pow2, 'x5pow3': Gate.x5pow3, 'x5pow4': Gate.x5pow4, 'y5': Gate.y5, 'y5pow1': Gate.y5, 'y5pow2': Gate.y5pow2, 'y5pow3': Gate.y5pow3, 'y5pow4': Gate.y5pow4, 'z5': Gate.z5}
A dictionary containing predefined gates for qudits of dimension 5, where the keys are the gate names, for example
CNOT5
, orX5
. Gates in this dictionary can be accessed usingtrueq.Gate.<gate_name>
, e.g.trueq.Gate.cnot5
.
 QUDIT_DIM7_ALIASES = {'cnot7': Gate.cx7, 'cx7': Gate.cx7, 'cz7': Gate.cz7, 'f7': Gate.f7, 'id7': Gate.id7, 'x7': Gate.x7, 'x7pow1': Gate.x7, 'x7pow2': Gate.x7pow2, 'x7pow3': Gate.x7pow3, 'x7pow4': Gate.x7pow4, 'x7pow5': Gate.x7pow5, 'x7pow6': Gate.x7pow6, 'y7': Gate.y7, 'y7pow1': Gate.y7, 'y7pow2': Gate.y7pow2, 'y7pow3': Gate.y7pow3, 'y7pow4': Gate.y7pow4, 'y7pow5': Gate.y7pow5, 'y7pow6': Gate.y7pow6, 'z7': Gate.z7}
A dictionary containing predefined gates for qudits of dimension 7, where the keys are the gate names, for example
CNOT7
, orX7
. Gates in this dictionary can be accessed usingtrueq.Gate.<gate_name>
, e.g.trueq.Gate.cnot7
.
 static from_generators(*args, dim=None)
Constructs a
Gate
instance from a compound rotation about Pauli axes. Specifically, the gate is constructed as \(exp(1j * \sum_i (\pi\theta_i/180) W_i/ 2)\) where \(W_i\) are the input Paulis (of Weyls for qudit systems) and \(\theta_i\) are the input rotation angles in degrees. This can be input either as an alternating sequence of Weyl strings and angles, or as adict
.import trueq as tq g1 = tq.Gate.from_generators("X", 90) g2 = tq.Gate.from_generators("XX", 90, "YY", 180) g3 = tq.Gate.from_generators({"XX": 90, "YY": 180}) g4 = tq.Gate.from_generators("W01W02", 90, "W10W20", 180, dim=3) g2
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 Gate(YY, XX)
 Likeness:

 CNOT
 Generators:

 'YY': 180.0
 'XX': 90.0
 Matrix:

Note
For rotations around a single Pauli term, it is faster to use
rp()
. Parameters
*args – A single dictionary mapping Pauli strings to angles in degrees, or an alternating sequence thereof.
dim (
int
) – The qudit dimension, which must be one ofSMALL_PRIMES
. The default value ofNone
will result in the default dimension obtained fromget_dim()
.
 Return type
 get_alias(default=None)
Returns the first matching alias of the gate if possible, or
default
if no aliases are found. Parameters
default (
NoneType
str
) – A string to return if no alias is found for this gate. If no string is specified,None
will be returned. Returns
The alias of this gate if it exists,
default
otherwise. Return type
str
NoneType
 property is_clifford
Whether or not this gate is Clifford.
If
width
is not a power of a small prime integer, implying that this gate does not act on \(n\) qudits of dimension inSMALL_PRIMES
, this is set toFalse
by convention. Type
bool
 property is_unitary
Whether or not this gate is unitary to numerical precision.
 Type
bool
 property is_identity
Whether or not this gate is the identity gate to numerical precision.
 Type
bool
 property mat
The matrix representation of this gate in the canonical basis.
 Type
numpy.ndarray
 property width
The number of columns of the matrix representation of this gate.
 Type
int
 property expansion
A sparse representation of this gate in terms of the Weyl operators; when this gate acts on qubits, a dictionary mapping Pauli strings to (complex) coefficients.
 Type
dict
 property generators
Assumes this gate acts on qubits and returns rotations about Pauli axes performed by this
Gate
. Specifically, outputs a description of this gate in the form: \(exp(1j * \sum_i (\pi\theta_i/180) P_i/ 2)\) where \(P_i\) are the Pauli strings and \(\theta_i\) are the rotation angles in degrees.import trueq as tq import numpy as np gens = tq.Gate.from_generators("X", 90, "Z", 20).generators assert np.isclose(gens["X"], 90) assert np.isclose(gens["Z"], 20)
 Type
dict
 property sqrt
A new
Gate
instance which is a square root of this gate, respecting the global phase. That is, this method guarantees that \(U^{1/2}U^{1/2}=U\) for any unitary gate \(U\). This differs from the general power method onGate
which chooses a certain scheme for global phases tailored for overrotation errors. Type
 controlled(control_on='1', pos=1, exclusive=True, dim=None)
A new
Gate
instance which is a controlled version of this gate.The gate is applied to the qudit(s) in positions
pos
if and only if the rest of the qudits are in the computational basis statecontrol_on
, assuming the default valueexclusive=True
is used.import trueq as tq # Toffoli gate tq.Gate.x.controlled("11")
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 Gate(ZZX, ZIX, ...)
 Generators:

 'ZZX': 45.0
 'ZIX': 45.0
 'IZX': 45.0
 'IIX': 45.0
 'ZZI': 45.0
 'ZII': 45.0
 'IZI': 45.0
 Matrix:

If
exclusive=False
, however, the alternate definition \(\sum_{k\in\mathbb{Z}_d^c} k+m\rangle\langle k+m \otimes U^{\sum_{i=1}^c k_i}\) is used, where \(m\) is given bycontrol_on
, and where \(c\) is the length of \(m\). For example, the CSUM (controlled sum) gate is given by \(\sum_{k=0}^{d1} k\rangle\langle k \otimes X^{k}\) where \(X\) is the qudit shift operator, orW10
in the string syntax ofWeylBase
. The subsystems are permuted to matchpos
.import trueq as tq # CSUM gate for qutrits tq.Gate(tq.math.Weyls("W10", dim=3)).controlled("0", exclusive=False)
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 Gate.cx3
 Aliases:

 Gate.cx3
 Gate.cnot3
 Matrix:

 Parameters
control_on (
Iterable
str
) – The ditstring specifying the computational state to condition on. This can be any iterable whose elements are castable to integers representing dits. The default setting conditions on the computational basis state \(1\rangle\).pos (
int
tuple
) – The position of the subsystem on which the controlled gate acts. The default is the last subsystem.exclusive (
int
None
) – IfTrue
,control_on
represents the exclusive ditstring state of the control qudits which cause this gate to be applied. IfFalse
, the definition provided earlier is applied.dim – The subsystem dimension. This can usually be left at the default value of
None
because the subsystem dimension can be inferred from the length ofpos
and the size of this gate. However, in the edge case that this gate is \(1\times 1\), this is not possible, in which case the value ofdim
is used, or taken fromget_dim()
ifdim
isNone
.
 Return type
 to_clifford()
Returns the
Clifford
representation of a Clifford gate.import trueq as tq tq.Gate.cz.to_clifford()
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 Clifford
 Dim:
 2
 Generator images:
X_{0} X_{1} Z_{0} Z_{1} ph. im(X_{0}) 1 0 0 1 0 im(X_{1}) 0 1 1 0 0 im(Z_{0}) 0 0 1 0 0 im(Z_{1}) 0 0 0 1 0  Return type
 Raises
RuntimeError – If
width
is not a power of a small prime integer inSMALL_PRIMES
.RuntimeError – If the gate is not Clifford.
 static cw(weyl, dim=None)
A convenience method to construct a controlledWeyl
Gate
defined as \(\sum_{k\in\mathbb{Z}_d} k\rangle\langle k \otimes W^{k}\), where \(W\) is the inputweyl
operator.import trueq as tq # this is a cz for qubits g1 = tq.Gate.cw("W01", 2) # this is the generalization of cz to qutrits g2 = tq.Gate.cw("W01", 3)
 Parameters
weyl (
Weyls
Iterable
) – AWeyls
, or a Weyl string and a dimension specifying the desired operation.dim (
int
) – The qudit dimension, which must be one ofSMALL_PRIMES
. The default value ofNone
will result in the default dimension obtained fromget_dim()
.
 Return type
 embed(extra_dim, n_sys=None)
Returns a
Gate
that is equal to this gate embedded in a larger Hilbert space. That is, if this gate acts on the space \((\mathbb{C}^d)^{\otimes n}\), then the returned gate will act on the space \((\mathbb{C}^{d+d'})^{\otimes n}\). Parameters
extra_dim (
int
) – The number of extra dimensions, \(d'\), to add to each subsystem.n_sys (
int
NoneType
) – The number of subsystems that this gate acts on. If unspecified, it is inferred automatically by usingauto_base()
.
 Return type
 static random(dim)
Generate a Haar random unitary gate of dimension
dim
. Parameters
dim (
int
) – The dimension of the gate. Return type
 plot(abs_max=None, ax=None)
Plots the matrix representation of this gate.
 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
) – An existing axis to plot on. If not given, a new figure will be created.
 static rp(pauli, angle, dim=None)
Construct a
Gate
from a Pauli or Weyl rotation by the specified angle.import trueq as tq import numpy as np gens = tq.Gate.rp("XX", 10).generators assert np.isclose(gens["XX"], 10) g_weyl = tq.Gate.rp("W02", 10, dim=3)
Note
For single pauli rotations this is faster numerically than
from_generators()
.Note
The methods
rp()
andrw()
are identical. They both exist to make searching the documentation easier. Parameters
pauli (
str
) – The Pauli axis of rotation, or the Weyl axis for qudit systems.angle (
float
) – The angle of the rotation in degrees.dim (
int
) – The qudit dimension, which must be one ofSMALL_PRIMES
. The default value ofNone
will result in the default dimension obtained fromget_dim()
.
 Return type
 static rw(pauli, angle, dim=None)
Construct a
Gate
from a Pauli or Weyl rotation by the specified angle.import trueq as tq import numpy as np gens = tq.Gate.rp("XX", 10).generators assert np.isclose(gens["XX"], 10) g_weyl = tq.Gate.rp("W02", 10, dim=3)
Note
For single pauli rotations this is faster numerically than
from_generators()
.Note
The methods
rp()
andrw()
are identical. They both exist to make searching the documentation easier. Parameters
pauli (
str
) – The Pauli axis of rotation, or the Weyl axis for qudit systems.angle (
float
) – The angle of the rotation in degrees.dim (
int
) – The qudit dimension, which must be one ofSMALL_PRIMES
. The default value ofNone
will result in the default dimension obtained fromget_dim()
.
 Return type
 static rx(angle)
Constructs a
Gate
from a PauliX
rotation by the specified angle.import trueq as tq import numpy as np gens = tq.Gate.rx(10).generators assert np.isclose(gens["X"], 10)
 Parameters
angle (
float
) – The angle of the rotation in degrees. Return type
 static ry(angle)
Constructs a
Gate
from a PauliY
rotation by the specified angle.import trueq as tq import numpy as np gens = tq.Gate.ry(10).generators assert np.isclose(gens["Y"], 10)
 Parameters
angle (
float
) – The angle of the rotation in degrees. Return type
 static rz(angle)
Constructs a
Gate
from a PauliZ
rotation by the specified angle.import trueq as tq import numpy as np gens = tq.Gate.rz(10).generators assert np.isclose(gens["Z"], 10)
 Parameters
angle (
float
) – The angle of the rotation in degrees. Return type
 static fourier(dim=None)
Returns a Fourier
Gate
acting on a qudit of dimensiondim
, as defined in [19]. The FourierGate
maps \(X \rightarrow Z, Z \rightarrow X^{1}\), and can thus be regarded as a generalization of the Hadamard gate to qudits. Parameters
dim (
int
) – The qudit dimension, which must be one ofSMALL_PRIMES
. The default value ofNone
will result in the default dimension obtained fromget_dim()
. Return type
 static w(weyl, dim=None)
Constructs a
Gate
implementing aWeyls
operation. The desired operation can be input either as Weyl strings and associated dimention, or as aWeyls
.import trueq as tq from trueq.math import Weyls g1 = tq.Gate.w("W02", 3) g2 = tq.Gate.w(Weyls("W02", dim=3))
 Parameters
weyl (
Weyls
str
) – AWeyls
, or a Weyl string and a dimension specifying the desired operation.dim (
int
) – The qudit dimension, which must be one ofSMALL_PRIMES
. The default value ofNone
will result in the default dimension obtained fromget_dim()
.
 Return type
Measurement
 class trueq.Meas
A measurement of a single qubit in the computational basis.
This class holds no information, it allows a
trueq.Circuit
to keep track of where a measurement has taken place.This is a singleton class; the constructor always yields the same instance.
Native Gate
 class trueq.NativeGate(name, u, params=None)
A subclass of
Gate
which introduces metadata attributes.name
is a string giving a name to the gate.(optional)
parameters
is a dictionary that maps parameter names to parameter values.
These attributes are entirely metadata; there are no automatic consistency checks between them and the matrix representation. The purpose of these attributes is to provide tools, such as transpilers to thirdparty circuit formats, with a fast method of identifying gates without introspecting matrices. This requires trust that these metadata were generated correctly and in good faith.
Native gates are typically (and most easily) constructed by
GateFactory
s, which are usually owned byConfig
objects.import trueq as tq config = tq.Config.basic() config.sx()
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 sx()
 Aliases:

 Gate.sx
 Gate.cliff5
 Generators:

 'X': 90.0
 Matrix:

Native gates are not intended to be mutable, and the parameters and names should not be changed after instantiation.
 Parameters
name (
str
) – Name of the gate operation.u (
dict
) – SeeGate
.params (
dict
) – Any parameters that the gate needs in order to be constructed.
 static from_generators(name, *args, params=None)
Constructs a
NativeGate
instance from rotations about Paulis. Also adds a name and any parameters used to construct the native gate. Specifically, the output gate in this case is given by \(exp(1j * \sum_i (\pi\theta_i/180) P_i/ 2)\) where \(P_i\) are the input Pauli strings and \(\theta_i\) are the input rotation angles in degrees. This can be input either as an alternating sequence of Pauli strings and angles, or as adict
.import trueq as tq g1 = tq.NativeGate.from_generators("name", "X", 90, params={"theta": 90}) g2 = tq.NativeGate.from_generators("name", "XX", 90, "YY", 180) g3 = tq.NativeGate.from_generators("name", {"XX": 90, "YY": 180}) g1
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 name(theta)
 Aliases:

 Gate.sx
 Gate.cliff5
 Parameters:

 theta = 90
 Generators:

 'X': 90.0
 Matrix:

 Parameters
name (
str
) – Name of the gate operation.*args – A single dictionary mapping Pauli strings to angles in degrees, or an alternating sequence thereof.
params (
dict
) – Any parameters that the gate needs in order to be constructed.
 Return type
 property parameters
The stored parameters.
 Type
dict
 property name
The name of this native gate.
 Type
str
Preparation
 class trueq.Prep
A preparation of a single qubit into the ground state of the computational basis.
This class holds no information, it allows a
trueq.Circuit
to keep track of where a state preparation has taken place.This is a singleton class; the constructor always yields the same instance.