# Simulation

 trueq.Simulator A (noisy) quantum simulator for Circuit objects that can be used to compute the final state(), the total effective operator(), or simply run() them to sample() shots. trueq.Simulator.add_cycle_noise A noise source which inserts or replaces entire cycles during simulation. trueq.Simulator.add_depolarizing Appends a noise source to this simulator that adds isotropic depolarizing noise to the simulation. trueq.Simulator.add_gate_replace Adds a noise source that replaces all matched Gates in the circuit with the returned gate or superoperator of the provided function fn. trueq.Simulator.add_knr_noise A noise source which emulates the noise measured in a KNR experiment. trueq.Simulator.add_kraus Appends a noise source to this simulator that adds Kraus noise to the simulation. trueq.Simulator.add_overrotation Appends a noise source to this simulator that performs gate over/underrotation. trueq.Simulator.add_povm add_povm() was deprecated in version 2.11.4 and will be removed no earlier than version 2.13.0. trueq.Simulator.add_prep Appends a state preparation description to this simulator that chooses which state to prepare whenever a preparation instruction is encountered in a circuit. trueq.Simulator.add_rcal_noise Appends measurement classification error to this simulator, where errors are specified by confusion matrices estimated by the readout calibration protocol (RCAL). trueq.Simulator.add_readout_error Appends measurement classification error and/or positive operator valued measurement (POVM) error to this simulator. trueq.Simulator.add_relaxation Appends a noise source to this simulator that adds $T_1$ and $T_2$ relaxation noise to the simulation. trueq.Simulator.add_stochastic_pauli Appends a noise source to this simulator that introduces stochastic Pauli noise. trueq.Simulator.dressed_noise Computes the dressed noise of the given cycle. trueq.Simulator.operator Returns the unitary or superoperator that results from simulating the given circuit. trueq.Simulator.predict_cb Predicts the estimates that would be generated by simulating make_cb() with this simulator. trueq.Simulator.predict_irb Predicts the estimates that would be generated by simulating make_irb() with this simulator. trueq.Simulator.predict_knr Predicts the estimates that would be generated by simulating make_knr() with this simulator. trueq.Simulator.predict_srb Predicts the estimates that would be generated by simulating make_srb() with this simulator. trueq.Simulator.run Updates the results attribute of each given circuit by simulating its final quantum state and sampling shots from it. trueq.Simulator.state Returns the quantum state that results from simulating the given circuit. trueq.simulation.backend.SimulationBackend Abstract parent class for all simulation backends. trueq.simulation.MeasPolicy Measurement policies for simulation backends. trueq.simulation.PrepPolicy Preparation policies for simulation backends. trueq.simulation.PropagationBackend A simulation backend that uses OperatorTensor or StateTensor to perform simulations. trueq.simulation.noise_source.NoiseSource Parent class for all built in noise sources which can be added to the simulator. trueq.simulation.match.Match A class whose purpose is to filter the operations presented to a given NoiseSource in a Simulator instance. trueq.simulation.match.CycleWrapper Wraps a Cycle into a list of OpWrappers, partitioning the cycle ops into three groups, Gate which contains all gates, Meas which contains all measurements, and Prep which contains all state preparations in the cycle being propagated by the simulator. trueq.simulation.match.OpWrapper Wraps labels and an Operation into one small object.

## Simulator

class trueq.Simulator(backend=None)

A (noisy) quantum simulator for Circuit objects that can be used to compute the final state(), the total effective operator(), or simply run() them to sample() shots.

import trueq as tq

# initialize a simulator with no noise present
sim = tq.Simulator()

# initialize a simulator with depolarizing noise at rate 1%

# initialize a simulator with depolarizing noise at rate 1%, and unitary
# overration by 5%

# make some circuits and populate their results with random shots
circuits = tq.make_srb([0], [4, 60])
sim.run(circuits, n_shots=100)
circuits[0].results

Results({'0': 4, '1': 96})

Parameters

backend (SimulationBackend) – A backend to perform simulations with.

property backend

The backend of this simulator instance.

Type

SimulationBackend

get_dim(circuit)

Returns the subsystem dimension this simulator would use for the given circuit. This dimension is determined by first checking whether any of the noise_sources demand a specific dimension, returning it if a consistent one is found. Otherwise, the subsystem dimension of the given circuit is used. In the edge case that the circuit is empty and no noise source specifies a dimension, trueq.settings.get_dim() is used.

Parameters

circuit (Circuit) – A circuit to be simulated.

Return type

py:class:int

property noise_sources

The noise sources that have been added to this simulator. They are ordered in the same way that they are applied to any given cycle.

An instance of the noiseless simulator NoiseSource always appears at the end, which handles the simulation of any gates which other noise sources ignored.

Type

list

append_noise_source(noise_source)

Appends a noise source to the simulator. A noise source in an instance of some subclass of NoiseSource, which includes custom user subclasses.

The order in which noise sources are added is the order in which they will be applied (cycle by cycle) during simulation.

Parameters

noise_source (NoiseSource) – A source of noise to be added to the simulator.

Returns

This simulator instance.

Return type

Simulator

Raises

ValueError – If the dimension of the noise source conflicts with an existing dimension.

process(circuit, prep_policy, meas_policy, n_shots=None)

A low-level interface to this simulator where backend policies can be specified manually. See operator(), state(), sample(), and run() for wrappers of this function that specify the policies automatically and invoke the final type.

This method loops over cycles of the given circuit, and for each one loops over noise sources, which are in turn given the oportunity to call methods of the backend such as process_gate(). This method does not necessarily perform the quantum simulation as some backends may elect to use the processing time to introspect and store the instructions rather than simulate them. See value.

Parameters
• circuit (Circuit | Cycle | dict Iterable) – A circuit to simulate. This can also be any valid first argument to the Circuit constructor, such as a Cycle, dict, or iterable of these two.

• prep_policy (PrepPolicy) – A policy specifying how to process preparation instructions.

• meas_policy (MeasPolicy) – A policy specifying how to process measurement instructions.

• n_shots (None | int | inf) – If relevant, the number of shots to sample from measurements of the circuit. This can be a positive integer, or something equal to float("inf") to report the probability of every outcome.

Return type

Runner

operator(circuit)

Returns the unitary or superoperator that results from simulating the given circuit.

If this simulator contains only unitary errors and pure state preparations, a unitary will be simulated and returned, otherwise, a superoperator will be simulated and returned.

import trueq as tq

# make a circuit with two clock cycles
circuit = tq.Circuit([{(0,): tq.Gate.h}, {(0, 1): tq.Gate.cnot}])

# simulate the circuit to find the final unitary
u = tq.Simulator().operator(circuit)
tq.plot_mat(u.mat())

# if we add depolarizing noise we get a superoperator
print("Super operator shape: ", s.mat().shape)

Super operator shape:  (16, 16)


Note

By default, this method skips Prep and Meas operators.

Parameters

circuit (Circuit) – A circuit to find the final operator of.

Returns

The effective operator of the circuit after simulation.

Return type

OperatorTensor

Raises

NotImplementedError – If the last cycle in the circuit contains measurements on some qubits but not all.

run(circuits, n_shots=50, overwrite=None, max_workers=1)

Updates the results attribute of each given circuit by simulating its final quantum state and sampling shots from it.

import trueq as tq

# make a circuit with two clock cycles and a measurement round
circuit = tq.Circuit([{(0,): tq.Gate.h}, {(0, 1): tq.Gate.cnot}])
circuit.measure_all()

# initialize a simulator with no noise
sim = tq.Simulator()

# run the circuit on the simulator 100 times to populate the results
sim.run(circuit, n_shots=100)
print(circuit.results)

# instantiate a simulator with depolarizing and overrotation noise

# we can also use run to evaluate circuit collections generated by protocols
# like CB, SRB, IRB, and XRB on a simulator:
circuits = tq.make_srb([0], [5, 50, 100], 30)
sim.run(circuits, n_shots=100)
circuits.plot.raw()

Results({'00': 46, '11': 54})

Parameters
• circuits (Circuit | Iterable) – A single circuit or an iterable of circuits.

• n_shots (int | float("inf")) – The number of shots to sample. The final state of the circuit is simulated once and shots are drawn from the resulting probability distribution. Or, if this value is infinity-like (e.g. float("inf") or numpy.inf) then results are populated with the exact simulated probabilities.

• overwrite (bool | NoneType) – If False, a circuit that already has results will have new simulation results added to the old results. If True or None, old results will be erased and replaced with new results, though a warning will be raised in the latter case of None (default).

• max_workers (int | NoneType) – The maximum number of workers to use when parallelizing this simulation over circuits. A value of None defaults to the number of processors on the machine. This feature is not available on Windows, which defaults to serial simulation.

sample(circuit, n_shots=50)

Samples ditstrings from the final state of the simulated circuit. In contrast to run(), this method does not update the results of the circuit.

import trueq as tq

# make a circuit with two clock cycles and a measurement round
circuit = tq.Circuit([{(0,): tq.Gate.h}, {(0, 1): tq.Gate.cnot}])
circuit.measure_all()

# instantiate a simulator with depolarizing and overrotation noise

# sample 100 shots from the final state of the circuit
print(sim.sample(circuit, 100))

Results({'00': 52, '11': 48})

Parameters
• circuit (Circuit) – A single circuit.

• n_shots (int | float("inf")) – The number of shots to sample. The final state of the circuit is simulated once and shots are drawn from the resulting probability distribution. Or, if this value is infinity-like (e.g. float("inf") or numpy.inf) then results are populated with the exact simulated probabilities.

Return type

Results

state(circuit)

Returns the quantum state that results from simulating the given circuit.

If this simulator contains only unitary errors and pure state preparations, a pure state will be simulated and returned, otherwise, a density matrix will be simulated and returned. Unless this simulator contains a special preparation (see e.g. add_prep()), every qubit in the circuit will be prepared with the state $|0\rangle$.

import trueq as tq

# make a circuit with two clock cycles
circuit = tq.Circuit([{(0,): tq.Gate.h}, {(0, 1): tq.Gate.cnot}])

# simulate the circuit to find the final pure state
psi = tq.Simulator().state(circuit)
print("Pure state: ", psi.mat())

# if we add depolarizing noise we get a density matrix
print("Density matrix: ", rho.mat())

# we can get outcome probabilities from the state
print("Pure state probabilities:     ", psi.probabilities())
print("Density matrix probabilities: ", rho.probabilities())

# and we can convert them to Result objects
print("Pure state as results:     ", psi.probabilities().to_results())
print("Density matrix as results: ", rho.probabilities().to_results())

Pure state:  [0.70710678+0.j 0.        +0.j 0.        +0.j 0.70710678+0.j]
Density matrix:  [[0.4975    +0.j 0.        +0.j 0.        +0.j 0.48759975+0.j]
[0.        +0.j 0.0025    +0.j 0.00245025+0.j 0.        +0.j]
[0.        +0.j 0.00245025+0.j 0.0025    +0.j 0.        +0.j]
[0.48759975+0.j 0.        +0.j 0.        +0.j 0.4975    +0.j]]
Pure state probabilities:      Tensor(<[(2,), ()] on labels [(0, 1)]>)
Density matrix probabilities:  Tensor(<[(2,), ()] on labels [(0, 1)]>)
Pure state as results:      Results({'00': 0.5000000000000001, '11': 0.5000000000000001})
Density matrix as results:  Results({'00': 0.4975000000000001, '01': 0.0025000000000000005, '10': 0.0025000000000000005, '11': 0.4975000000000001})

Parameters

circuit (Circuit) – A circuit to find the final state of.

Return type

StateTensor

Raises

NotImplementedError – If the circuit contains measurements before the final cycle.

A noise source which inserts or replaces entire cycles during simulation. The replacement cycle can be specified as a cycle-like object, or a function which takes in a cycle and returns a cycle-like. A cycle-like is either a cycle or a dictionary mapping labels to operations. The operations contained in this dictionary can be Gates and /or Superops.

The relative location of cycle insertion or replacement is determined by cycle_offset; a value of -1 inserts the noisy cycle before the matched cycle, 0 inserts the cycle in place of the matched cycle, and +1 inserts the cycle after the matched cycle. These are the only allowed values. The match owned by this instance must be a child class of BaseCycleMatch. This restriction allows the noise source to match on past cycles and thus has the ability to simulate non-Markovian noise.

Some examples are provided below:

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

cyc_h = tq.Cycle({0: tq.Gate.h, 3: tq.Gate.h})
cyc_cx = tq.Cycle({(0, 1): tq.Gate.cx})
cyc_cz = tq.Cycle({(0, 1): tq.Gate.cz})

m_h = tqs.CycleMatch(cyc_h, lag=1)
m_cx = tqs.CycleMatch(cyc_cx)
m_cz = tqs.CycleMatch(cyc_cz)

# insert a small flip angle cycle before any CNOT cycles
circuit1 = tq.Circuit([cyc_h, cyc_cx])
small_flip = {(0, 1): tq.Gate.rx(10)}
sim1 = tq.Simulator()
sim1.sample(circuit1, n_shots=100).plot()

# replace any CZ cycle which follows a Hadamard cycle with a CNOT cycle
circuit2 = tq.Circuit([cyc_cz, cyc_h, cyc_cz])
m_past = tqs.AndCycleMatch(m_h, m_cz)
print(sim2.state(circuit2).mat())

# insert a cycle-like which does a small X rotation on qubit 0 and
# depolarizes qubit 1 after any CNOT cycle
superop = tqm.Superop.from_ptm(np.diag([1, 0.99, 0.99, 0.99]))
cyc_like = {0: tq.Gate.rx(9), 1: superop}
circuit3 = tq.Circuit([cyc_h, cyc_cx, cyc_h, cyc_cx])
mixed_state = sim3.state(circuit3)
assert mixed_state.is_mixed

# replace any CNOT cycle with a noisy version of it, using a function in
# place of a cycle-like
def two_qubit_error(cycle, all_labels):
cycle_like = {}

# loop through gates in given cycle and perturb them
pert = tq.Gate.rp("ZZ", 5)
for labels, gate in cycle.gates.items():
cycle_like[labels] = pert @ gate**1.05 @ pert.adj

# loop through qubits idling in this cycle and add random phase noise
idle_qubits = set(all_labels).difference(cycle.labels)
rng = tq.settings.get_rng()
for label in idle_qubits:
cycle_like[label] = tq.Gate.rz(5 * rng.standard_normal())

return cycle_like

sim4 = tq.Simulator()
pure_state = sim4.state(circuit3)

[0.5+0.j 0.5+0.j 0. +0.j 0. +0.j 0. +0.j 0. +0.j 0.5+0.j 0.5+0.j]

Parameters
• replacement (Cycle | dict | function) – A Cycle-like, or a function which takes a Cycle (and, optionally as a second argument, a tuple of all labels in the current circuit being simulated) and returns a Cycle-like, to be inserted when the noise source matches.

• match (BaseCycleMatch | NoneType) – A match object that specifies which cycles this noise source applies to.

• cycle_offset (int) – Whether to insert the replacement cycle before (-1), in place of (0), or after (+1) the cycles matched by this noise source.

Returns

This simulator instance so that add_*() calls can be chained.

Return type

Simulator

Raises

ValueError – If the match is not a cycle match instance, the cycle_offset is not -1, 0 or 1, or if replace_with generates a cycle with invalid dimensions.

Appends a noise source to this simulator that adds isotropic depolarizing noise to the simulation. By default, noise is added every time a gate is encountered, however, by specifying match, the user can design a noise profile which is gate- and/or qubit-dependent. See Match for more details.

Depolarizing noise is applied to every system being acted on by a gate in a given cycle. For example, if qubits (0,1) get a CNOT, and qubit 3 gets an X in a given cycle, then depolarizing noise is applied to only these three qubits, even if the preceeding cycle contained a gate which acted on qubit 2. This noise source works for any subsystem dimension.

This noise source only applies noise—it does not try to implement the cycle in question, but only adds noise to it.

By default, if a two-qubit gate is encountered, then the tensor product of two one-qubit depolarizing channels will be applied to the two qubits the gate acts on, see the documentation of depolarizing() and also the following example:

import trueq as tq
import trueq.simulation as tqs

circuit = tq.Circuit([{(0, 1): tq.Gate.cx, 2: tq.Gate.h}])

# make a simulator with local depolarizing noise with a depolarizing
# strength of p=0.02 applied to each single qubit a gate acts on
print(sim.sample(circuit, 1000))

# make a simulator where depolarizing noise acts only on locations where
# gates act on qubit 5
print(sim.sample(circuit, 1000))

Results({'000': 481, '001': 492, '010': 7, '011': 7, '110': 8, '111': 5})
Results({'000': 513, '001': 487})


By setting local=False, a global depolarizing channel is applied to all qubits that a matched gate acts on. That is, with respect to the formula above, we will have $D=2$ for single qubit gates, $D=4$ for two-qubit gates, $D=27$ for three-qutrit gates, and so on. This is shown in the following example.

import trueq as tq
import trueq.simulation as tqs

circuit = tq.Circuit([{(0, 1): tq.Gate(np.eye(4))}])

# make a simulator with depolarizing noise acting globally on each gate that
# is found
tq.math.Superop(sim.operator(circuit).mat()).plot_ptm()

Parameters
• p (float) – The depolarizing parameter, as defined in the equation above.

• local (bool) – Whether local depolarizing noise is applied to every qubit that is encountered on each matched gate, or whether global depolarizing noise is applied to all qubits encounterned on each matched gate.

• match (Match | NoneType) – A match object that specifies which operations or cycles this noise source applies to.

Returns

This simulator instance so that add_*() calls can be chained.

Return type

Simulator

Adds a noise source that replaces all matched Gates in the circuit with the returned gate or superoperator of the provided function fn. Each new gate-like object returned by fn must act on the same dimensional subsystems as the gate it replaces.

See below for an example of gate replacement noise, which overrotates native gates matching a specific name by a random value within $[-5, +5]$ degrees.

import trueq as tq
import trueq.simulation as tqs
import random

def rand_noise(gate):
# creates a small amount of random parameter noise. we use a match below
# to guarantee any provided gate has a parameter called "theta"
theta = gate.parameters["theta"]
return tq.Gate.rp("X", theta + random.uniform(-5, 5))

# this match will ensure that rand_noise only ever applies to native
# gates with the name "rx_noisy"
match = tqs.NativeMatch("rx_noisy")

# create and simulate a circuit with non-trivial noise application, where
# any native gate with name "rx_noisy" also has a parameter named "theta"
g = tq.NativeGate.from_generators("rx_noisy", "X", 90, params={"theta": 90})
circuit = tq.Circuit([{(0,): g}])

# The 90 degree X rotation has been overrotated by a random amount
sim.state(circuit).mat()

array([0.72658947+0.j        , 0.        -0.68707186j])


No caching is performed on the outputs of fn; consider memoizing your function with functools.lru_cache() to increase speed.

Parameters
• fn (function) – A function which takes in a gate and returns a Gate or Superop to replace it with when simulating.

• match (Match | NoneType) – A match object that specifies which operations or cycles this noise source applies to.

Returns

This simulator instance so that add_*() calls can be chained.

Return type

Simulator

Raises

ValueError – If the new gate does not match the dimensions of the gate that it is replacing.

A noise source which emulates the noise measured in a KNR experiment.

For each unique Cycle with results contained in fit_or_circuits, we extract the NormalEstimates for each gate-body within the cycle. For each gate-body, we construct the Superop which models the one-body noise, as measured by the KNR protocol.

In the example below, we instantiate a noisy simulator and estimate the noise seen by an example cycle via the KNR protocol. We then use the resulting fit to instantiate a new simulator which reconstructs the noise of the first simulator.

import trueq as tq
import trueq.simulation as tqs

cycle = tq.Cycle({0: tq.Gate.h, 1: tq.Gate.z, (2, 3): tq.Gate.cx})
knr_circuits = tq.make_knr(
cycle, n_random_cycles=[2, 6, 10], n_circuits=30, subsystems=1
)

sim.run(knr_circuits)

circuit = tq.Circuit(cycle)
circuit.measure_all()

knr_sim.sample(circuit, 100).plot()

sim.sample(circuit, 100).plot()


Running the KNR protocol on a cycle with explicitly defined identities will estimate the noise seen by idle qubits. For convenience, setting the flag ignore_identity to True will allow the reconstructed noise operator to match with cycles that do not contain explicit identity operators, so long as another cycle within the circuit being simulated contains operations which act on those qubits.

Similarly, KNR noise can be set to apply to only those cycles which are equal to the cycle estimated by the KNR protocol and have equal marker values, by setting ignore_marker to False. The matching occurs via an internally instantiated CycleMatch.

Parameters
• fit_or_circuits (EstimateCollection | CircuitCollection) – An estimate collection or a set of circuits which contain estimates of, or circuits for, the KNR protocol.

• exclusive (bool) – If True, all gates within the targeted cycles will be marked as no_more_noise during simulation. This is False by default.

• ignore_identity (bool) – Whether or not to match this noise to cycles containing identical operations, ignoring any identity operators. This is False by default.

• ignore_marker (bool) – If True, match this noise to cycles which contain the same operations but have differet markers. This is True by default.

Returns

This simulator instance so that add_*() calls can be chained.

Return type

Simulator

Raises

ValueError – If a cycle in the fit_or_circuits appears in the fit multiple times, or if the fit is empty.

Appends a noise source to this simulator that adds Kraus noise to the simulation. The noise is added to each gate matched by match, so long as the Kraus operators’ dimensions allow it to be applied to the matched gates. Single-qubit Kraus noise will apply to all subsystems of any matched multi-qubit gate, whereas multi-qubit Kraus noise will only be applied to gates of the same dimension; this behaviour can be changed by using a match which only matches single-qubit gates such as NQubitMatch. The noise map is defined by

$\mathcal{K}(\rho) = \sum_{i} K_i \rho K_i^\dagger$

Note that this applies only noise—it does not try to implement a given cycle in question, but only adds noise to it.

In every cycle, Kraus noise will only affect non-idling qubits. For example, if in a given cycle qubits (0, 1) get a CNOT and no gate acts on qubit 2, then Kraus noise is only applied to qubits (0, 1).

import trueq as tq
import trueq.simulation as tqs
import numpy as np

circuit = tq.Circuit([{(0, 1): tq.Gate.cx, 2: tq.Gate.h}])

# define qubit dephasing kraus operators
kraus_ops = [np.sqrt(0.99) * np.eye(2), np.sqrt(0.01) * np.diag([1, -1])]

# initialize a simulator in which qubit dephasing noise acts on every
# location where gates are applied
print(sim.sample(circuit, 1000))

# define a set of Kraus operators acting on single-qubit gates, and set
# of Kraus operators to explicitly act on two-qubit gates
kraus_ops1 = [np.sqrt(0.999) * np.eye(2), np.sqrt(0.001) * np.diag([1, -1])]
kraus_ops2 = [
np.sqrt(0.99) * np.eye(4),
np.sqrt(0.01) * np.fliplr(np.diag([1, 1, 1, 1])),
]

# initialize a simulator with the noise defined above acting at
# every location where a gate acts
print(sim.sample(circuit, 1000))

# initialize a simulator which applies the above noise when a gate acts
# on qubit 0
noisy_label = tqs.LabelMatch(0, strict=False)
print(sim.sample(circuit, 1000))

# initialize a simulator which applies the above noise when an X gate acts
# on qubit 2 or 3
# no 2 qubit gates will be matched under these conditions, no need to add
# the two-qubit kraus channel to the simulator
sim = tq.Simulator()
noise_match = tqs.LabelMatch((2, 3)) & tqs.GateMatch(tq.Gate.x)
print(sim.sample(circuit, 1000))

Results({'000': 499, '001': 501})
Results({'000': 472, '100': 5, '001': 520, '101': 3})
Results({'000': 500, '001': 488, '100': 6, '101': 6})
Results({'000': 518, '001': 482})

Parameters
• kraus_ops (iterable) – An iterable of square matrices, each of which must be the same size, and act on subsystems of dimension dim.

• match (Match | NoneType) – A match object that specifies which operations or cycles this noise source applies to.

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

Returns

This simulator instance so that add_*() calls can be chained.

Return type

Simulator

Raises

ValueError – If kraus_ops has the wrong dimension or is not a list of square matrices.

Appends a noise source to this simulator that performs gate over/underrotation. This is done by propagating by the matrix power $U_{err}=U^{(1+\epsilon)}$ instead of the ideal gate $U$.

import trueq as tq
import trueq.simulation as tqs

circuit = tq.Circuit({(0, 1): tq.Gate.cx, 2: tq.Gate.h})

# initialize a simulator in which every single-qubit gate is overrotated by
# epsilon=0.01
print(sim.sample(circuit, 1000))

# initialize a simulator in which every single-qubit gate is overrotated by
# epsilon=0.02 and every multi-qubit gate by epsilon=0.04
print(sim.sample(circuit, 1000))

# initialize a simulator in which overrotations are applied whenever a gate
# acts on any qubits in {0, 2}
sim = tq.Simulator()
noisy_labels = tqs.LabelMatch((0, 2))
print(sim.sample(circuit, 1000))

Results({'000': 505, '001': 495})
Results({'000': 477, '001': 523})
Results({'000': 503, '001': 497})

Parameters
• single_sys (float) – The amount of over/underrotation to apply to gates that act on a single subsystem (ie. single-qubit). A value of 0 corresponds to no overrotation, a negative value corresponds to underrotation, and a positive value corresponds to overrotation.

• multi_sys (float) – The amount of over/underrotation to apply to gates that act on a multiple subsystems (ie. multi-qubit). A value of 0 corresponds to no overrotation, a negative value corresponds to underrotation, and a positive value corresponds to overrotation.

• match (Match | NoneType) – A match object that specifies which operations or cycles this noise source applies to.

Returns

This simulator instance, so that add_*() calls can be chained.

Return type

Simulator

add_povm() was deprecated in version 2.11.4 and will be removed no earlier than version 2.13.0. Please use add_readout_error() instead, which now supports mixed representations, including POVMs and classification error matrices.

Appends measurement noise in the form of a positive operator-valued measurement (POVM). This method allows two types of POVMs: “true” POVMs, and “classification” POVMs, described below.

A true POVM applies to density matrices, where the probability of a particular outcome $x$ is given by $p_x=\operatorname{Tr}(\rho E_x)$. Here, $x$ is typically a bitstring and $E_x$ is the corresponding positive-semidefinite measurement operator. For measurement noise that is uncorrelated between subsystems, $E_x$ will be a tensor product $E_x=E_{x_1}\otimes \cdots \otimes E_{x_n}$. This is entered as a Tensor with shape ((k,), (d, d)) where k is the number of outcomes per qubit, and d is the Hilbert state dimension, which must match the simulation. A sum over the k axis must produce the identity matrix for a probability preserving POVM. Note that there is no requirement that k == d, for example, k = 3; d = 2 represents a qubit simulation being classified into three possible results.

This method additionally allows for what we call classification POVMs. This is a less general form of measurement noise, which can be expressed as inner products against the density matrix as described above, but which are more efficiently described as a transformation of the ideal measurement probability vector in the computational basis. In this case, we have $p=Cq$ where $p$ is the vector of outcome probabilities $p_x$, $C$ is a classification error matrix, and $q$ is the vector of ideal measurement probabilities of $\rho$ in the computational basis. For measurement noise that is uncorrelated between subsystems, $C=C_1\otimes\cdots\otimes C_n$. This is entered as a Tensor with shape ((k,), (d,)) where k is the number of outcomes per qubit, and d is the Hilbert state dimension, which must match the simulation. Each column must sum to 1 for a probability preserving measurement.

Parameters
• povm (Tensor) – A tensor describing a POVM.

• match (Match | NoneType) – A match object that specifies which operations or cycles this noise source applies to.

Returns

This simulator instance so that add_*() calls can be chained.

Return type

Simulator

Appends a state preparation description to this simulator that chooses which state to prepare whenever a preparation instruction is encountered in a circuit.

A preparation state can be specified in any of the following formats:

• A number between 0 and 1, representing a bitflip error probability of the ideal preparation $|0\rangle$.

• A length-$d$ vector, representing the diagonal of a density matrix.

• A length-$d$ vector, representing a pure state.

• A $d\times d$ matrix, representing a density matrix.

If only one of the above is provided, it is used every time a Prep object is encountered. One can also specify that different subsystems get different errors by providing a dictionary mapping whose values are a combination of the above formats. See the examples below.

import trueq as tq

circuit = tq.Circuit({i: tq.Prep() for i in range(3)})

# add 5% bitflip error to any qubit
print(sim.sample(circuit, float("inf")))

# add 2% bitflip error by default, but 5% for qubits 1 and 2
sim = tq.Simulator().add_prep(0.02, {2: 0.05, 1: 0.05})
print(sim.sample(circuit, float("inf")))

# add specific density matrix to qubit 1, but 1% bitflip error by default
sim = tq.Simulator().add_prep(0.01, {1: [[0.8, 0.1], [0.1, 0.2]]})
print(sim.sample(circuit, float("inf")))

Results({'000': 0.8573749999999999, '001': 0.045125, '010': 0.045125, '011': 0.0023750000000000004, '100': 0.045125, '101': 0.0023750000000000004, '110': 0.0023750000000000004, '111': 0.00012500000000000003})
Results({'000': 0.88445, '100': 0.01805, '001': 0.04655, '101': 0.00095, '010': 0.04655, '110': 0.00095, '011': 0.0024500000000000004, '111': 5.000000000000001e-05})
Results({'000': 0.78408, '001': 0.00792, '100': 0.00792, '101': 8e-05, '010': 0.19602, '011': 0.00198, '110': 0.00198, '111': 2e-05})


For state simulation, if a Prep operator is encountered on a subsystem that has previously been initialized, then the register is traced out and the state is replaced with a new one.

Parameters
• state (float | numpy.ndarray-like | dict) – A description of quantum states to initialize with.

• match (Match | NoneType) – A match object that specifies which operations or cycles this noise source applies to.

Returns

This simulator instance so that add_*() calls can be chained.

Return type

Simulator

Raises

ValueError – If inconsistent dimensions are provided.

Appends measurement classification error to this simulator, where errors are specified by confusion matrices estimated by the readout calibration protocol (RCAL).

fit_or_circuits should be either a circuit collection with results from the RCAL protocol, or an estimate collection containing the fit for an RCAL estimate. If no RCAL fitting information is found, or if there are multiple fits present in fit_or_circuits, an error will be raised.

import trueq as tq
import numpy as np

# generate RCAL circuits to measure the readout errors on qubits [0, 1]
circuits = tq.make_rcal([0, 1])

# initialize a mock device with a 20% readout error on every qubit

# run the circuits on the mock device to populate their results and
sim.run(circuits, n_shots=1000)
fit = circuits.fit()

# instantiate a simulator with the RCAL estimate and report the results
# of running an example circuit with the new simulator
circuit = tq.Circuit({(0, 1): tq.Gate.x}).measure_all()
sim.run(circuit, n_shots=np.inf)
circuit.results.plot()

Parameters

fit_or_circuits (EstimateCollection | CircuitCollection) – An estimate collection or a set of circuits which contain estimates of or circuits for the KNR protocol.

Raises

ValueError – If fit_or_circuits does not contain exactly one RCAL estimate.

Returns

This simulator instance so that add_*() calls can be chained.

Return type

Simulator

Appends measurement classification error and/or positive operator valued measurement (POVM) error to this simulator.

For a qubit, a readout error of $0.1$ will result in the confusion matrix $\left(\begin{smallmatrix}0.9&0.1\\0.1&0.9\end{smallmatrix}\right)$, and the readout error of $(0.1, 0.02)$ in a confusion matrix $\left(\begin{smallmatrix}0.9&0.0.2\\0.1&0.98\end{smallmatrix}\right)$. See trueq.math.general.make_confusion_matrix() for more details about the allowed input formats for constructing confusion matrices.

A POVM is specified as a 3D array $E_{ijk}$ where each matrix $E_i$ is positive semi-definite and encodes the probability of observing the $i^{th}$ outcome. If $\rho$ is the pre-measurement density matrix during simulation, then $\operatorname{Tr}(\rho E_i)$ is the probability of outcome $i$.

Errors can differ on individual qubits by singling out their labels in the second argument, see the example below. Similarly, correlated errors can be added by specifying larger confusion matrices and the qubits they act on.

The number of outcomes into which quantum states are classified does not need to match the subsystem dimension; confusion matrices can be rectangular so long as columns sum to unity.

The subsystem dimension and number of outcomes are inferred automatically from the given default_error and errors. When this is not possible, then the value of get_dim() is used.

import trueq as tq

# make a test circuit that measures 4 qubits
circuit = tq.Circuit([{range(4): tq.Meas()}])

# all qubits get 1% readout error
print(sim.sample(circuit, 200))

# all qubits get 1% readout error, except qubit 1 gets 5% readout error
print(sim.sample(circuit, 200))

# all qubits get 1% readout error, except qubit 1 get 5% readout error,
# and qubit 3 get asymmetric readout error of 1% on |0> and 7% on |1>
print(sim.sample(circuit, 200))

# we specify correlated measurement errors on qubits (3, 2) with a 4x4
# confusion matrix (row/column order is 00, 01, 10, 11). all other qubits
# get 1% symmetric classification error.
confusion = [
[0.81, 0.72, 0.72, 0.64],
[0.09, 0.18, 0.08, 0.16],
[0.06, 0.08, 0.18, 0.16],
[0.04, 0.02, 0.02, 0.04],
]
print(sim.sample(circuit, 200))

# here we add asymmetric qutrit classification error. the default error is
# given as 0.02, so that for each i=0,1,2, a measurement |i> is
# misclassified as |i+1> or |i+2> with probability 1% each. however, on
# qubit 2, we have specified that |0> is reported as a '0' 99% of the time,
# |1> is reported as a '1' 92% of the time, and |2> is reported as a '2' 80%
# of the time. off-diagonals are the probabilities of misclassification
# into specific outcomes corresponding to the row they are located in. we
# add this error specifically to qutrit 2, the rest of the qutrits have
# ideal measurements.
confusion = [[0.99, 0.03, 0.02], [0.005, 0.92, 0.18], [0.005, 0.05, 0.8]]
print(sim.sample(circuit, 200))

# classify qutrit measurements into binary outcomes. this situation arises,
# for example, in superconducting qubits when there is leakage into the
# third level of the anharmonic oscillator, but readout always reports a '0'
# or a '1'. we use a rectangular confusion matrix where |2> is classified as
# a '0' 70% of the time, and as a '1' 30% of the time.
confusion = [[0.99, 0.08, 0.7], [0.01, 0.92, 0.3]]
print(sim.sample(circuit, 200))

Results({'0000': 197, '0010': 1, '0100': 1, '1000': 1})
Results({'0000': 186, '0001': 4, '0010': 3, '1000': 1, '0100': 6})
Results({'0000': 189, '1000': 2, '0001': 3, '0100': 6})
Results({'0000': 144, '0100': 3, '1000': 3, '0010': 21, '0110': 1, '0001': 15, '0011': 13})
Results({'0000': 192, '0100': 3, '0200': 1, '1000': 2, '2000': 2}, dim=3)
Results({'0000': 194, '0001': 1, '0010': 1, '0100': 3, '1000': 1})

Parameters
• default_error (NoneType | numpy.ndarray-like | float) – Default readout error to be applied to all qubits that don’t receive special errors in errors. See trueq.math.general.make_confusion_matrix() for the allowed input formats.

• errors (dict) – A dictionary mapping qubit labels or tuples of qubit labels to any of the input formats allowed by default_error. Targeting multiple qubit labels in an entry allows for correlated readout error, though no qubit labels may overlap.

• match (Match | NoneType) – A match object that specifies which operations or cycles this noise source applies to.

Returns

This simulator instance so that add_*() calls can be chained.

Return type

Simulator

Raises

ValueError – For any inconsistencies in the input arguments.

add_relaxation(t1, t2, t_single, t_multi, excited_pop=0, match=None)

Appends a noise source to this simulator that adds $T_1$ and $T_2$ relaxation noise to the simulation. Specifically, for every cycle in a circuit, this noise source adds error to every qubit in the state defined by the Choi matrix:

$\begin{split}C = \begin{pmatrix} 1-p(1-e^{-t/T_1}) & 0 & 0 & e^{-t/T_2} \\ 0 & (1-p)e^{-t/T_1} & 0 & 0 \\ 0 & 0 & pe^{-t/T_1} & 0 \\ e^{-t/T_2} & 0 & 0 & 1-(1-p)e^{-t/T_1} \end{pmatrix}\end{split}$

where $t$ is the time of the longest gate in the cycle (i.e. either t_single or t_multi for an entire cycle).

import trueq

circuit = tq.Circuit([{(0, 1): tq.Gate.cx, 2: tq.Gate.h}])

# make a simulator with relaxation noise with t1=100e-6, t2=50e-6,
# single-qubit gate time 25e-9, two-qubit gate time 100e-9 and an excited
# equilibrium population 0.01.
sim = tq.Simulator().add_relaxation(100e-6, 50e-6, 25e-9, 100e-9, 0.01)
print(sim.sample(circuit, 1000))

# make a simulator with relaxation noise with t1=10us, single-qubit gate
# time 25ns and two-qubit gate time 100ns. t2=5us on all qubits except qubit
# 0 which has t2=2.5us.
t2 = {None: 5e-6, 0: 2.5e-6}
sim = tq.Simulator().add_relaxation(10e-6, t2, 25e-9, 100e-9)
# plot the final state
tq.plot_mat(sim.state(circuit).mat())

Results({'000': 515, '001': 485})

Parameters
• t1 (float | dict) – The $T_1$ time, i.e. the characteristic time of relaxation. These can vary from qubit to qubit, see the example above.

• t2 (float | dict) – The $T_2$ time, i.e. the characteristic time of dephasing. These can vary from qubit to qubit, see the example above.

• t_single (float) – The time it takes to perform a single-qubit gate.

• t_multi (float) – The time it takes to perform a multi-qubit gate.

• excited_pop (float | dict) – The excited state population at equilibrium.

• match (Match | NoneType) – A match object that specifies which operations or cycles this noise source applies to.

Returns

This simulator instance so that add_*() calls can be chained.

Return type

Simulator

Raises

ValueError – If invalid $T_1$ and $T_2$ arguments are provided.

Appends a noise source to this simulator that introduces stochastic Pauli noise. This is done by applying superoperators with the Kraus representation

$\Epsilon(\rho) = \sum{i}p_i P_i \rho P_i^\dagger$

for some probability vector $p$, and where $P_i$ are n-qubit Pauli operators (or n-qudit Weyl operators). Therefore, this method is a convenience wrapper for add_superop() restricted to a certain family of superoperators. Note that this channel only applies noise—it does not try to implement the cycle in question, but only adds noise to it.

Like add_superop(), in the single-qubit case, this noise is applied to every system being acted on by a gate in a given cycle. For example, if qubits (0,1) get a CNOT, and qubit 3 gets an X in a given cycle, then stochastic Pauli noise is applied to only these three qubits, even if the preceeding cycle acted on qubit 2. The match argument can be used to specify under which conditions the stochastic Pauli noise should be applied, see Match for more information.

import numpy as np
import trueq as tq

circuit = tq.Circuit([{(0, 1): tq.Gate.cx, 2: tq.Gate.x, 3: tq.Gate.h}])

# initialize a simulator in which every location where a gate acts undergoes
# Pauli noise with px=0.01 and py=0.04
print(sim.sample(circuit, 1000))

# initialize a simulator where every X gate undergoes Pauli noise with
# py=pz=0 and px=0.04
noisy_gate = tqs.GateMatch(tq.Gate.x)
print(sim.sample(circuit, 1000))

# initialize a simulator with stochastic XX and YX noise on two-qubit gates
print(sim.sample(circuit, 1000))

# initialize a simulator with stochastic W11 and W22 qutrit noise
circuit = tq.Circuit({0: tq.Gate(np.diag([1, 1, -1]))})
print(sim.sample(circuit, 1000))

Results({'0000': 18, '0001': 23, '0010': 440, '0011': 419, '0100': 1, '0101': 4, '0110': 22, '0111': 21, '1010': 1, '1011': 2, '1100': 1, '1101': 2, '1110': 24, '1111': 22})
Results({'0000': 25, '0001': 18, '0010': 472, '0011': 485})
Results({'0010': 466, '0011': 461, '1010': 34, '1011': 39})

Results({'0': 966, '1': 10, '2': 24}, dim=3)


Note

The methods add_stochastic_pauli() and add_stochastic_weyl() are identical. They both exist to make searching the documentation easier.

Parameters
• *args – (Deprecated, please only keyword arguments.) Probabilities for qubit X, Y, and Z errors respectively.

• match (Match | NoneType) – A match object that specifies which operations or cycles this noise source applies to.

• **probabilities – Keyword names describe which Pauli or Weyl error to introduce, and their value is the corresponding probability. The names can be Pauli or Weyl strings (see the constructor of WeylBase for allowed formats), or px, py, and pz for legacy support.

Returns

This simulator instance so that add_*() calls can be chained.

Return type

Simulator

Raises
• ValueError – If the probabilities do not sum to one, or any are negative.

• ValueError – If the same Pauli/Weyl is specified in multiple ways.

• ValueError – If different provided Pauli/Weyl strings represent different numbers of subsystems.

Appends a noise source to this simulator that introduces stochastic Pauli noise. This is done by applying superoperators with the Kraus representation

$\Epsilon(\rho) = \sum{i}p_i P_i \rho P_i^\dagger$

for some probability vector $p$, and where $P_i$ are n-qubit Pauli operators (or n-qudit Weyl operators). Therefore, this method is a convenience wrapper for add_superop() restricted to a certain family of superoperators. Note that this channel only applies noise—it does not try to implement the cycle in question, but only adds noise to it.

Like add_superop(), in the single-qubit case, this noise is applied to every system being acted on by a gate in a given cycle. For example, if qubits (0,1) get a CNOT, and qubit 3 gets an X in a given cycle, then stochastic Pauli noise is applied to only these three qubits, even if the preceeding cycle acted on qubit 2. The match argument can be used to specify under which conditions the stochastic Pauli noise should be applied, see Match for more information.

import numpy as np
import trueq as tq

circuit = tq.Circuit([{(0, 1): tq.Gate.cx, 2: tq.Gate.x, 3: tq.Gate.h}])

# initialize a simulator in which every location where a gate acts undergoes
# Pauli noise with px=0.01 and py=0.04
print(sim.sample(circuit, 1000))

# initialize a simulator where every X gate undergoes Pauli noise with
# py=pz=0 and px=0.04
noisy_gate = tqs.GateMatch(tq.Gate.x)
print(sim.sample(circuit, 1000))

# initialize a simulator with stochastic XX and YX noise on two-qubit gates
print(sim.sample(circuit, 1000))

# initialize a simulator with stochastic W11 and W22 qutrit noise
circuit = tq.Circuit({0: tq.Gate(np.diag([1, 1, -1]))})
print(sim.sample(circuit, 1000))

Results({'0000': 25, '0001': 33, '0010': 430, '0011': 433, '0100': 1, '0101': 2, '0110': 15, '0111': 15, '1100': 1, '1110': 24, '1111': 21})
Results({'0000': 21, '0001': 21, '0010': 472, '0011': 486})
Results({'0010': 453, '0011': 484, '1010': 28, '1011': 35})
Results({'0': 970, '1': 11, '2': 19}, dim=3)


Note

The methods add_stochastic_pauli() and add_stochastic_weyl() are identical. They both exist to make searching the documentation easier.

Parameters
• *args – (Deprecated, please only keyword arguments.) Probabilities for qubit X, Y, and Z errors respectively.

• match (Match | NoneType) – A match object that specifies which operations or cycles this noise source applies to.

• **probabilities – Keyword names describe which Pauli or Weyl error to introduce, and their value is the corresponding probability. The names can be Pauli or Weyl strings (see the constructor of WeylBase for allowed formats), or px, py, and pz for legacy support.

Returns

This simulator instance so that add_*() calls can be chained.

Return type

Simulator

Raises
• ValueError – If the probabilities do not sum to one, or any are negative.

• ValueError – If the same Pauli/Weyl is specified in multiple ways.

• ValueError – If different provided Pauli/Weyl strings represent different numbers of subsystems.

Appends a noise source to this simulator that adds generic superoperator noise to the simulation. The noise is added to each gate matched by match with an appropriate size. Specifically, single-qubit noise will apply to all subsystems of any matched multi-qubit gate, whereas multi-qubit noise will only be applied to gates of exactly the same dimension; this behaviour can be changed by using a match which only matches single-qubit gates such as NQubitMatch.

Note that this applies only noise—it does not try to implement a given cycle in question, but only adds noise to it.

In every cycle, noise will only affect non-idling qubits. For example, if in a given cycle qubits (0, 1) get a CNOT and no gate acts on qubit 2, then noise is only applied to qubits (0, 1).

import trueq as tq
import trueq.simulation as tqs
import numpy as np

circuit = tq.Circuit([{(0, 1): tq.Gate.cx, 2: tq.Gate.h}])

# choose a random superoperator with a process infidelity of 0.02
superop = tq.math.Superop.random_constrained_cptp(2, 0.02, 0.4)

# initialize a simulator in which qubit noise acts on every
# location where gates are applied
print(sim.sample(circuit, 1000))

# initialize a simulator where single qubits get one superoperator,
# and two-qubit gates get another
superop1 = tq.math.Superop.random_constrained_cptp(2, 0.01, 0.5)
superop2 = tq.math.Superop.random_constrained_cptp(4, 0.04, 0.5)
print(sim.sample(circuit, 1000))

Results({'000': 504, '001': 469, '010': 5, '011': 8, '110': 8, '111': 6})
Results({'000': 393, '010': 7, '100': 4, '110': 5, '001': 571, '011': 8, '101': 7, '111': 5})

Parameters
• superop – The superoperator noise.

• match (Match | NoneType) – A match object that specifies which operations or cycles this noise source applies to.

Returns

This simulator instance so that add_*() calls can be chained.

Return type

Simulator

dressed_noise(cycle, n_randomizations=None, twirl='P', compiler=None, mix_orbits=True, probabilities=False)

Computes the dressed noise of the given cycle. The dressed noise is defined as

$\tilde{\mathcal{E}} = \mathbb{E}[ \mathcal{T}^\dagger \mathcal{C}^\dagger \tilde{\mathcal{C}} \tilde{\mathcal{T}} ]$

where $\mathcal{C}$ is the cycle of interest, $\tilde{\mathcal{C}}$ its noisy simulation, $\mathcal{T}$ an element of the twirling group, $\tilde{\mathcal{T}}$ its noisy simulation, and where the expectation is taken uniformly over the twirling group.

The expectation value is taken in one of two ways. If n_randomizations=None, twirling operations are assumed to be perfect so that the expectation value is a simple function of the diagonal elements of the superoperator $\mathcal{C}^\dagger \tilde{\mathcal{C}}$ in the Pauli vectorization basis. Otherwise, the expectation is computed using n_randomizations Monte-Carlo samples from the twirling group, each of which is simulated with this simulator.

Additionally, if mix_orbits=True, error mixing via the orbit of the provided cycle is included. This mixing happens in practice during protocols like CB whose circuits consist of the cycle of interest repeated many times interleaved with samples of the twirling group. For instance, if a cycle contains a Hadamard, the effect of multiple dressed Hadamards back to back under Pauli twirling will result in the $Z$ and $X$ errors being mixed together. In particular, given the dressed noise $\tilde{\mathcal{E}}$ defined above, the orbit-mixed error returned by this method is

$P[(\mathcal{C}\tilde{\mathcal{E}})^o]^{\frac{1}{o}}$

where $o$ is such that $\mathcal{C}^o \propto I$.

Parameters
• cycle (Cycle) – The cycle to find the dressed noise of.

• n_randomizations (int | NoneType) – The number of random twirls to average over. If None, the twirling group is assumed to have a perfect implementation.

• twirl (Twirl | str) – The twirling group to use.

• compiler (Compiler) – A compiler to run all cycles through before simulating them.

• mix_orbits (bool) – Whether to mix the dressed noise under the cycle.

• probabilities (bool) – Whether to return the Pauli/Weyl probabilities of the noise channel as a trueq.math.tensor.Tensor instead of the channel itself.

Return type
predict_cb(cycle, labels=None, n_randomizations=50, targeted_errors=None, twirl='P', compiler=None)

Predicts the estimates that would be generated by simulating make_cb() with this simulator.

import trueq as tq

sim.predict_cb({0: tq.Gate.id, (1, 2): tq.Gate.cnot})

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".
 CB Cycle Benchmarking Paulis (0,) : Gate.id (1, 2) : Gate.cx Key: cycles: (Cycle((0,): Gate.id, (1, 2): Gate.cx),) labels: (0, 1, 2) name: protocol: CB twirl: Paulis on [0, 1, 2] ${e}_{F}$ The probability of an error acting on the specified labels during a dressed cycle of interest. 6.7e-02 (0.0e+00) 0.0665238362366557, 0.0 ${e}_{III}$ The probability of the subscripted error acting on the specified labels. 9.3e-01 (0.0e+00) 0.9334761637633443, 0.0 ${p}_{ZZZ}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 8.9e-01 (0.0e+00) 0.8942540810859934, 0.0 ${p}_{ZZY}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 8.9e-01 (0.0e+00) 0.8948220734257166, 0.0 ${p}_{ZZX}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.1e-01 (0.0e+00) 0.9122239749756353, 0.0 ${p}_{ZZI}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.2e-01 (0.0e+00) 0.915109257535822, 0.0 ${p}_{ZYZ}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 8.9e-01 (0.0e+00) 0.8934147343952171, 0.0 ${p}_{ZYY}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 8.9e-01 (0.0e+00) 0.8934163940424799, 0.0 ${p}_{ZYX}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.1e-01 (0.0e+00) 0.9145301936081857, 0.0 ${p}_{ZYI}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.1e-01 (0.0e+00) 0.9145301936081857, 0.0 ${p}_{ZXZ}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 8.9e-01 (0.0e+00) 0.8934163940424802, 0.0 ${p}_{ZXY}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 8.9e-01 (0.0e+00) 0.8934147343952172, 0.0 ${p}_{ZXX}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.5e-01 (0.0e+00) 0.9521004530202912, 0.0 ${p}_{ZXI}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.5e-01 (0.0e+00) 0.952100453020291, 0.0 ${p}_{ZIZ}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 8.9e-01 (0.0e+00) 0.8942540810859934, 0.0 ${p}_{ZIY}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 8.9e-01 (0.0e+00) 0.8948220734257164, 0.0 ${p}_{ZIX}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.5e-01 (0.0e+00) 0.9538981725492792, 0.0 ${p}_{ZII}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.6e-01 (0.0e+00) 0.9569164012055114, 0.0 ${p}_{YZZ}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 8.9e-01 (0.0e+00) 0.8938294532281422, 0.0 ${p}_{YZY}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 8.9e-01 (0.0e+00) 0.8943971758622239, 0.0 ${p}_{YZX}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.1e-01 (0.0e+00) 0.9117908142883461, 0.0 ${p}_{YZI}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.1e-01 (0.0e+00) 0.914674726800155, 0.0 ${p}_{YYZ}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 8.9e-01 (0.0e+00) 0.8929905050929832, 0.0 ${p}_{YYY}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 8.9e-01 (0.0e+00) 0.8929921639521786, 0.0 ${p}_{YYX}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.1e-01 (0.0e+00) 0.9140959378353963, 0.0 ${p}_{YYI}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.1e-01 (0.0e+00) 0.9140959378353962, 0.0 ${p}_{YXZ}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 8.9e-01 (0.0e+00) 0.8929921639521787, 0.0 ${p}_{YXY}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 8.9e-01 (0.0e+00) 0.8929905050929832, 0.0 ${p}_{YXX}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.5e-01 (0.0e+00) 0.9516483573750196, 0.0 ${p}_{YXI}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.5e-01 (0.0e+00) 0.9516483573750194, 0.0 ${p}_{YIZ}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 8.9e-01 (0.0e+00) 0.8938294532281421, 0.0 ${p}_{YIY}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 8.9e-01 (0.0e+00) 0.8943971758622238, 0.0 ${p}_{YIX}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.5e-01 (0.0e+00) 0.953445223274364, 0.0 ${p}_{YII}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.6e-01 (0.0e+00) 0.9564620187540573, 0.0 ${p}_{XZZ}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.3e-01 (0.0e+00) 0.9308319406991108, 0.0 ${p}_{XZY}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.3e-01 (0.0e+00) 0.9314231657469682, 0.0 ${p}_{XZX}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.5e-01 (0.0e+00) 0.9495368608747494, 0.0 ${p}_{XZI}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.5e-01 (0.0e+00) 0.9525401607441805, 0.0 ${p}_{XYZ}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.3e-01 (0.0e+00) 0.9299582620370626, 0.0 ${p}_{XYY}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.3e-01 (0.0e+00) 0.9299599895692205, 0.0 ${p}_{XYX}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.5e-01 (0.0e+00) 0.9519374112449603, 0.0 ${p}_{XYI}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.5e-01 (0.0e+00) 0.9519374112449601, 0.0 ${p}_{XXZ}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.3e-01 (0.0e+00) 0.9299599895692207, 0.0 ${p}_{XXY}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.3e-01 (0.0e+00) 0.9299582620370627, 0.0 ${p}_{XXX}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.9e-01 (0.0e+00) 0.9910444147474425, 0.0 ${p}_{XXI}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.9e-01 (0.0e+00) 0.9910444147474423, 0.0 ${p}_{XIZ}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.3e-01 (0.0e+00) 0.9308319406991106, 0.0 ${p}_{XIY}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.3e-01 (0.0e+00) 0.9314231657469678, 0.0 ${p}_{XIX}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.9e-01 (0.0e+00) 0.9929156667700982, 0.0 ${p}_{XII}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 1.0e+00 (0.0e+00) 0.9960573506572352, 0.0 ${p}_{IZZ}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.3e-01 (0.0e+00) 0.9345164112135812, 0.0 ${p}_{IZY}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.4e-01 (0.0e+00) 0.9351099764811505, 0.0 ${p}_{IZX}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.5e-01 (0.0e+00) 0.9532953702396817, 0.0 ${p}_{IZI}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.6e-01 (0.0e+00) 0.9563105579369054, 0.0 ${p}_{IYZ}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.3e-01 (0.0e+00) 0.9336392743082896, 0.0 ${p}_{IYY}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.3e-01 (0.0e+00) 0.9336410086784609, 0.0 ${p}_{IYX}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.6e-01 (0.0e+00) 0.9557054226012519, 0.0 ${p}_{IYI}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.6e-01 (0.0e+00) 0.9557054226012517, 0.0 ${p}_{IXZ}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.3e-01 (0.0e+00) 0.9336410086784607, 0.0 ${p}_{IXY}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.3e-01 (0.0e+00) 0.9336392743082893, 0.0 ${p}_{IXX}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.9e-01 (0.0e+00) 0.9949672216097903, 0.0 ${p}_{IXI}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.9e-01 (0.0e+00) 0.9949672216097902, 0.0 ${p}_{IIZ}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.3e-01 (0.0e+00) 0.9345164112135809, 0.0 ${p}_{IIY}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 9.4e-01 (0.0e+00) 0.9351099764811504, 0.0 ${p}_{IIX}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 1.0e+00 (0.0e+00) 0.9968458805257869, 0.0 ${p}_{III}$ Decay parameter of the exponential decay $Ap^m$ for the given Pauli term. 1.0e+00 (0.0e+00) 0.999999999999996, 0.0
Parameters
• cycle (dict | Cycle) – The cycle to make the prediction for.

• labels (Iterable) – A list of which sets of system labels are to be twirled together in each circuit, e.g. [3, [1, 2], 4].

• n_randomizations (int | NoneType) – The number of random twirls to average over. If None, the twirling group is assumed to have a perfect implementation.

• targeted_errors (Iterable) – A list of Pauli strings, e.g. ["ZIZIZ", "XYXYX"] that specify which errors to target. By default, the Pauli "I" * n_qubits is used to estimate the probability of no error, i.e. the process fidelity.

• twirl (Twirl | str) – The twirling group to use.

• compiler (Compiler) – A compiler to run all cycles through before simulating them.

Return type

EstimateCollection

Raises

ValueError – If targeted_errors are not all Pauli strings of the correct length.

predict_irb(cycle, n_randomizations=50, twirl='C', compiler=None)

Predicts the estimates that would be generated by simulating make_irb() with this simulator.

import trueq as tq

sim.predict_irb({0: tq.Gate.id, (1, 2): tq.Gate.cnot})

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".
 IRB Interleaved Randomized Benchmarking Cliffords (0,) : Gate.id (1, 2) : Gate.cx Key: cycles: (Cycle((0,): Gate.id, (1, 2): Gate.cx),) labels: (0,) name: protocol: IRB twirl: Cliffords on [0, (1, 2)] Cliffords (0,) : Gate.id (1, 2) : Gate.cx Key: cycles: (Cycle((0,): Gate.id, (1, 2): Gate.cx),) labels: (1, 2) name: protocol: IRB twirl: Cliffords on [0, (1, 2)] ${e}_{F}$ The probability of an error acting on the targeted systems during a dressed gate of interest. 2.0e-02 (0.0e+00) 0.019870000000000942, 0.0 8.1e-02 (0.0e+00) 0.0808220704876671, 0.0 ${p}$ Decay parameter of the exponential decay $Ap^m$. 9.7e-01 (0.0e+00) 0.9735066666666654, 0.0 9.1e-01 (0.0e+00) 0.9137897914798218, 0.0
Parameters
• cycle (dict | Cycle) – The cycle to make the prediction for.

• n_randomizations (int | NoneType) – The number of random twirls to average over. If None, the twirling group is assumed to have a perfect implementation.

• twirl (Twirl | str) – The twirling group to use.

• compiler (Compiler) – A compiler to run all cycles through before simulating them.

Return type

EstimateCollection

predict_knr(cycle, subsystems=1, n_randomizations=50, twirl='P', compiler=None)

Predicts the estimates that would be generated by simulating make_knr() with this simulator.

import trueq as tq

cycle = {0: tq.Gate.h, (1, 2): tq.Gate.cnot}
prediction = sim.predict_knr(cycle, subsystems=2)
prediction.plot.knr_heatmap()

Parameters
• cycle (dict | Cycle) – The cycle to make the prediction for.

• subsystems (Iterable | :py:class~trueq.Subsystems | int) – A list of labels of combinations of gate-bodies to reconstruct the marginal probabilities for, which can be specified using a Subsystems object. Also accepts a positive integer to instantiate all combinations of gate bodies with up to and including that many elements.

• n_randomizations (int | NoneType) – The number of random twirls to average over. If None, the twirling group is assumed to have a perfect implementation.

• twirl (Twirl | str) – The twirling group to use.

• compiler (Compiler) – A compiler to run all cycles through before simulating them.

Return type

EstimateCollection

Raises

ValueError – If a subsystem partially overlaps with one of the labels of the cycle.

predict_srb(labels, n_randomizations=50, compiler=None)

Predicts the estimates that would be generated by simulating make_srb() with this simulator.

import trueq as tq

prediction = sim.predict_srb([0, [1, 2]])
prediction += sim.predict_srb([0])
prediction += sim.predict_srb([[1, 2]])
prediction.plot.compare_rb()

Parameters
• labels (Iterable | Twirl) – A list specifying sets of system labels to be twirled together by Clifford gates in each circuit, e.g. [3, [1, 2], 4] for mixed single and two-qubit twirling. Or, for advanced usage (including to specify a non-default dimension), a Twirl instance.

• n_randomizations (int | NoneType) – The number of random twirls to average over. If None, the twirling group is assumed to have a perfect implementation.

• compiler (Compiler) – A compiler to run all cycles through before simulating them.

Return type

EstimateCollection

## Simulation Backends

class trueq.simulation.backend.SimulationBackend

Abstract parent class for all simulation backends. A instance of a subclass stores backend specific options and, when requested, constructs new Runner instances, each of which can be used to simulate a single quantum circuit.

new_runner(prep_policy, meas_policy, dim, n_shots=inf)

Instantiates a new backend Runner that can be used to simulate a single quantum circuit.

Parameters
• prep_policy (PrepPolicy) – A policy specifying how to process preparation instructions.

• meas_policy (MeasPolicy) – A policy specifying how to process measurement instructions.

• dim (int) – The subsystem dimension to use during simulation.

• n_shots (int) – The number of shots to sample if and when requested. This will only be relevant when meas_policy is RequireMeas.

class Runner(prep_policy, meas_policy, dim, n_shots=inf)

An instance of a subclass performs the simulation of a single quantum circuit. Instructions are fed to a backend instance to be processed one at a time. The instance can either simulate it on the fly, or store instuctions and simulate when completed.

Parameters
property dim

The subsystem dimension this backend is using.

Type

int

property meas_policy

The measurement policy used during simulation.

Type

MeasPolicy

property n_shots

The number of shots to sample if and when requested. This will only be relevant when meas_policy is RequireMeas.

Type

int

property prep_policy

The measurement policy used during simulation.

Type

MeasPolicy

divide()

Instructs this backend that a new cycle of noise is about to begin. This exists in case it is helpful to a particular backend’s implementation; it should not affect the simulation result.

process_confusion(labels, confusion)

Processes a confusion matrix instruction on the given labels.

Parameters
• labels (tuple) – The (ordered) labels on which the measurement acts.

• confusion (numpy.ndarray) – A real array of shape (n_outcomes ** n, dim ** n) where n = len(labels) and where each column sums to unity.

process_gate(labels, gate)

Processes a single gate on the given labels.

Parameters
• labels (tuple) – The (ordered) labels on which the gate acts.

• gate (Gate) – A gate whose shape is consistent with the labels and the dimension of this backend.

process_meas(labels)

Processes an ideal measurement instruction on each of the given labels.

Parameters

labels (tuple) – The labels on which the measurement acts.

process_mixed_prep(labels, prep=None)

Processes a mixed state preparation instruction on the given labels.

Parameters
• labels (tuple) – The (ordered) labels on which the preparation acts.

• prep (numpy.ndarray) – A density matrix with width dim ** len(labels).

process_prep(labels)

Processes an ideal preparation instruction on the given labels.

Parameters

labels (tuple) – The labels on which the preparation acts.

process_povm(labels, povm)

Processes a positive operator valued measurement instruction on the given labels.

Parameters
• labels (tuple) – The (ordered) labels on which the measurement acts.

• povm (numpy.ndarray) – A complex array of shape (n_outcomes ** n, dim ** n, dim ** n) where n = len(labels), and where povm[i] is the i’th POVM element in lexicographical order with respect to the order of the labels.

process_pure_prep(labels, prep)

Processes a pure state preparation instruction on the given labels.

Parameters
• labels (tuple) – The (ordered) labels on which the preparation acts.

• prep (numpy.ndarray) – A pure state with shape dim ** len(labels).

process_superop(labels, superop)

Processes a single superoperator on the given labels.

Parameters
• labels (tuple) – The (ordered) labels on which the matrix acts.

• mat (Superop) – A superoperator.

property value

The final result of simulation. The type is completely dependent on the implementation.

Type

object

class trueq.simulation.MeasPolicy(value)

Measurement policies for simulation backends.

ForceDangling = 1

Ignore measurement instructions.

This policy is used to force a backend to compute the quantum state or operator of a circuit that contains measurement instructions.

Note

In the edge case that a measurement instruction is applied to a subsystem that hasn’t been addressed yet, a backend should include the label of the measurement in the simulation so that the total number of systems is consistent with that of the circuit.

This policy may be nonsensical for circuits with mid-circuit measurements; it is up to each backend type to raise appropriate errors or warnings.

RequireMeas = 2

If no measurement instructions are specified in the circuit, finish the simulation by processing a measurement instruction on every qubit. Otherwise, process measurement instructions just as they appear.

This policy is used to allow a backend to sample outcomes from circuits irrespective of whether they were provided with measurement instructions.

class trueq.simulation.PrepPolicy(value)

Preparation policies for simulation backends.

ForceDangling = 1

Ignore preparation instructions.

This policy is typically used in conjugation with MeasPolicy.ForceDangling to force a backend to compute the total operator (unitary or superoperator) of a circuit that contains preparation instructions.

Note

In the edge case that a preparation instruction is applied to a subsystem that hasn’t been addressed yet, a backend should include the label of the preparation in the simulation so that the total number of systems is consintent with that of the circuit.

ForcePrep = 2

Whenever a new qubit label is first encountered with a non-preparation instruction, preceed it by an ideal preparation instruction.

This policy is used to allow a backend to simulate quantum states of circuits, or sample outcomes from circuits, irrespective of whether they begin with preparation instructions.

class trueq.simulation.PropagationBackend

A simulation backend that uses OperatorTensor or StateTensor to perform simulations.

class Runner(prep_policy, meas_policy, dim, n_shots=inf)

Instantiates a OperatorTensor or StateTensor during construction and mutates it through the course of a simulation.

Parameters
property meas_labels

All of the labels that explicitly received a Meas instruction.

Return type

tuple

process_gate(labels, gate)

Processes a single gate on the given labels.

Parameters
• labels (tuple) – The (ordered) labels on which the gate acts.

• gate (Gate) – A gate whose shape is consistent with the labels and the dimension of this backend.

process_meas(labels, error=0)

Processes an ideal measurement instruction on each of the given labels.

Parameters

labels (tuple) – The labels on which the measurement acts.

process_povm(labels, error=0)

Processes an ideal measurement instruction on each of the given labels.

Parameters

labels (tuple) – The labels on which the measurement acts.

process_confusion(labels, error=0)

Processes a measurement instruction with errors on the given labels.

Parameters
• labels (tuple) – The (ordered) labels on which the measurement acts.

• error (numpy.ndarray) – A 2D matrix specifying a real confusion matrix or a 3D complex array specifying a POVM.

process_prep(labels, prep=None)

Processes an ideal preparation instruction on the given labels.

Parameters

labels (tuple) – The labels on which the preparation acts.

process_pure_prep(labels, prep=None)

Processes an ideal preparation instruction on the given labels.

Parameters

labels (tuple) – The labels on which the preparation acts.

process_mixed_prep(labels, prep=None)

Processes a preparation instruction on the given labels.

Parameters
• labels (tuple) – The labels on which the preparation acts.

• prep (numpy.ndarray) – The pure state or mixed state to prepare on the given labels.

property dim

The subsystem dimension this backend is using.

Type

int

divide()

Instructs this backend that a new cycle of noise is about to begin. This exists in case it is helpful to a particular backend’s implementation; it should not affect the simulation result.

property meas_policy

The measurement policy used during simulation.

Type

MeasPolicy

property n_shots

The number of shots to sample if and when requested. This will only be relevant when meas_policy is RequireMeas.

Type

int

property prep_policy

The measurement policy used during simulation.

Type

MeasPolicy

process_superop(labels, superop)

Processes a single superoperator on the given labels.

Parameters
• labels (tuple) – The (ordered) labels on which the matrix acts.

• superop (Superop) – A superoperator.

property value

The final result of simulation. The type depends on whether it was an operator, state, or sampling simulation.

Type
new_runner(prep_policy, meas_policy, dim, n_shots=inf)

Instantiates a new backend Runner that can be used to simulate a single quantum circuit.

Parameters
• prep_policy (PrepPolicy) – A policy specifying how to process preparation instructions.

• meas_policy (MeasPolicy) – A policy specifying how to process measurement instructions.

• dim (int) – The subsystem dimension to use during simulation.

• n_shots (int) – The number of shots to sample if and when requested. This will only be relevant when meas_policy is RequireMeas.

## Match

class trueq.simulation.match.Match(exclusive=False)

A class whose purpose is to filter the operations presented to a given NoiseSource in a Simulator instance. This is typically used by a noise source to specify, for example, which gate, label, and cycle combinations to apply noise to. This base class is a pass-through that does not do any filtering except the following which is common to all noise sources:

The match also defines iter_* functions, which are called by NoiseSources, and return an iterator of pairs of Operations and labels (tuples). The noise source function called during propagation, apply(), does not call iter() directly, which returns an iterator of OpWrappers, but instead calls to the match via the iter_* wrapper functions, meaning that a NoiseSource does not (and should not) need to interact with a OpWrapper.

Two or more matches can be joined together with & (AndMatch) and | (OrMatch) that restrict operations to the intersection of matches, or the union of matches, respectively. This is seen in the following examples:

import trueq as tq
import trueq.simulation as tqs

# Create a match that matches only X gates.
noisy_x = tqs.GateMatch(tq.Gate.x)

# Create a match that matches only Y gates.
noisy_y = tqs.GateMatch(tq.Gate.y)

# Create a match that matches any operation acting on a subset of labels
# (2, 4).
noisy_labels = tqs.LabelMatch((2, 4))

# Single label matches can send in the label as an integer instead of a tuple
# of length 1. This match matches all operations on qubit 0.
zero_match = tqs.LabelMatch(0)

# A match which prevents subsequent noise sources from seeing operations
# that are matched with this one.
no_more_match = tqs.Match(exclusive=True)

# Create a match that restricts to X gates and any gate on qubits 2 and 4
noise_restriction = tqs.OrMatch(noisy_x, noisy_labels)

# Equivalently, we can use |
noise_restriction = noisy_x | noisy_labels

# Create a match that restricts to X gates only on qubits 2 and 4
noise_restriction = tqs.AndMatch(noisy_x, noisy_labels)

# Equivalently, we can use &
noise_restriction = noisy_x & noisy_labels

# Create a match that restricts to Y gates on qubits 2 and 4, and any X gates
compound_match = (noisy_y & noisy_labels) | noisy_x

# The following produce equivalent match behaviour:
equiv1 = noisy_x | noisy_y
equiv2 = tqs.GateMatch([tq.Gate.x, tq.Gate.y])

iter(op_type, cycle_wrappers, noise_only=True, update_flags=True)

Iterates through the operations in the most recent cycle in cycle_wrappers, yielding every operation of type op_type that is matched by this instance and that has not yet been marked for no further simulation via Opwrapper.no_more_noise.

Parameters
Returns

An iterator of OpWrappers with operations of type op_type

Return type

Iterable

iter_gates(cycle_wrappers, noise_only=True)

Yields pairs (labels, gate) in the most recent cycle in cycle_wrappers where gate is a Gate that is matched by this instance.

Parameters
• cycle_wrappers (list) – A list of CycleWrappers, with the last one being the most recent.

• noise_only (bool) – Whether to mark the OpWrapper as has_been_simulated after it has been yielded.

Returns

An iterator of labels and operators.

Return type

Iterable

iter_meas(cycle_wrappers, noise_only=True)

Yields pairs (labels, meas) in the most recent cycle in cycle_wrappers where meas is a Meas that is matched by this instance.

Parameters
• cycle_wrappers (list) – A list of CycleWrappers, with the last one being the most recent.

• noise_only (bool) – Whether to mark the OpWrapper as has_been_simulated after it has been yielded.

Returns

An iterator of labels and operators.

Return type

Iterable

iter_prep(cycle_wrappers, noise_only=True)

Yields pairs (labels, prep) in the most recent cycle in cycle_wrappers where prep is a Prep that is matched by this instance.

Parameters
• cycle_wrappers (list) – A list of CycleWrappers, with the last one being the most recent.

• noise_only (bool) – Whether to mark the OpWrapper as has_been_simulated after it has been yielded.

Returns

An iterator of labels and operators.

Return type

Iterable

class trueq.simulation.match.AndMatch(*matches, exclusive=None)

Instances of this class match operations that match the intersection of the given matches. Operations are matched as follows:

• Any operation that matches all of the given matches.

• However, any presented OpWrapper where no_more_noise is True is skipped.

Parameters
• *matches – One or more Matches of any type.

• exclusive (bool | NoneType) – Whether to mark every operation yielded during iteration as no_more_noise. In any future iterations by any Match an operation with this marking will be skipped.

class trueq.simulation.match.AssociativeMatch(*matches, exclusive=None)

Abstract parent class for children C that stores a list of Matches and have the property that C([A, C([B, C])) and C([A, B, C]) share the same behaviour. This flattening process is performed automatically by the constructor when AssociativeMatches of the same type are encountered.

Parameters
• *matches – One or more Matches of any type.

• exclusive (bool | NoneType) – Whether to mark every operation yielded during iteration as no_more_noise. In any future iterations by any Match an operation with this marking will be skipped.

class trueq.simulation.match.GateMatch(gates, exclusive=False)

Instances of this class match a fixed set of gates. Operations are matched as follows:

• Any gate in gates.

• However, any presented OpWrapper where no_more_noise is True is skipped.

Parameters
• gates (Gate | Iterable) – A gate or iterable of gates.

• exclusive (bool) – Whether to mark every operation yielded during iteration as no_more_noise. In any future iterations by any Match an operation with this marking will be skipped.

class trueq.simulation.match.LabelMatch(labels, strict=True, exclusive=False)

Instances of this class match operations whose qubit labels correspond to a specified set of labels. Operations are matched as follows:

• If strict is true, operations whose labels are a strict subset of this instance’s labels are yielded.

• If strict is False, operations whose labels have some intersection with this instance’s labels are yielded.

• However, any presented OpWrapper where no_more_noise is True is skipped.

For example, a LabelMatch initialized with the label 0 will match to single-qubit operations on label 0, regardless of the value of strict. If strict is False, it will also match to any multi-qubit operations on label 0, like a CNOT gate over labels 0 and 1.

Parameters
• labels (tuple) – A tuple of integers representing the labels to be matched.

• strict (bool) – Determines if label matching is performed by subset or by intersection.

• exclusive (bool) – Whether to mark every operation yielded during iteration as no_more_noise. In any future iterations by any Match an operation with this marking will be skipped.

class trueq.simulation.match.NQubitMatch(n_sys, exclusive=False)

Instances of this class match operations that act on only n_sys qubit labels. Operations are matched as follows:

• All operations that act on a n_sys qubit labels.

• Any presented OpWrapper where no_more_noise is True is skipped.

Parameters
• n_sys (int) – An integer defining the number of qubit labels upon which an operator must act on order to be matched by this instance.

• exclusive (bool) – Whether to mark every operation yielded during iteration as no_more_noise. In any future iterations by any Match an operation with this marking will be skipped.

class trueq.simulation.match.NativeMatch(name, exclusive=False)

Instances of this class match operations that are NativeGates with a specified name. Operations are matched as follows:

• Any native gate with a name that matches name is returned.

• However, any presented OpWrapper where no_more_noise is True is skipped.

The following example showcases this behaviour:

import trueq as tq
import trueq.simulation as tqs

# match all native gates with the name "rz"
match = tqs.NativeMatch("rz")

Parameters
• name – A string for matching to names of native gates.

• exclusive (bool) – Whether to mark every operation yielded during iteration as no_more_noise. In any future iterations by any Match an operation with this marking will be skipped.

class trueq.simulation.match.OrMatch(*matches, exclusive=None)

Instances of this class match operations that match the union of the specified set of matches. Operations are matched as follows:

• Any operation that matches any of the given matches.

• However, any presented OpWrapper where no_more_noise is True is skipped.

Parameters
• *matches – One or more Matches of any type.

• exclusive (bool | NoneType) – Whether to mark every operation yielded during iteration as no_more_noise. In any future iterations by any Match an operation with this marking will be skipped.

class trueq.simulation.match.R90Match(exclusive=False)

Instances of this class match qubit gates that can be represented as a 90 degree rotation about an axis within the XY plane. Operations are matched as follows:

• Any qubit gate that is a 90 degree rotation about an axis within the XY plane.

• However, any presented OpWrapper where no_more_noise is True is skipped.

Parameters

exclusive (bool) – Whether to mark every operation yielded during iteration as no_more_noise. In any future iterations by any Match an operation with this marking will be skipped.

Raises

RuntimeError – Throws an error if the cycle is not from a qubit circuit.

class trueq.simulation.match.SingleQubitMatch(exclusive=False)

Instances of this class match operations that act on only a single qubit label. Operations are matched as follows:

• All operations that act on a single qubit label.

• However, any presented OpWrapper where no_more_noise is True is skipped.

Parameters

exclusive (bool) – Whether to mark every operation yielded during iteration as no_more_noise. In any future iterations by any Match an operation with this marking will be skipped.

## NoiseSource

class trueq.simulation.noise_source.NoiseSource(match=None, dim=None)

Parent class for all built in noise sources which can be added to the simulator.

A noise source comprises a method apply() that mutates the state of simulation by inspecting a given cycle, a Match instance for filtering which operations in the cycle to consider, a cache for storing saved calculations that persists for the lifetime of the noise source, and an optional attribute dim for explicitly defining the allowed subsystem dimension.

Generally, a NoiseSource is added to a simulator, and then during circuit propagation it has an opportunity to apply noise cycle by cycle. Within the simulation of each cycle, a noise source can either simply add noise, or additionally attempt to simulate some of the operations within a cycle. A default noise source is always included in the Simulator to ensure that all gates within a cycle that are not simulated by a user-appended noise source are eventually simulated ideally.

Parameters
• match (Match | NoneType) – A match object which determines the conditions required for a noise source to attempt to apply itself to the state during simulation.

• dim (int | NoneType) – The subspace dimension expected by a noise source, or None, if there is no needed restriction to a particular subspace dimension.

property dim

The Hilbert space dimension of each subsystem expected by this propagator. If the dimension is set to None, then the noise source does not care about the dimension of a given subsystem. Otherwise, the dimension is an integer, and the noise source can use this information to assert that it can be applied correctly during simulation.

Returns

The expected dimension of subsystems as an integer, or None.

Type

int | NoneType

property match

The Match instance owned by this noise source. This object is typically used by apply() to filter which cycle operations are presented to it.

For example, the noise source can be initialized with a match on gates, defined by an instance of GateMatch initialized with a list of gates. In this case, the noise source will only have the opportunity to apply itself to the state whenever one of these gates appears.

Returns

The Match which determines the conditions required for a noise source to attempt to apply itself to the state during simulation.

Type

Match

make_circuit_cache(circuit)

Instantiates a cache object that will be made available to apply() as circuit_cache for each call within the circuit. The cache is typically a dictionary that stores information about the circuit that is needed by apply() but not available in individual cycles of the circuit, such as all of the labels the circuit acts on.

Parameters

circuit (Circuit) – The circuit being simulated by the (noisy) simulator.

Returns

A dictionary of circuit information.

Return type

dict

apply(cycle_wrappers, backend, circuit_cache)

Processes all operations in the last of the given cycle_wrappers that have not yet been completely processed by the given backend. This method defines ideal simulation, which children will typically overload to add noise.

Parameters
• cycle_wrappers (list) – All CycleWrappers from the beginning of the circuit up until the current cycle in the simulation. These are all given to allow non-Markovian noise; typically only the last will be used.

• backend (Runner) – The simulation backend.

• circuit_cache (dict) – Cached information about the circuit.

A noise source that adds depolarizing noise; see add_depolarizing() for details.

Parameters
• p (float) – The depolarizing parameter, as defined in the above referenced function.

• local (bool) – Whether local depolarizing noise is applied to every qubit that is encountered on each matched gate, or whether global depolarizing noise is applied to all qubits encounterned on each matched gate.

• match (Match | NoneType) – A match object that specifies which operations or cycles this noise source applies to.

apply(cycle_wrappers, backend, cache)

Allows the backend to process the depolarizing noise.

Parameters
• cycle_wrappers (list) – All CycleWrappers from the beginning of the circuit up until the current cycle in the simulation. This noise source only uses the last one.

• backend (Runner) – The simulation backend.

• cache (dict) – A dictionary to cache results in.

A noise source that replaces all valid Gates present in the circuit with the returned valued of fn; see add_gate_replace() for details.

Parameters
Raises

ValueError – If the new gate does not match the dimensions of the gate that it is replacing.

apply(cycle_wrappers, backend, cache)

Allows the backend to process the gate replacement noise.

Parameters
• cycle_wrappers (list) – All CycleWrappers from the beginning of the circuit up until the current cycle in the simulation. This noise source only uses the last one.

• backend (Runner) – The simulation backend.

• cache (dict) – A dictionary to cache results in.

Raises

ValueError – If the new gate-like object does not act on the same dimensional subsystems as the gate that it is replacing.

A noise source that adds unitary overrotation error; see add_overrotation() for details.

Parameters
• single_sys (float) – The amount of overrotation to add to single-qubit gates..

• multi_sys (float) – The amount of overrotation to add to multi-qubit gates.

• match (Match | NoneType) – A match object that specifies which operations or cycles this noise source applies to.

apply(cycle_wrappers, backend, cache)

Allows the backend to process the overrotation noise.

Parameters
• cycle_wrappers (list) – All CycleWrappers from the beginning of the circuit up until the current cycle in the simulation. This noise source only uses the last one.

• backend (Runner) – The simulation backend.

• cache (dict) – A dictionary to cache results in.

class trueq.simulation.add_basic.RelaxationNoise(t1, t2, t_single, t_multi, excited_pop=0, match=None)

A noise source that adds relaxation noise, a noise restricted to qubits; see add_relaxation() for details.

Relaxation noise simulates a $T_1$ and $T_2$ process on the qubits, in an idealized manner assuming that gates happen instantaneously, and that the relaxation occurs before any ideal operation is simulated.

Parameters
• t1 (float | dict) – The $T_1$ time, i.e. the characteristic time of relaxation.

• t2 (float | dict) – The $T_2$ time, i.e. the characteristic time of dephasing.

• t_single (float) – The idealized time it takes for single-qubit cycles to run.

• t_multi (float) – The idealized time it takes for multi-qubit cycles to run.

• excited_pop (float | dict) – The excited state population at equilibrium.

• match (Match | NoneType) – A match object that specifies which operations or cycles this noise source applies to.

make_circuit_cache(circuit)

Instantiates a cache object that will be made available to apply().

Parameters

circuit (Circuit) – The circuit being simulated.

Returns

All of the labels the circuit acts on.

Return type

tuple

static relaxation_channel(t1, t2, t, excited_pop)

Determines and returns the superoperator for a relaxation channel by computing the Choi representation, and casting it to a superoperator.

Parameters
• t1 (float | dict) – The $T_1$ time, i.e. the characteristic time of relaxation.

• t2 (float | dict) – The $T_2$ time, i.e. the characteristic time of dephasing.

• t (float) – The time that the channel is applied for.

• excited_pop (float | dict) – The excited state population at equilibrium.

Return type

Superop

Raises

ValueError – Raises when $T_1$ and $T_2$ are physically invalid, or if excited_pop is a not a valid probability.

apply(cycle_wrappers, state, cache)

Allows the backend to process the relaxation noise.

Parameters
• cycle_wrappers (list) – All CycleWrappers from the beginning of the circuit up until the current cycle in the simulation. This noise source only uses the last one.

• backend (Runner) – The simulation backend.

• cache (dict) – A dictionary to cache results in.

A noise source that adds superoperator noise; see add_superop() for details.

Parameters
• superop (Superop) – A superoperator to use as noise.

• match (Match | NoneType) – A match object that specifies which operations or cycles this noise source applies to.

apply(cycle_wrappers, backend, cache)

Allows the backend to process the superoperator noise.

Parameters
• cycle_wrappers (list) – All CycleWrappers from the beginning of the circuit up until the current cycle in the simulation. This noise source only uses the last one.

• backend (Runner) – The backend to propagate.

• cache (dict) – A dictionary to cache results in.

Raises

ValueError – If a multi-qubit superoperator matches to a multi-qubit gate with a larger dimension.

A noise source which inserts or replaces entire cycles during simulation; see add_cycle_noise() for more information.

Parameters
• replacement (Cycle | dict | function) – A Cycle-like, or a function which takes a Cycle (and, optionally as a second argument, a tuple of all labels in the current circuit being simulated) and returns a Cycle-like, to be inserted when the noise source matches.

• match (BaseCycleMatch | NoneType) – A match object that specifies which cycles this noise source applies to.

• cycle_offset (int) – Whether to insert the replacement cycle before (-1), in place of (0), or after (+1) the cycles matched by this noise source.

Raises

ValueError – If the match is not a cycle match instance, the cycle_offset is not -1, 0, or 1, or if replacement generates a cycle with invalid dimensions.

make_circuit_cache(circuit)

Caches the circuit labels to be used during apply().

Parameters

circuit (Circuit) – The circuit being simulated by the (noisy) simulator.

Return type

tuple

apply(cycle_wrappers, backend, cache)

Allows the backend to process the replacement noise.

Parameters
• cycle_wrappers (list) – All CycleWrappers from the beginning of the circuit up until the current cycle in the simulation.

• backend (Runner) – The simulation backend.

• cache (dict) – A dictionary to cache results in.

A noise source which reconstructs the noise of each cycle for each KNR experiment in fit_or_circuits; see the KNR guide and add_knr_noise() for details.

Parameters
• fit_or_circuits (EstimateCollection | CircuitCollection) – An estimate collection or a set of circuits which contain estimates of, or circuits for, the KNR protocol.

• exclusive (bool) – If True, all gates within the targeted cycles will be marked as no_more_noise during simulation. This is False by default.

• ignore_identity (bool) – Whether or not to match this noise to cycles containing identical operations, ignoring any identity operators. This is False by default.

• ignore_marker (bool) – If True, match this noise to cycles which contain the same operations but have differet markers. This is True by default.

Raises

ValueError – If a cycle in the fit_or_circuits appears in the fit multiple times, or if the fit is empty.

static knr_superop(estimate)

Returns a Superop representing the one-body noise acting on the gate-body estimated by estimate, as fitted by the KNR protocol. The superoperator is given by a Pauli transfer matrix, see from_ptm() for more information.

Parameters

estimate (NormalEstimate) – The estimate containing the fit information for the gate-body.

Returns

The reconstructed superoperator for the KNR estimate.

Return type

Superop

apply(cycle_wrappers, backend, cache)

Allows the backend to process the KNR noise.

Parameters
• cycle_wrappers (list) – All CycleWrappers from the beginning of the circuit up until the current cycle in the simulation. This noise source only uses the last one.

• backend (Runner) – The simulation backend.

• cache (dict) – A dictionary to cache results in.

A noise source that adds state preparation error; see add_prep() for details.

A noise source that adds readout error; see add_readout_error() for details.

Abstract parent class for state preparation and measurement (SPAM) noise sources.

Parameters
• default (object) – The default SPAM noise to apply.

• custom (dict) – A dictionary mapping qubit labels or label tuples to SPAM noise.

• match (Match | NoneType) – A match object that specifies which operations or cycles this noise source applies to.

apply(cycle_wrappers, backend, cache)

Allows the backend to process this SPAM noise.

Parameters
• cycle_wrappers (list) – All CycleWrappers from the beginning of the circuit up until the current cycle in the simulation. This noise source only uses the last one.

• backend (Runner) – The simulation backend.

• cache (dict) – A dictionary to cache results in.

## Cycle Matches

class trueq.simulation.match.BaseCycleMatch(lag=0, exclusive=False)

Abstract parent class for children that match against all operations in a Cycle as a whole, rather than single operations at a time. Children of this class can specify custom cycle matching functions by overwriting check_match(). Two or more matches can be joined together with & (AndCycleMatch) and | (OrCycleMatch) that restrict operations to the intersection of matches, or the union of matches, respectively. Cycle matches can be initialized such that they match on the past cycles via the lag parameter, allowing for the possibility of non-Markovian noise.

import trueq as tq
import trueq.simulation as tqs

cyc_h = tq.Cycle({0: tq.Gate.h})
cyc_cx = tq.Cycle({(0, 1): tq.Gate.cx})
cyc_cx_alt = tq.Cycle({(0, 2): tq.Gate.cx})

# create a match which matches if the previous cycle was cyc_h
m_past = tqs.CycleMatch(cyc_h, lag=1)

# create a match which matches if the current cycle is cyc_cx
m_present = tqs.CycleMatch(cyc_cx)

# create equivalent matches which match if the previous cycle was cyc_h
# and the current cycle is cyc_cx
m_joint = m_past & m_present
m_joint_alt = tqs.AndCycleMatch(m_past, m_present)

# create a match which matches if the previous cycle was an H gate on
# 0 and the current cycle is a CNOT on either (0,1) or (0,2)
m_cx_2 = tqs.CycleMatch(cyc_cx_alt)
m_entangling = m_past & (m_present | m_cx_2)

Parameters
• lag (int) – This instance will match to the cycle which occured lag cycles in the past. By default, this is 0, and matches to the present cycle.

• exclusive (bool) – Whether to mark every operation yielded during iteration as no_more_noise. In any future iterations by any Match an operation with this marking will be skipped.

check_match(cycle_wrappers)

Abstract method to be overwritten by children which defines the cycle matching function. This function should return True if the desired cycle within cycle_wrappers matches the conditions of the match, and False otherwise.

Parameters

cycle_wrappers (list) – A list of CycleWrappers to match against.

property lag

Returns the number of cycles this match instance lags the present cycle by.

Returns

The number of cycles behind the current cycle being matched to.

Type

int

find_matches(cycle_wrappers)

If the cycle_wrappers match to this instance, this method returns this match object in a list, or an empty list otherwise. This instance’s matching function is given by check_match().

Parameters

cycle_wrappers (list) – A list of CycleWrappers to match against.

Returns

A list containing this match instance if the cycle matches, and an empty list otherwise.

Return type

list

class trueq.simulation.match.AndCycleMatch(*matches, exclusive=None)

Instances of this class match operations that match the intersection of the given matches. All matches must be children of BaseCycleMatch. Operations are matched as follows:

• Any operation that matches all of the given matches.

• However, any presented OpWrapper where no_more_noise is True is skipped.

Parameters
• *matches – One or more BaseCycleMatches of any type.

• exclusive (bool | NoneType) – Whether to mark every operation yielded during iteration as no_more_noise. In any future iterations by any Match an operation with this marking will be skipped.

class trueq.simulation.match.AssociativeCycleMatch(*matches, exclusive=None)

Abstract parent class for children C that stores a list of BaseCycleMatches and have the property that C([A, C([B, C])) and C([A, B, C]) share the same behaviour. This flattening process is performed automatically by the constructor when AssociativeCycleMatches of the same type are encountered.

Parameters
• *matches – One or more BaseCycleMatches of any type.

• exclusive (bool | NoneType) – Whether to mark every operation yielded during iteration as no_more_noise. In any future iterations by any Match an operation with this marking will be skipped.

Raises

ValueError – If any match in matches is not a child class of BaseCycleMatch.

class trueq.simulation.match.CycleMatch(cycle, ignore_marker=True, ignore_id=False, lag=0, exclusive=False)

Instances of this class match all operations in a given cycle equivalent to the cycle used to instantiate this instance. By default, cycle equivalence means cycle equality, but the boolean arguments ignore_marker and ignore_id allow three more variants of cycle equivalence. Operations are matched as follows:

• All operations in a cycle if the cycle is equivalent to one of cycles.

• However, any presented OpWrapper where no_more_noise is True is skipped.

Additionally, this match instance has the ability to match on previously simulated cycles, allowing for the possibility of non-Markovian noise sources. By initializing an instance of this match with lag > 0, the instance will match on the cycle simulated lag cycles ago. By default, lag is 0 and matches on the cycle currently being simulated. This match will not match until lag cycles have been simulated.

Parameters
• cycle (Cycle | dict) – A Cycle-like to match against.

• ignore_marker (bool) – Whether to ignore the marker values of cycles. Default is True.

• ignore_id (bool) – Whether to treat all identity gates as though they are not present. Default is False.

• lag (int) – This instance will match to the cycle which occured lag cycles in the past. By default, this is 0, and matches to the present cycle.

• exclusive (bool) – Whether to mark every operation yielded during iteration as no_more_noise. In any future iterations by any Match an operation with this marking will be skipped.

class trueq.simulation.match.OrCycleMatch(*matches, exclusive=None)

Instances of this class match operations that match the union of the given matches. All matches must be children of BaseCycleMatch. Operations are matched as follows:

• Any operation that matches any of the given matches.

• However, any presented OpWrapper where no_more_noise is True is skipped.

Parameters
• *matches – One or more BaseCycleMatches of any type.

• exclusive (bool | NoneType) – Whether to mark every operation yielded during iteration as no_more_noise. In any future iterations by any Match an operation with this marking will be skipped.

## CycleWrapper

class trueq.simulation.match.CycleWrapper(cycle)

Wraps a Cycle into a list of OpWrappers, partitioning the cycle ops into three groups, Gate which contains all gates, Meas which contains all measurements, and Prep which contains all state preparations in the cycle being propagated by the simulator. The wrapper also contains the Cycle itself, as well as its dimension via dim.

A CycleWrapper can be asked to return all gates which have not yet been simulated via final_iter(), serving as a final catch during cycle propagation, and is utilized by the parent class of noise sources, NoiseSource during the call of apply().

Parameters

cycle (Cycle) – The cycle being wrapped.

property all_ops

Yields all operators within the cycle wrapped by this instance. Each operator in the cycle is yielded as an OpWrapper.

Type

generator

## OpWrapper

class trueq.simulation.match.OpWrapper(labels, op)

Wraps labels and an Operation into one small object. The wrapper keeps track of whether or not it has been simulated via has_been_simulated, which is marked as True during cycle propagation within a simulation by a NoiseSource's Match when it simulates the action of the operation on the state. Further, it keeps track of whether some noise source has indicated that no further noise should be applied via no_more_noise, and is primarily modified by setting the exclusive variable defined in a Match to True.

Two OpWrappers are equal if and only if all four of their properties are equal. That is, the labels and operations must be equal, as well as the state of their flags, has_been_simulated and no_more_noise.

Parameters