# Zero Noise Extrapolation¶

Zero noise extrapolation is an error mitigation technique in which an expectation value is computed at different noise levels and, as a second step, the ideal expectation value is inferred extrapolating the measured results to the zero-noise limit.

Both steps—noise scaling and extrapolation—can be applied with Mitiq. The corresponding sub-modules are mitiq.zne.scaling and mitiq.zne.inference.

## Noise scaling by unitary folding¶

Unitary folding is a method for noise scaling that operates directly at the gate level. This makes it easy to use with current quantum computing libraries. It is especially appropriate when the underlying noise scales with the depth and/or the number of gates of a quantum program. More details can be found in [2] where the unitary folding framework was introduced.

At the gate level, noise is amplified by mapping gates (or groups of gates) G to

$G \mapsto G G^\dagger G .$

This makes the circuit longer (adding more noise) while keeping its effect unchanged (because $$G^\dagger = G^{-1}$$ for unitary gates). We refer to this process as unitary folding. If $$G$$ is a subset of the gates in a circuit, we call it local folding. If $$G$$ is the entire circuit, we call it global folding.

In Mitiq, folding functions input a circuit and a scale factor, i.e., a floating point value which corresponds to (approximately) how much the length of the circuit is scaled. The minimum scale factor is one (which corresponds to folding no gates). A scale factor of three corresponds to folding all gates locally. Scale factors beyond three begin to fold gates more than once.

### Local folding methods¶

For local folding, there is a degree of freedom for which gates to fold first. The order in which gates are folded can have an important effect on how the noise is scaled. As such, Mitiq defines several local folding methods.

1. mitiq.zne.scaling.fold_gates_from_left()

2. mitiq.zne.scaling.fold_gates_from_right()

3. mitiq.zne.scaling.fold_gates_at_random()

The function fold_gates_from_left will fold gates from the left (or start) of the circuit until the desired scale factor is reached.

>>> import cirq
>>> from mitiq.zne.scaling import fold_gates_from_left

# Get a circuit to fold
>>> qreg = cirq.LineQubit.range(2)
>>> circ = cirq.Circuit(cirq.ops.H.on(qreg[0]), cirq.ops.CNOT.on(qreg[0], qreg[1]))
>>> print("Original circuit:", circ, sep="\n")
Original circuit:
0: ───H───@───
│
1: ───────X───

# Fold the circuit
>>> folded = fold_gates_from_left(circ, scale_factor=2.)
>>> print("Folded circuit:", folded, sep="\n")
Folded circuit:
0: ───H───H───H───@───
│
1: ───────────────X───


In this example, we see that the folded circuit has the first (Hadamard) gate folded.

Note

Mitiq folding functions do not modify the input circuit.

Because input circuits are not modified, we can reuse this circuit for the next example. In the following code, we use the fold_gates_from_right function on the same input circuit.

>>> from mitiq.zne.scaling import fold_gates_from_right

# Fold the circuit
>>> folded = fold_gates_from_right(circ, scale_factor=2.)
>>> print("Folded circuit:", folded, sep="\n")
Folded circuit:
0: ───H───@───@───@───
│   │   │
1: ───────X───X───X───


We see the second (CNOT) gate in the circuit is folded, as expected when we start folding from the right (or end) of the circuit instead of the left (or start).

Finally, we mention fold_gates_at_random which folds gates according to the following rules.

1. Gates are selected at random and folded until the input scale factor is reached.

2. No gate is folded more than once for any scale_factor <= 3.

3. “Virtual gates” (i.e., gates appearing from folding) are never folded.

All of these local folding methods can be called with any scale_factor >= 1.

### Any supported circuits can be folded¶

Any program types supported by Mitiq can be folded, and the interface for all folding functions is the same. In the following example, we fold a Qiskit circuit.

Note

This example assumes you have Qiskit installed. Mitiq can interface with Qiskit, but Qiskit is not a core Mitiq requirement and is not installed by default.

>>> import qiskit
>>> from mitiq.zne.scaling import fold_gates_from_left

# Get a circuit to fold
>>> qreg = qiskit.QuantumRegister(2)
>>> circ = qiskit.QuantumCircuit(qreg)
>>> _ = circ.h(qreg[0])
>>> _ = circ.cnot(qreg[0], qreg[1])
>>> print("Original circuit:", circ, sep="\n")
Original circuit:
┌───┐
q31_0: ┤ H ├──■──
└───┘┌─┴─┐
q31_1: ─────┤ X ├
└───┘


We can now fold this circuit as follows.

>>> folded = fold_gates_from_left(circ, scale_factor=2.)
>>> print("Folded circuit:", folded, sep="\n")
Folded circuit:
┌───┐┌───┐┌───┐
q_0: ┤ H ├┤ H ├┤ H ├──■──
└───┘└───┘└───┘┌─┴─┐
q_1: ───────────────┤ X ├
└───┘


By default, the folded circuit has the same type as the input circuit. To return an internal Mitiq representation of the folded circuit (a Cirq circuit), one can use the keyword argument return_mitiq=True.

#### Folding gates by fidelity¶

In local folding methods, gates can be folded according to custom fidelities by passing the keyword argument fidelities into a local folding method. This argument should be a dictionary where each key is a string which specifies the gate and the value of the key is the fidelity of that gate. An example is shown below where we set the fidelity of all single qubit gates to be 1.0, meaning that these gates introduce no errors in the computation.

from cirq import Circuit, LineQubit, ops
from mitiq.zne.scaling import fold_gates_at_random

qreg = LineQubit.range(3)
circ = Circuit(
ops.H.on_each(*qreg),
ops.CNOT.on(qreg[0], qreg[1]),
ops.T.on(qreg[2]),
ops.TOFFOLI.on(*qreg)
)
print(circ)
# 0: ───H───@───@───
#           │   │
# 1: ───H───X───@───
#               │
# 2: ───H───T───X───

folded = fold_gates_at_random(
circ, scale_factor=3., fidelities={"single": 1.0,
"CNOT": 0.99,
"TOFFOLI": 0.95}
)
print(folded)
# 0: ───H───@───@───@───@───@───@───
#           │   │   │   │   │   │
# 1: ───H───X───X───X───@───@───@───
#                       │   │   │
# 2: ───H───T───────────X───X───X───


We can see that only the two-qubit gates and three-qubit gates have been folded in the folded circuit.

Specific gate keys override the global “single”, “double”, or “triple” options. For example, the dictionary fidelities = {"single": 1.0, "H": 0.99} sets all single qubit gates to fidelity one except the Hadamard gate.

A full list of string keys for gates can be found with help(fold_method) where fold_method is a valid local folding method. Fidelity values must be between zero and one.

### Global folding¶

As mentioned, global folding methods fold the entire circuit instead of individual gates. An example using the same Cirq circuit above is shown below.

>>> import cirq
>>> from mitiq.zne.scaling import fold_global

# Get a circuit to fold
>>> qreg = cirq.LineQubit.range(2)
>>> circ = cirq.Circuit(cirq.ops.H.on(qreg[0]), cirq.ops.CNOT.on(qreg[0], qreg[1]))
>>> print("Original circuit:", circ, sep="\n")
Original circuit:
0: ───H───@───
│
1: ───────X───

# Fold the circuit
>>> folded = fold_global(circ, scale_factor=3.)
>>> print("Folded circuit:", folded, sep="\n")
Folded circuit:
0: ───H───@───@───H───H───@───
│   │           │
1: ───────X───X───────────X───


Notice that this circuit is still logically equivalent to the input circuit, but the global folding strategy folds the entire circuit until the input scale factor is reached. As with local folding methods, global folding can be called with any scale_factor >= 3.

### Custom folding methods¶

Custom folding methods can be defined and used with Mitiq (e.g., with mitiq.zne.execute_with_zne). The signature of this function must be as follows.

import cirq
from mitiq.zne.scaling import converter

@converter
def my_custom_folding_function(circuit: cirq.Circuit, scale_factor: float) -> cirq.Circuit:
# Insert custom folding method here
return folded_circuit


Note

The converter decorator makes it so my_custom_folding_function can be used with any supported circuit type, not just Cirq circuits. The body of the my_custom_folding_function should assume the input circuit is a Cirq circuit, however.

This function can then be used with mitiq.zne.execute_with_zne as an option to scale the noise:

# Variables circ and scale are a circuit to fold and a scale factor, respectively
zne = mitiq.zne.execute_with_zne(circuit, executor, scale_noise=my_custom_folding_function)


## Classical fitting and extrapolation: Factory Objects¶

A Factory object is a self-contained representation of an error mitigation method.

This representation is not just hardware-agnostic, it is even quantum-agnostic, in the sense that it mainly deals with classical data: the classical input and the classical output of a noisy computation. Nonetheless, a factory can easily interact with a quantum system via its self.run method which is the only interface between the “classical world” of a factory and the “quantum world” of a circuit.

The typical tasks of a factory are:

1. Record the result of the computation executed at the chosen noise level;

2. Determine the noise scale factor at which the next computation should be run;

3. Given the history of noise scale factors and results, evaluate the associated zero-noise extrapolation.

The structure of the Factory class is adaptive by construction, since the choice of the next noise level can depend on the history of these values. Obviously, non-adaptive methods are supported too and they actually represent the most common choice. Non-adaptive factories are instances of BatchedFactory objects. Adaptive factories are instances of AdaptiveFactory objects.

Specific classes derived from the abstract class Factory represent different zero-noise extrapolation methods. All the built-in factories can be found in the module mitiq.zne.inference and are summarized in the following table.

 mitiq.zne.inference.LinearFactory Factory object implementing zero-noise extrapolation based on a linear fit. mitiq.zne.inference.RichardsonFactory Factory object implementing Richardson extrapolation. mitiq.zne.inference.PolyFactory Factory object implementing a zero-noise extrapolation algorithm based on a polynomial fit. mitiq.zne.inference.ExpFactory Factory object implementing a zero-noise extrapolation algorithm assuming an exponential ansatz y(x) = a + b * exp(-c * x), with c > 0. mitiq.zne.inference.PolyExpFactory Factory object implementing a zero-noise extrapolation algorithm assuming an (almost) exponential ansatz with a non linear exponent y(x) = a + sign * exp(z(x)), where z(x) is a polynomial of a given order. mitiq.zne.inference.AdaExpFactory Factory object implementing an adaptive zero-noise extrapolation algorithm assuming an exponential ansatz y(x) = a + b * exp(-c * x), with c > 0.

Once instantiated, a factory can be passed as an argument to the high-level functions contained in the module mitiq.zne. Alternatively, a factory can be directly used to implement a zero-noise extrapolation procedure in a fully self-contained way.

To clarify this aspect, we now perform the same zero-noise extrapolation with both methods.

### Using a factory object with the mitiq.zne module¶

Let us consider an executor function which is similar to the one used in the getting started section.

import numpy as np
from cirq import Circuit, depolarize, DensityMatrixSimulator

# initialize a backend
SIMULATOR = DensityMatrixSimulator()
# 5% depolarizing noise
NOISE = 0.05

def executor(circ: Circuit) -> float:
"""Executes a circuit with depolarizing noise and
returns the expectation value of the projector |0><0|."""
circuit = circ.with_noise(depolarize(p=NOISE))
rho = SIMULATOR.simulate(circuit).final_density_matrix
obs = np.diag([1, 0])
expectation = np.real(np.trace(rho @ obs))
return expectation


Note

In this example we used Cirq but other quantum software platforms can be used, as shown in the getting started section.

We also define a simple quantum circuit whose ideal expectation value is by construction equal to 1.0.

from cirq import LineQubit, X, H

q = LineQubit(0)
circuit = Circuit(H(q), X(q), H(q), H(q), X(q), H(q))
expval = executor(circuit)
exact = 1.0
print(f"The ideal result should be {exact}")
print(f"The real result is {expval:.4f}")
print(f"The abslute error is {abs(exact - expval):.4f}")

The ideal result should be 1.0
The real result is 0.8305
The abslute error is 0.1695


Now we are going to initialize three factory objects, each one encapsulating a different zero-noise extrapolation method.

from mitiq.zne.inference import LinearFactory, RichardsonFactory, PolyFactory

# method: scale noise by 1 and 2, then extrapolate linearly to the zero noise limit.
linear_fac = LinearFactory(scale_factors=[1.0, 2.0])

# method: scale noise by 1, 2 and 3, then evaluate the Richardson extrapolation.
richardson_fac = RichardsonFactory(scale_factors=[1.0, 2.0, 3.0])

# method: scale noise by 1, 2, 3, and 4, then extrapolate quadratically to the zero noise limit.
poly_fac = PolyFactory(scale_factors=[1.0, 2.0, 3.0, 4.0], order=2)


The previous factory objects can be passed as arguments to the high-level functions in mitiq.zne. For example:

from mitiq.zne.zne import execute_with_zne

zne_expval = execute_with_zne(circuit, executor, factory=linear_fac)
print(f"Error with linear_fac: {abs(exact - zne_expval):.4f}")

zne_expval = execute_with_zne(circuit, executor, factory=richardson_fac)
print(f"Error with richardson_fac: {abs(exact - zne_expval):.4f}")

zne_expval = execute_with_zne(circuit, executor, factory=poly_fac)
print(f"Error with poly_fac: {abs(exact - zne_expval):.4f}")

Error with linear_fac: 0.0575
Error with richardson_fac: 0.0195
Error with poly_fac: 0.0291


We can also specify the number of shots to use for each noise-scaled circuit.

from mitiq.zne.inference import LinearFactory

# Specify the number of shots for each scale factor.
factory_with_shots = LinearFactory(scale_factors=[1.0, 2.0], shot_list=[100, 200])


In this case the factory will pass the number of shots from the shot_list to the executor. Accordingly, the executor should support a shots keyword argument, otherwise the shot values will go unused.

### Using batched executors with BatchedFactory¶

As mentioned, BatchedFactory objects are such that all circuits to execute can be precomputed. This is in contrast to AdapativeFactory objects in which the next circuit to execute depends on the result of the previous circuit execution.

If the quantum processor is costly to access (e.g., in a queue-based system), executing circuits sequentially can result in high runtimes for zero-noise extrapolation. To deal with this, all classical inference techniques which inherit from a BatchedFactory can use a “batched executor.” In contrast to the previous executor which inputs a single circuit and outputs a single expectation value, a batched executor inputs a list of circuits and outputs a list of expectation values (one for each circuit).

To indicate that an executor is batched, one must provide a return annotation which is either a numpy.ndarray, List[float], Tuple[float], Sequence[float], or Iterable[float]. For example:

from typing import List
from mitiq import QPROGRAM

def batched_executor(circuits: List[QPROGRAM]) -> List[float]:
pass


A :class::.BatchedFactory will detect from the return annotation if an executor is batched or not. If no annotation is provided, the executor is assumed to be sequential (i.e., not batched).

### Directly using a factory for error mitigation¶

Zero-noise extrapolation can also be implemented by directly using the methods self.run and self.reduce of a Factory object.

The method self.run evaluates different expectation values at different noise levels until a sufficient amount of data is collected.

The method self.reduce instead returns the final zero-noise extrapolation which, in practice, corresponds to a statistical inference based on the measured data.

# we import one of the built-in noise scaling function
from mitiq.zne.scaling import fold_gates_at_random

linear_fac.run(circuit, executor, scale_noise=fold_gates_at_random)
zne_expval = linear_fac.reduce()
print(f"Error with linear_fac: {abs(exact - zne_expval):.4f}")

richardson_fac.run(circuit, executor, scale_noise=fold_gates_at_random)
zne_expval = richardson_fac.reduce()
print(f"Error with richardson_fac: {abs(exact - zne_expval):.4f}")

poly_fac.run(circuit, executor, scale_noise=fold_gates_at_random)
zne_expval = poly_fac.reduce()
print(f"Error with poly_fac: {abs(exact - zne_expval):.4f}")

Error with linear_fac: 0.0575
Error with richardson_fac: 0.0195
Error with poly_fac: 0.0291


Behind the scenes, a factory object collects different expectation values at different scale factors. After running a factory, this information can be accessed with appropriate get methods. For example:

scale_factors = poly_fac.get_scale_factors()
print("Scale factors:", scale_factors)
exp_values = poly_fac.get_expectation_values()
print("Expectation values:", np.round(exp_values, 2))

Scale factors: [1. 2. 3. 4.]
Expectation values: [0.83 0.72 0.64 0.6 ]


If the user has manually evaluated a list of expectation values associated to a list of scale factors, the simplest way to estimate the corresponding zero-noise limit is to directly call the static extrapolate method of the desired factory class (in this case initializing a factory object is unnecessary). For example:

zero_limit = PolyFactory.extrapolate(scale_factors, exp_values, order=2)
print(f"Error with PolyFactory.extrapolate method: {abs(exact - zero_limit):.4f}")

Error with PolyFactory.extrapolate method: 0.0291


Beyond the zero-noise limit, additional information about the fit (e.g., optimal parameters, errors, extrapolation curve, etc.) can be returned from extrapolate by specifying full_output = True.

There are also a number of methods to get additional information calculated by the factory class:

from mitiq.zne.inference import LinearFactory
from mitiq.zne.zne import execute_with_zne

fac = LinearFactory(scale_factors=[1.0, 2.0, 3.0])
_ = execute_with_zne(circuit, executor, factory=fac)
print(f"Zero-noise limit: {fac.get_zero_noise_limit():.4f}")
print(f"Fit error on zero-noise limit: {fac.get_zero_noise_limit_error():.4f}")
print(f"Covariance of fitted model parameters: {np.round(fac.get_parameters_covariance(), 5)}")
print(f"Fitted model parameters: {np.round(fac.get_optimal_parameters(), 4)}")
print(f"Extrapolation curve evaluated at zero: {fac.get_extrapolation_curve()(0):.4f}")

Zero-noise limit: 0.9172
Fit error on zero-noise limit: 0.0237
Covariance of fitted model parameters: [[ 0.00012 -0.00024]
[-0.00024  0.00056]]
Fitted model parameters: [-0.093   0.9172]
Extrapolation curve evaluated at zero: 0.9172


### Advanced usage of a factory¶

Note

This section can be safely skipped by all the readers who are interested in a standard usage of Mitiq. On the other hand, more experienced users and Mitiq contributors may find this content useful to understand how a factory object actually works at a deeper level.

In this advanced section we present a low-level usage of a factory and we also discuss how the user can easily define a custom factory class.

#### Low-level usage: the run_classical method.¶

The self.run method takes as arguments a circuit and other “quantum” objects. On the other hand, the core computation performed by any factory corresponds to a some classical computation applied to the measurement results.

At a lower level, it is possible to clearly separate the quantum and the classical steps of a zero-noise extrapolation procedure. This can be done by defining a function which maps a noise scale factor to the corresponding expectation value.

def noise_to_expval(scale_factor: float) -> float:
"""Function returning an expectation value for a given scale_factor."""
# apply noise scaling
scaled_circuit = fold_gates_at_random(circuit, scale_factor)
# return the corresponding expectation value
return executor(scaled_circuit)


Note

The body of the previous function contains the execution of a quantum circuit. However, if we see it as a “black-box”, it is just a classical function mapping real numbers to real numbers.

The function noise_to_expval encapsulate the “quantum part” of the problem. The “classical part” of the problem can be solved by passing noise_to_expval to the self.run_classical method of a factory. This method will repeatedly call noise_to_expval for different noise levels, so one can view self.run_classical as the classical counterpart of the quantum method self.run.

linear_fac.run_classical(noise_to_expval)
zne_expval = linear_fac.reduce()
print(f"Error with linear_fac: {abs(exact - zne_expval):.4f}")

richardson_fac.run_classical(noise_to_expval)
zne_expval = richardson_fac.reduce()
print(f"Error with richardson_fac: {abs(exact - zne_expval):.4f}")

poly_fac.run_classical(noise_to_expval)
zne_expval = poly_fac.reduce()
print(f"Error with poly_fac: {abs(exact - zne_expval):.4f}")

Error with linear_fac: 0.0575
Error with richardson_fac: 0.0195
Error with poly_fac: 0.0291


Note

With respect to self.run, the self.run_classical method is much more flexible and can be applied whenever the user is able to autonomously scale the noise level associated to an expectation value. Indeed, the function noise_to_expval can represent any experiment or any simulation in which noise can be artificially increased. The scenario is therefore not restricted to quantum circuits but can be easily extended to annealing devices or to gates which are controllable at a pulse level. In principle, one could even use the self.run_classical method to mitigate experiments which are unrelated to quantum computing.

#### Defining a custom factory¶

If necessary, the user can modify an existing extrapolation methods by subclassing one of the built-in factories.

Alternatively, a new adaptive extrapolation method can be derived from the abstract class Factory. In this case its core methods must be implemented: self.next, self.push, self.is_converged, self.reduce, etc. Typically, the self.__init__ method must be overridden.

A new non-adaptive method can instead be derived from the abstract BatchedFactory class. In this case it is usually sufficient to override only the self.__init__ and the self.extrapolate methods, which are responsible for the initialization and for the final zero-noise extrapolation, respectively.

### Example: a simple custom factory¶

Assume that, from physical considerations, we know that the ideal expectation value (measured by some quantum circuit) must always be within two limits: min_expval and max_expval. For example, this is a typical situation whenever the measured observable has a bounded spectrum.

We can define a linear non-adaptive factory which takes into account this information and clips the result if it falls outside its physical domain.

from mitiq.zne.inference import BatchedFactory, LinearFactory
import numpy as np

class MyFactory(BatchedFactory):
"""Factory object implementing a linear extrapolation taking
into account that the expectation value must be within a given
interval. If the zero-noise limit falls outside the
interval, its value is clipped.
"""

def __init__(self, scale_factors, min_expval, max_expval):
"""
Args:
scale_factors: The noise scale factors at which
expectation values should be measured.
min_expval: The lower bound for the expectation value.
min_expval: The upper bound for the expectation value.
"""
super(MyFactory, self).__init__(scale_factors)
self._options = {"min_expval": min_expval, "max_expval": max_expval}

@staticmethod
def extrapolate(
scale_factors, exp_values, min_expval, max_expval, full_output = False,
):
"""Fit a linear model and clip its zero-noise limit."""

# Perform standard linear extrapolation
result = LinearFactory.extrapolate(
scale_factors, exp_values, full_output,
)

# Return the clipped zero-noise extrapolation.
if not full_output:
return np.clip(result, min_expval, max_expval)

if full_output:
# In this case "result" is a tuple of extrapolation data
zne_limit = np.clip(result[0], min_expval, max_expval)
return (zne_limit, *result[1:])


This custom factory can be used in exactly the same way as we have shown in the previous section. By simply replacing LinearFactory with MyFactory in all the previous code snippets, the new extrapolation method will be applied.

### Regression tools in mitiq.zne.inference¶

In the body of the previous MyFactory example, we imported and used the mitiq_polyfit() function. This is simply a wrap of numpy.polyfit(), slightly adapted to the notion and to the error types of Mitiq. This function can be used to fit a polynomial ansatz to the measured expectation values. This function performs a least squares minimization which is linear (with respect to the coefficients) and therefore admits an algebraic solution.

Similarly, from mitiq.zne.inference one can also import mitiq_curve_fit(), which is instead a wrap of scipy.optimize.curve_fit(). Differently from mitiq_polyfit(), mitiq_curve_fit() can be used with a generic (user-defined) ansatz. Since the fit is based on a numerical non-linear least squares minimization, this method may fail to converge or could be subject to numerical instabilities.