How to generate exact sampling coefficients

This how-to guide is intended to show users how they can generate exact sampling coefficients to be used in reconstructing the simulated expectation value of the original circuit.

First, we set up a simple cutting problem following the first tutorial.

[1]:

import numpy as np
from qiskit.circuit.library import EfficientSU2
from qiskit.quantum_info import SparsePauliOp

from circuit_knitting.cutting import (
    partition_problem,
    generate_cutting_experiments,
)

[2]:

circuit = EfficientSU2(4, entanglement="linear", reps=2).decompose()
circuit.assign_parameters([0.8] * len(circuit.parameters), inplace=True)
observable = SparsePauliOp(["ZZZZ"])
circuit.draw("mpl", scale=0.8)

[2]:
../../_images/circuit_cutting_how-tos_how_to_generate_exact_sampling_coefficients_2_0.png

Partition the circuit between qubits 1 and 2 by cutting 2 CNOT gates.

[3]:

partitioned_problem = partition_problem(
    circuit=circuit, partition_labels="AABB", observables=observable.paulis
)
subcircuits = partitioned_problem.subcircuits
bases = partitioned_problem.bases
subobservables = partitioned_problem.subobservables
subcircuits["A"].draw("mpl", scale=0.6)

[3]:
../../_images/circuit_cutting_how-tos_how_to_generate_exact_sampling_coefficients_4_0.png 
[4]:

subcircuits["B"].draw("mpl", scale=0.6)

[4]:
../../_images/circuit_cutting_how-tos_how_to_generate_exact_sampling_coefficients_5_0.png

Demonstrate how to obtain all weights exactly

If you wish to calculate all weights exactly, no matter how small, you can achieve this by passing infinity (np.inf) to num_samples:

[5]:

subexperiments, coefficients = generate_cutting_experiments(
    circuits=subcircuits,
    observables=subobservables,
    num_samples=np.inf,
)
coefficients

[5]:

[(0.24999999999999992, <WeightType.EXACT: 1>),
 (0.24999999999999992, <WeightType.EXACT: 1>),
 (0.24999999999999992, <WeightType.EXACT: 1>),
 (-0.24999999999999992, <WeightType.EXACT: 1>),
 (0.24999999999999992, <WeightType.EXACT: 1>),
 (-0.24999999999999992, <WeightType.EXACT: 1>),
 (0.24999999999999992, <WeightType.EXACT: 1>),
 (0.24999999999999992, <WeightType.EXACT: 1>),
 (0.24999999999999992, <WeightType.EXACT: 1>),
 (-0.24999999999999992, <WeightType.EXACT: 1>),
 (0.24999999999999992, <WeightType.EXACT: 1>),
 (-0.24999999999999992, <WeightType.EXACT: 1>),
 (0.24999999999999992, <WeightType.EXACT: 1>),
 (0.24999999999999992, <WeightType.EXACT: 1>),
 (0.24999999999999992, <WeightType.EXACT: 1>),
 (-0.24999999999999992, <WeightType.EXACT: 1>),
 (0.24999999999999992, <WeightType.EXACT: 1>),
 (-0.24999999999999992, <WeightType.EXACT: 1>),
 (-0.24999999999999992, <WeightType.EXACT: 1>),
 (-0.24999999999999992, <WeightType.EXACT: 1>),
 (-0.24999999999999992, <WeightType.EXACT: 1>),
 (0.24999999999999992, <WeightType.EXACT: 1>),
 (-0.24999999999999992, <WeightType.EXACT: 1>),
 (0.24999999999999992, <WeightType.EXACT: 1>),
 (0.24999999999999992, <WeightType.EXACT: 1>),
 (0.24999999999999992, <WeightType.EXACT: 1>),
 (0.24999999999999992, <WeightType.EXACT: 1>),
 (-0.24999999999999992, <WeightType.EXACT: 1>),
 (0.24999999999999992, <WeightType.EXACT: 1>),
 (-0.24999999999999992, <WeightType.EXACT: 1>),
 (-0.24999999999999992, <WeightType.EXACT: 1>),
 (-0.24999999999999992, <WeightType.EXACT: 1>),
 (-0.24999999999999992, <WeightType.EXACT: 1>),
 (0.24999999999999992, <WeightType.EXACT: 1>),
 (-0.24999999999999992, <WeightType.EXACT: 1>),
 (0.24999999999999992, <WeightType.EXACT: 1>)]

Demonstrate how to find the minimum num_samples needed to retrieve all exact weights for 2 CNOT cuts

When num_samples is set to a finite number, each weight whose absolute value is above a threshold of 1 / num_samples will be evaluated exactly. The remaining weights – those in the tail of the distribution – will then be sampled from, resulting in at most num_samples unique weights.

In the case of a CNOT gate – or any gate equivalent to it up to single-qubit unitaries – each of the six weights of the quasi-probability decomposition have the same magnitude, so each gets sampled with a probability of \(1/6\):

[6]:

print(f"Mapping probabilities for a CNOT decomposition: {bases[0].probabilities}")

Mapping probabilities for a CNOT decomposition: [0.16666667 0.16666667 0.16666667 0.16666667 0.16666667 0.16666667]

In this example, we have cut two CNOT gates. Given that the probability of any given mapping in a CNOT decomposition is \(1/6\), the probability of any given mapping in the joint distribution combining the two cut CNOT gates is \((1/6)^2\). Therefore, we need to take at least \(6^2\) weights in order to retrieve all exact weights from generate_cutting_experiments.

[7]:

from circuit_knitting.cutting.qpd import QPDBasis
from qiskit.circuit.library.standard_gates import CXGate

qpd_basis_cx = QPDBasis.from_instruction(CXGate())


def _min_nonzero(seq, /):
    """Return the minimum value in a sequence, ignoring values near zero."""
    return min(x for x in seq if not np.isclose(x, 0))


num_cx_cuts = 2

print(
    f"Number of samples needed to retrieve exact weights: {1 / _min_nonzero(qpd_basis_cx.probabilities)**num_cx_cuts}"
)

Number of samples needed to retrieve exact weights: 36.0

Observe the coefficient weights returned from generate_cutting_experiments are WeightType.EXACT

Above, we determined 36 samples would trigger the coefficients to be returned as exact. Here we set num_samples to exactly 36 to test this.

[8]:

subexperiments, coefficients = generate_cutting_experiments(
    circuits=subcircuits,
    observables=subobservables,
    num_samples=36,
)
coefficients

[8]:

[(0.25, <WeightType.EXACT: 1>),
 (0.25, <WeightType.EXACT: 1>),
 (0.25, <WeightType.EXACT: 1>),
 (-0.25, <WeightType.EXACT: 1>),
 (0.25, <WeightType.EXACT: 1>),
 (-0.25, <WeightType.EXACT: 1>),
 (0.25, <WeightType.EXACT: 1>),
 (0.25, <WeightType.EXACT: 1>),
 (0.25, <WeightType.EXACT: 1>),
 (-0.25, <WeightType.EXACT: 1>),
 (0.25, <WeightType.EXACT: 1>),
 (-0.25, <WeightType.EXACT: 1>),
 (0.25, <WeightType.EXACT: 1>),
 (0.25, <WeightType.EXACT: 1>),
 (0.25, <WeightType.EXACT: 1>),
 (-0.25, <WeightType.EXACT: 1>),
 (0.25, <WeightType.EXACT: 1>),
 (-0.25, <WeightType.EXACT: 1>),
 (-0.25, <WeightType.EXACT: 1>),
 (-0.25, <WeightType.EXACT: 1>),
 (-0.25, <WeightType.EXACT: 1>),
 (0.25, <WeightType.EXACT: 1>),
 (-0.25, <WeightType.EXACT: 1>),
 (0.25, <WeightType.EXACT: 1>),
 (0.25, <WeightType.EXACT: 1>),
 (0.25, <WeightType.EXACT: 1>),
 (0.25, <WeightType.EXACT: 1>),
 (-0.25, <WeightType.EXACT: 1>),
 (0.25, <WeightType.EXACT: 1>),
 (-0.25, <WeightType.EXACT: 1>),
 (-0.25, <WeightType.EXACT: 1>),
 (-0.25, <WeightType.EXACT: 1>),
 (-0.25, <WeightType.EXACT: 1>),
 (0.25, <WeightType.EXACT: 1>),
 (-0.25, <WeightType.EXACT: 1>),
 (0.25, <WeightType.EXACT: 1>)]