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Scalable error mitigation for noisy quantum circuits produces competitive expectation values

Youngseok Kim, Christopher J. Wood, Theodore J. Yoder, Seth T. Merkel, Jay M. Gambetta, Kristan Temme, Abhinav Kandala

arXiv:2108.09197v1quant-phcond-mat.mes-hall

TL;DR

Coherent errors can make noise scaling inaccurate and even produce unphysical mitigated expectation values. The paper combines Pauli twirling with zero-noise extrapolation and circuit-time reduction strategies, finding that eight twirled instances suffice in one native-decomposition quench setting, while weight-2 observables do not show a definitive advantage over PEPS.

Abstract

from arXiv · show

Noise in existing quantum processors only enables an approximation to ideal quantum computation. However, these approximations can be vastly improved by error mitigation, for the computation of expectation values, as shown by small-scale experimental demonstrations. However, the practical scaling of these methods to larger system sizes remains unknown. Here, we demonstrate the utility of zero-noise extrapolation for relevant quantum circuits using up to 26 qubits, circuit depths of 60, and 1080 CNOT gates. We study the scaling of the method for canonical examples of product states and entangling Clifford circuits of increasing size, and extend it to the quench dynamics of 2-D Ising spin lattices with varying couplings. We show that the efficacy of the error mitigation is greatly enhanced by additional error suppression techniques and native gate decomposition that reduce the circuit time. By combining these methods, we demonstrate an accuracy in the approximate quantum simulation of the quench dynamics that surpasses the classical approximations obtained from a state-of-the-art 2-D tensor network method. These results reveal a path to a relevant quantum advantage with noisy, digital, quantum processors.

Supplementary Information: Scalable error mitigation for noisy quantum circuits

The supplementary material is titled “Scalable error mitigation for noisy quantum circuits.”

DEVICE CHARACTERISTICS

The experiments use a 27-qubit heavy-hex superconducting processor with microwave control and cross-resonance-based entangling gates. CNOTs and fractional RZZ gates are constructed from calibrated RZX interactions.

DEVICE CHARACTERISTICS

SII. EXPERIMENTAL CONSIDERATIONS FOR OPTIMAL PERFORMANCE OF ERROR MITIGATION

Error-mitigation performance is limited by circuit noise, motivating both shorter gate decompositions and experimental error-suppression measures.

SII. EXPERIMENTAL CONSIDERATIONS FOR OPTIMAL PERFORMANCE OF ERROR MITIGATION

A. Dynamical decoupling

Dynamical decoupling is considered for GHZ and quench circuits because their connectivity and structure create idling periods that can introduce dephasing and coherent ZZ-crosstalk errors.

A. Dynamical decoupling

B. Pauli twirling

Pauli twirling suppresses coherent errors that can distort noise scaling in quench circuits, complementing dynamical decoupling and enabling effective mitigation with few randomized instances.

Pauli twirling

  • Pauli twirling is important after dynamical decoupling because residual coherent errors can produce inaccurate noise scaling and unphysical mitigated expectation values.This issue is demonstrated for post-DD quench circuits with J = 0.1.Source: B. Pauli twirling · p. 10

Dynamical decoupling

  • Dynamical decoupling extends coherent evolution and refocuses coherent error in 26-qubit quench circuits at J = 0.1.The examined RX(π) −RX(−π) sequence showed no discernible improvement from adding more decoupling pulses.Source: Figure S2 · p. 10

Pauli twirling

  • Pauli twirling converts arbitrary noise into a stochastic Pauli channel by suppressing off-diagonal coherent-error contributions.For CNOT-decomposed RZZ operations, twirling gates are sampled per CNOT and averaged across instances without increasing circuit time.Source: B. Pauli twirling · p. 11

Native gate decomposition

  • For native RZZ(θ) decomposition, twirling uses the commuting subset GZZ = {[I, I], [X, X], [Y, Y], [Z, Z]} on each control-target pair.The same sampled twirling gates are applied before and after RZZ(θ).Source: B. Pauli twirling · p. 11

Pauli twirling

  • 8 twirled circuit instances suffice for the native decomposition and J = 0.5236, with no further Euclidean-distance improvement beyond that number.The comparison concerns the error-mitigated magnetization distance from the ideal vector.Source: B. Pauli twirling · p. 11

C. Noise model dependence of the identity insertion noise amplification method

Identity insertion amplifies noise by repeating identity-equivalent gates, but the intended scaling depends on the noise model and can fail for non-uniform Pauli channels.

Identity insertion

  • Odd repetitions of identity-equivalent gates, such as 2k + 1 CNOTs, target noise amplification by a factor of 2k + 1.The method is easier to implement than precisely calibrated gate stretching because it operates directly at the circuit level.Source: C. Noise model dependence of the… · pp. 11–12

Depolarizing noise

  • For a uniform two-qubit depolarizing channel, repeating a noisy CNOT produces the desired amplification from λ to (2k + 1)λ.The depolarizing factor changes as (1 − ϵ)^(2k+1).Source: C. Noise model dependence of the… · p. 11

Pauli noise

  • For Pauli channels, Pauli matrices form an eigenbasis with channel eigenvalues µα, while a CNOT maps Pα to Pαcx.This transformation changes which eigenvalue governs an observable after the gate.Source: C. Noise model dependence of the… · p. 12

Noise-model dependence

  • Identity insertion yields the correct amplification only when the relevant Pauli-channel eigenvalues satisfy the required transformed-eigenvalue relation.The paper contrasts the desired µαcx → µαcx^3 scaling with the result generated by the insertion trick.Source: C. Noise model dependence of the… · p. 12
  • Slightly more complicated noise models can make identity insertion produce uncontrolled zero-noise extrapolation estimates.The paper illustrates this using a Pauli-error generalization of the depolarizing channel.Source: C. Noise model dependence of the… · pp. 12–13

D. Two level systems

Resonant interactions with defect two-level systems create coherence and performance fluctuations that challenge stable error-mitigation experiments.

Two-level systems

  • Resonant interactions between qubits and defect two-level systems fluctuate coherence and system performance.The study maps the TLS spectral environment and its dynamics near the qubit frequency through energy-relaxation measurements.Source: D. Two level systems · p. 13

SIII. LOCAL WEIGHT-2 OBSERVABLES IN QUENCH DYNAMICS OF 2D ISING SPIN LATTICES

This section evaluates local weight-2 ⟨ZZ⟩ observables during 2-D Ising quench dynamics and compares error-mitigated experiments with PEPS approximations. For these observables, the experiment does not show the definitive advantage observed for weight-1 observables.

Observable and error definition

  • The averaged local ⟨ZZ⟩ observable is computed over adjacent qubit pairs, with Nadj = 27 for the 26-qubit layout.The observable is defined as the average of ⟨ZiZj⟩ over adjacent pairs.Source: SIII. LOCAL WEIGHT-2 OBSERVABLES IN… · p. 14
  • The relative error is emethod_avg = |⟨ZZ⟩ideal − ⟨ZZ⟩method|/|⟨ZZ⟩ideal|, where the method is PEPS or the error-mitigated experiment.This metric is used to compare both approximation methods against exact numerical results.Source: SIII. LOCAL WEIGHT-2 OBSERVABLES IN… · p. 14

Comparison with PEPS

SIV. INDIVIDUAL QUBIT RESULTS FOR QUENCH DYNAMICS OF A 2-D ISING SPIN LATTICE

The supplementary quench results report individual-qubit ⟨Z⟩ and adjacent-pair ⟨ZZ⟩ measurements using dynamical decoupling, random twirling, and linear zero-noise extrapolation. Each observable is measured from 100,000 shots across stretch factors c = 1, 1.6, 2.

Individual-qubit ⟨Z⟩ results

  • Individual-qubit ⟨Z⟩ evolution is measured with 100,000 shots at each stretch factor for J = 0.5236.The qubit indices follow the labels in Fig. 3(a).Source: Figure S7 · p. 15

Error suppression and extrapolation

  • Both measurements use dynamical decoupling for idling qubits and eight random twirling gate instances.These error-suppression settings are applied in the individual-observable quench experiments.Source: Figure S7 · p. 15, Figure S8 · p. 16
  • Both measurements apply linear extrapolation using stretch factors c = 1, 1.6, 2.The figures also compare the measurements with exact numerical results and numerical approximations.Source: Figure S7 · p. 15, Figure S8 · p. 16

Local ⟨ZZ⟩ results

  • Adjacent-pair local ⟨ZZ⟩ observables are measured with 100,000 shots for J = 0.5236.The qubit-pair indices follow the labels in Fig. 3(a).Source: Figure S8 · p. 16

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