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Accounting for Errors in Quantum Algorithms via Individual Error Reduction

Matthew Otten, Stephen Gray

arXiv:1804.06969v1quant-ph

TL;DR

Near-term quantum computers cannot yet provide full fault-tolerant correction, so the paper develops a lower-overhead way to reduce errors in measured observables. It measures with all noise active and with each source separately reduced, then applies the corrections to VQE simulations. The approach achieves chemical accuracy at error rates up to two orders of magnitude larger in the reported simulations, with only O(nqubits) additional evaluations.

  • Problem

    Near-term quantum computers have limited, noisy qubits, while full fault-tolerant error correction is still too expensive or distant for these systems.

    Source: INTRODUCTION · p. 1

  • Method

    Measure an observable with all errors active and with each error source removed separately, then sum the resulting corrections using limited single-qubit correction.

    Source: INTRODUCTION · p. 1, RESULTS · pp. 2–3

  • Results

    Error rates can be up to two orders of magnitude larger while achieving chemical accuracy in the reported VQE simulations.

    Source: INTRODUCTION · p. 1, CONCLUSION · p. 5

  • Takeaways & Limitations

    The scheme offers significantly more accurate near-term quantum calculations without requiring full error correction, with only O(nqubits) additional evaluations.

    Source: RESULTS · p. 3, CONCLUSION · p. 5

    The correction relies on correctly understanding how each noise source is removed; correlated terms removed twice require halving the calculated correction.

    Source: RESULTS · p. 4

Abstract

from arXiv · show

We discuss a surprisingly simple scheme for accounting (and removal) of error in observables determined from quantum algorithms. A correction to the value of the observable is calculated by first measuring the observable with all error sources active and subsequently measuring the observable with each error source removed separately. We apply this scheme to the variational quantum eigensolver, simulating the calculation of the ground state energy of equilibrium H$_2$ and LiH in the presence of several noise sources, including amplitude damping, dephasing, thermal noise, and correlated noise. We show that this scheme provides a decrease in the needed quality of the qubits by up to two orders of magnitude. In near-term quantum computers, where full fault-tolerant error correction is too expensive, this scheme provides a route to significantly more accurate calculation

INTRODUCTION

Near-term quantum computers are noisy and unlikely to support full fault-tolerant correction soon, motivating methods that reduce errors with limited correction. The paper presents a scheme that removes each noise source separately and demonstrates it on VQE calculations for H2 and LiH.

  • Near-term quantum computers will have limited, noisy qubits before full fault-tolerant error correction becomes available.Source: INTRODUCTION · p. 1
  • The scheme removes each noise source separately and sums the resulting corrections to reduce errors in measured observables.Source: INTRODUCTION · p. 1
  • The method uses single-qubit correction within VQE simulations of ground-state energies for H2 and LiH.Source: INTRODUCTION · p. 1
  • The simulations include amplitude damping, dephasing, thermal noise, and correlated noise, and the scheme also applies to other repeatedly measured observables.Source: INTRODUCTION · p. 1

Time Evolution and Noise Modeling

The model represents gate operations as unitary jumps separated by Lindblad evolution, with separate terms encoding distinct error sources. It simulates amplitude damping, dephasing, thermal, and correlated noise while assuming one qubit’s noise can be strongly reduced.

Time Evolution and Noise Modeling

Variational Quantum Eigensolver

VQE optimizes a parameterized wavefunction’s energy classically while evaluating the ansatz on a quantum computer. For quantum chemistry, the method uses qubit Hamiltonians and a UCCSD ansatz with short circuits and noise robustness.

Variational Quantum Eigensolver

RESULTS

The scheme corrects noisy observables by measuring with all noise active and then removing each noise source separately. In VQE simulations, it substantially improves H2 energy estimates across environmental and correlated noise sources.

Method

  • The correction combines the fully noisy observable with measurements where each Lindblad noise source is removed separately.For VQE, the procedure requires the original energy evaluation plus O(n) single-qubit error-corrected evaluations.Source: RESULTS · pp. 2–3
  • The corrected observable approximates the error-free value without first-order errors in the gate interval, whereas the uncorrected observable has first-order errors.This cancellation arises from subtracting contributions isolated by separately removing noise sources.Source: RESULTS · p. 2

H2

  • The H2 study used a four-qubit UCCSD circuit at equilibrium bond length 0.74 Å, with noise optimized on every qubit before single-qubit removal sweeps.The simulations included amplitude damping and dephasing across regimes associated with different quantum-computing architectures.Source: RESULTS · p. 3
  • 45× larger error rates still reached chemical accuracy for H2, with corrections of 60–70 mHa.The scheme continually moved results closer to the fully error-corrected answer across tested rates.Source: RESULTS · p. 3

Additional noise sources

  • The scheme applies beyond amplitude damping and dephasing to thermal and correlated noise when each source can be isolated and removed independently.Thermal-noise corrections of around 70 mHa brought H2 within chemical accuracy at error rates nearly 50 times larger than otherwise needed.Source: RESULTS · pp. 3–4

Correlated noise

  • For correlated noise involving two qubits, removing each qubit separately double-counts every correlated term, so the calculated correction is halved.The method depends on knowing how often each noise term is removed; controlled partial removal can instead be rescaled.Source: RESULTS · p. 4

LiH

LiH provides a larger-circuit test of the scheme using 12 qubits and more than 12,000 gates. Corrections of 100–200 mHa achieved chemical accuracy while tolerating error rates over two orders of magnitude higher.

LiH results

  • Error rates up to 128 times higher still reached chemical accuracy for LiH, with corrections of 100–200 mHa.The reported factors ranged from 68 for amplitude damping alone to 128 for dephasing alone.Source: LiH · p. 4

Scaling

  • The authors report that the correction works even better for LiH than for the smaller H2 circuit as circuit size increases.They relate this expectation to the growing number of approximately corrected first-order errors, while second-order errors scale as γ^2 and τ^2.Source: LiH · pp. 4–5

LiH setup

  • The LiH simulation used the UCCSD ansatz under amplitude damping, dephasing, and their combination, with chemical accuracy defined as 1.6 mHa.The figure compares correction magnitudes and residual differences from the noiseless energy across these noise conditions.Source: Figure 3 · p. 4, LiH · p. 4

CONCLUSION

The scheme reduces quantum-algorithm error without full error correction by correcting each qubit separately and combining the resulting observable differences. It improves chemical-accuracy calculations at low overhead and applies beyond VQE to any repeatedly measured observable.

CONCLUSION

  • Error rates can be up to two orders of magnitude larger while still achieving chemical accuracy, without full error correction.The demonstrated correction greatly reduces the coherence requirements for near-term quantum computations.Source: CONCLUSION · p. 5
  • The method requires only an additional O(n_qubits) evaluations with single-qubit error correction, plus the original uncorrected calculation.Its correction relies on cancellation between expectation values, so each value must be measured sufficiently accurately.Source: CONCLUSION · p. 5
  • The approach can reduce error in any measured observable, although further study is needed for algorithms beyond VQE.The paper specifically identifies quantum phase estimation and quantum approximate optimization as future applications.Source: CONCLUSION · p. 5
  • The correction magnitude can indicate proximity to the true answer and compare architectures without knowing that answer.A smaller correction is presented as an indicator of smaller environmental-noise effects.Source: CONCLUSION · p. 5

Derivation of the Observable Correction Formula

The derivation constructs noisy and selectively error-removed evolutions, then combines their observable values so first-order error terms cancel. The resulting corrected observable is accurate through first order in the gate interval, leaving errors of order τ^2 and higher.

Derivation of the Observable Correction Formula

  • The evolution sequence applies each gate followed by Lindblad evolution, with the resulting density matrix determined by the ordered gate and noise operations.The evolution operators are not symmetric because the noise propagators act consistently on one side of the gate products.Source: Derivation of the Observable… · p. 5
  • The noisy evolution contains all error sources, while separate evolutions remove one error source at a time for comparison.The formalism represents noise with Lindblad evolution between successive gate operations.Source: Derivation of the Observable… · p. 5
  • The corrected density matrix cancels all first-order error terms, leaving remaining errors of order τ^2 and higher.The uncorrected density matrix retains first-order error terms.Source: Derivation of the Observable… · p. 6
  • Tracing the corrected density matrix with observable A yields a corrected expectation value that is accurate to order τ, unlike the uncorrected observable.The observable formula follows directly from the density-matrix correction construction.Source: Derivation of the Observable… · pp. 5–6

Entangling Ansatz for H2

The entangling ansatz uses layered ring entanglement and produces H2 error-cancellation results nearly matching UCCSD. Corrections reach chemical accuracy while tolerating substantially larger error rates.

Entangling Ansatz for H2

  • The ansatz begins with parameterized rotations and alternates ring-wide Hadamard-CNOT entangling layers with parameterized rotations.The reported H2 results use four layers of this circuit.Source: Entangling Ansatz for H2 · p. 6
  • Figure 5 compares correction amounts and residual corrected-energy differences under amplitude damping, dephasing, and combined noise.Triangle-marked dashed lines show corrections, while circle-marked solid lines show differences from the noiseless energy.Source: Entangling Ansatz for H2 · p. 6, Figure 5 · p. 6
  • For the entangling circuit, four-layer results are almost indistinguishable from UCCSD results, with corrections of up to 70 mHa applied to reach chemical accuracy.Source: Entangling Ansatz for H2 · p. 6
  • Corrections of up to 70 mHa bring the four-layer entangling-ansatz H2 calculation to chemical accuracy, with error rates up to 50× larger.The comparison uses chemical accuracy of 1.6 mHa as the target.Source: Entangling Ansatz for H2 · p. 6, Figure 5 · p. 6
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