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Decoding quantum errors with subspace expansions

Jarrod R. McClean, Zhang Jiang, Nicholas C. Rubin, Ryan Babbush, Hartmut Neven

arXiv:1903.05786v1quant-phphysics.chem-ph

TL;DR

Near-term quantum devices may need error mitigation before fully fault-tolerant computation, while conventional code experiments face demanding syndrome measurements and feedback. The paper uses post-processing with existing codes and subspace expansions, reporting a p ≈ 0.50 pseudo-threshold for the [[5, 1, 3]] code and improved hydrogen-molecule results across depolarizing strengths.

  • Problem

    Near-term devices may precede fully fault-tolerant quantum computers, and complicated syndrome measurements and fast feedback make experimental study of many codes difficult.

    Source: §I · p. 1

  • Method

    The paper constructs post-processing projectors from stabilizer codes, extends them with subspace expansions, and uses stochastic sampling to mitigate errors without direct syndrome measurement.

    Source: §I · p. 2, §II · p. 3

  • Results

    The [[5, 1, 3]] code has a p = 0.50 pseudo-threshold under symmetric depolarizing noise, and hydrogen-molecule logical infidelity improves across the full range of depolarizing strengths.

    Source: A. [[5, 1, 3]] code · p. 7, B. Hydrogen Molecule · p. 8

  • Takeaways & Limitations

    Existing quantum codes can be used with classical post-processing and additional measurements to mitigate errors and study code performance under experimental conditions.

    Source: §I · p. 2

Abstract

from arXiv · show

With the rapid developments in quantum hardware comes a push towards the first practical applications on these devices. While fully fault-tolerant quantum computers may still be years away, one may ask if there exist intermediate forms of error correction or mitigation that might enable practical applications before then. In this work, we consider the idea of post-processing error decoders using existing quantum codes, which are capable of mitigating errors on encoded logical qubits using classical post-processing with no complicated syndrome measurements or additional qubits beyond those used for the logical qubits. This greatly simplifies the experimental exploration of quantum codes on near-term devices, removing the need for locality of syndromes or fast feed-forward, allowing one to study performance aspects of codes on real devices. We provide a general construction equipped with a simple stochastic sampling scheme that does not depend explicitly on a number of terms that we extend to approximate projectors within a subspace. This theory then allows one to generalize to the correction of some logical errors in the code space, correction of some physical unencoded Hamiltonians without engineered symmetries, and corrections derived from approximate symmetries. In this work, we develop the theory of the method and demonstrate it on a simple example with the perfect $[[5,1,3]]$ code, which exhibits a pseudo-threshold of $p \approx 0.50$ under a single qubit depolarizing channel applied to all qubits. We also provide a demonstration under the application of a logical operation and performance on an unencoded hydrogen molecule, which exhibits a significant improvement over the entire range of possible errors incurred under a depolarizing channel.

I. INTRODUCTION

Near-term quantum devices may support useful applications before fault tolerance, but noise and the practical demands of syndrome measurement, feedback, and decoding constrain error correction experiments. This work proposes post-processing existing quantum codes to mitigate errors and study code performance, with demonstrations on an encoded code and an unencoded hydrogen molecule.

I. INTRODUCTION

  • NISQ devices may enable meaningful computation, but extending quantum-supremacy demonstrations to applications beyond pure computation remains open.The introduction notes that fully fault-tolerant hardware for non-trivial algorithms may still be years away.Source: §I · p. 1
  • Practical error correction on NISQ devices is not yet specified, while complicated syndrome measurements, fast feedback, and decoding make experimental study of most codes difficult.The introduction presents studying and optimizing codes on real devices as a potential early use of quantum computers.Source: §I · p. 1
  • The paper uses existing quantum codes with classical post-processing and additional measurements, generalizing quantum subspace expansions to circuits within codes, logical errors, and approximate symmetries.Earlier quantum subspace expansion work addressed application-specific contexts [34] and had experimental verification [35].Source: §I · p. 2
  • The demonstrations include a [[5, 1, 3]] code with a p ≈ 0.50 pseudo-threshold under uncorrelated depolarizing noise and an unencoded hydrogen molecule tested across depolarizing errors.The introduction also describes a logical-error correction extension within the code space.Source: §I · p. 2

II. CORRECTING LOGICAL OBSERVABLES IN POST-PROCESSING

The section develops post-processing projection of errors from stabilizer-code states using projectors built from stabilizers. It avoids direct syndrome measurement and pairs the projection formalism with stochastic sampling to make large expansions practical.

II. CORRECTING LOGICAL OBSERVABLES IN POST-PROCESSING

  • Stabilizer-code logical states are simultaneous +1 eigenstates of a commutative stabilizer group, while check operators can provide syndrome information.Codes are denoted [[n, k, d]], where n is physical qubits, k logical qubits, and distance d is the minimum Pauli weight inducing a logical error.Source: §II · p. 2
  • Stabilizer elements define projectors that remove state components outside the code space, but this projection cannot remove logical errors within that space.The method avoids direct syndrome measurements, which can be cumbersome on geometrically local layouts and challenging to perform fault-tolerantly for complex codes.Source: §II · pp. 2–3
  • The complete projector is a sum over stabilizer-group elements; although its terms grow exponentially, stochastic correction cost depends on state weight outside the code space.Commuting generator projectors can be multiplied, and only generator-associated projectors are needed for the complete projection.Source: §II · p. 3
  • Stochastic sampling should exploit that a state fully inside the code space gives c = 1 with zero variance, so convergence depends on state quality rather than simply term count.The authors reject measuring every expansion term to fixed precision as poorly scaling.Source: §II · p. 3
  • Repeated state preparations permit unbiased estimators from qubitwise Pauli measurements, avoiding ancilla-based stabilizer measurements and simplifying non-local stabilizers.The procedure may destroy the encoded state during measurement because it is performed in post-processing.Source: §II · p. 3

III. CORRECTIONS WITH RECOVERY OPERATIONS

The section extends projection to include recovery operations for a specified set of correctable Pauli errors. Recovery can increase the space contributing signal, but it trades projection quality against finite-sample performance.

III. CORRECTIONS WITH RECOVERY OPERATIONS

  • For known correctable Pauli errors and recovery operations, the method projects onto each error subspace and applies its recovery before combining the corrected projections.The error subspace is identified by the error’s stabilizer syndrome; the set includes the identity for no error.Source: §III · p. 4
  • Including recovery enlarges the Hilbert space over which signal is attained and can lower errors with small finite samples, but may reduce projection quality.The recovered projection may reduce to a stochastically sampled sum of Pauli operators.Source: §III · p. 4
  • For a distance-d code, errors above weight (d−1)/2 may be recovered into logical errors that projection cannot remove, whereas strict projection can remove errors through weight d−1.The finite-sampling benefit and potential correction must be balanced in implementations.Source: §III · p. 4

IV. RELAXING PROJECTORS TO SUBSPACE EXPANSIONS

The section generalizes code-space projection to flexible subspace expansions, including truncated projector sets, and uses these constructions to address logical errors when a problem Hamiltonian distinguishes code states.

IV. RELAXING PROJECTORS TO SUBSPACE EXPANSIONS

  • Truncated projector expansions require nonuniform coefficients, which can be optimized to approximate the code-space ground state subject to normalization.The solution is a generalized eigenvalue problem; its overlap matrix may be singular, so zero-eigenvalue directions are discarded before diagonalization.Source: §IV · pp. 4–5
  • When check operators come from stabilizer generators, the generalized-eigenvalue formulation recovers the earlier projector formalism, and measurement elements scale linearly with the number of checks.The stabilizer commutation relation with the code Hamiltonian permits precomputation that reduces the matrix-element count.Source: §IV · pp. 4–5
  • The code Hamiltonian's ground-state degeneracy prevents this procedure alone from correcting logical errors, but a problem Hamiltonian can break that degeneracy for eigenstate preparation or energy minimization.A stable option is to estimate corrected matrix elements in the logical-operator basis, then diagonalize the problem Hamiltonian.Source: §IV · p. 5
  • Including logical operators becomes beneficial after the encoded problem Hamiltonian breaks code-space degeneracy and also permits access to excited problem-Hamiltonian states.Using logical operators with the code Hamiltonian alone can introduce errors through mixing within the code space.Source: §IV · p. 5

V. CORRECTIONS WITH UNENCODED SYSTEMS

The approach extends to unencoded physical Hamiltonians, where QSE can choose among exact or approximate symmetry-derived projectors, including cases where the desired symmetry sector is unknown.

V. CORRECTIONS WITH UNENCODED SYSTEMS

  • The unencoded-system construction applies the developed machinery directly to a physical problem Hamiltonian, avoiding the overhead of encoded gates in some near-term experiments.The section identifies pair-occupation projectors on high-energy orbitals as a possible future direction, not one explored in its examples.Source: §V · pp. 5–6
  • For unencoded systems, known exact symmetries such as particle number and spin can constrain the desired state, but selecting one particle number may require a projector with many terms.Two-eigenvalue Z2 symmetries, including number parities, offer a simpler alternative.Source: §V · p. 5
  • When the desired ground-state sector of a known Z2 symmetry is unknown, the subspace expansion selects it automatically.For fermionic occupation projectors, QSE can balance a site's contribution to the exact wavefunction against energetic damage from noise-induced occupation.Source: §V · pp. 5–6
  • QSE can automatically select between symmetry-projector choices that may otherwise produce drastically different results, including concatenated symmetries.The section also proposes projectors from approximate symmetries or structured operators that need not commute with the Hamiltonian.Source: §V · p. 6

VI. EXAMPLE DEMONSTRATIONS

The demonstrations cover a general error-correcting code and problem Hamiltonians with symmetries, using a single-qubit depolarizing noise model.

VI. EXAMPLE DEMONSTRATIONS

  • The numerical examples study both a general error-correcting code and specific problem-Hamiltonian systems with symmetries.The supplied passage describes the demonstrations' scope, not their outcomes.Source: §VI · p. 6
  • The numerical studies use a single-qubit depolarizing channel.The excerpt does not specify a channel probability or performance result for these demonstrations.Source: §VI · p. 6
  • Under the stated depolarizing-channel convention, the totally mixed state occurs at p = 3/4.The passage notes that another convention places it at p = 1 and that the conventions can be converted.Source: §VI · p. 6

A. [[5, 1, 3]] code

The [[5, 1, 3]] code is tested as a post-processing recovery method under depolarizing noise, using stabilizer-derived expansions. Recovery reaches a pseudo-threshold near p = 0.50, and a further test examines a transversal logical X operation.

A. [[5, 1, 3]] code

  • The example uses a distance-3 code encoding one logical qubit in five physical qubits, with a hierarchy of checks built from stabilizer generators.The hierarchy S(l) is generated by products of the ordered stabilizer generators, and the expansion operators are applied to the noisy encoded state to evaluate logical fidelity.Source: A. [[5, 1, 3]] code · pp. 6–7
  • The test applies an uncorrelated depolarizing channel to all five qubits, including error patterns up to five-qubit errors.The passage notes that errors beyond the code's expected recovery capacity can still occur, with five-qubit errors having probability p^5.Source: A. [[5, 1, 3]] code · p. 6
  • p = 0.50 is the pseudo-threshold for S(4) recovery with the [[5, 1, 3]] code under a symmetric depolarizing channel.The comparison plots logical infidelity against per-qubit depolarizing probability; the pseudo-threshold is where encoded infidelity becomes lower than unencoded physical infidelity.Source: Figure 3 · p. 7, A. [[5, 1, 3]] code · p. 7
  • A decisive pseudo-threshold is also observed when applying the transversal logical X gate under depolarizing noise.The figure reports a p ≈ 0.45 pseudo-threshold for this logical-operation test.Source: A. [[5, 1, 3]] code · p. 7, Figure 4 · p. 7

B. Hydrogen Molecule

The hydrogen-molecule test applies subspace projection to an unencoded, four-qubit ground state under independent depolarizing noise. The method improves fidelity across the full noise range, with improvements up to 3x in logical infidelity.

B. Hydrogen Molecule

  • Projection uses α- and β-number parity operators plus the total X operator, which need not be an exact Hamiltonian symmetry.In the Jordan-Wigner encoding, the selected set is S = {Z0Z2, Z1Z3, X0X1X2X3}.Source: Figure 5 · p. 8, B. Hydrogen Molecule · p. 8
  • The experiment uses an H2 molecule at a bond length of 1.50 Å, whose ground-state wavefunction requires considerable entanglement for qualitative accuracy.The physical encoding has no error-correction overhead, so the caption states that improvement over the uncorrected solution is always positive.Source: Figure 5 · p. 8
  • The number-parity projectors are chosen as a more efficient alternative to constructing projectors directly from the fermionic number operator.The text notes that direct number-operator projectors can be cumbersome because they must remove all incorrect particle-number sectors and involve many Pauli operators.Source: B. Hydrogen Molecule · p. 8
  • The unencoded H2 test improves logical fidelity across the full range of depolarizing strengths, with up to 3x improvement in logical infidelity.The test prepares the exact ground state and applies independent depolarizing noise to all four qubits; the stretched geometry makes it a sensitive test of performance.Source: B. Hydrogen Molecule · p. 8, Figure 5 · p. 8

VII. STOCHASTIC SAMPLING FOR CORRECTIONS

The section presents a stochastic estimator for correction measurements that samples expansion terms rather than requiring explicit enumeration. It analyzes how state quality affects variance and extends the sampling scheme to recovery and importance sampling.

VII. STOCHASTIC SAMPLING FOR CORRECTIONS

  • The estimator samples terms from a normalized probability distribution and averages ±1 Pauli measurement outcomes to estimate the corrected expectation value.The scheme samples expansion and Pauli-decomposition terms jointly; its variance is the estimator variance divided by the number of samples.Source: §VII · pp. 8–9
  • For a perfect code-space state, the sampling statistics match the original measurement and do not depend on the number of stabilizer terms.At w = 0, the sum over projector terms collapses; the passage states that this adds no average cost for a perfect state.Source: §VII · p. 9
  • The example variance is minimal at w = 0 and increases as state quality degrades.For a totally mixed state, the single Pauli operator is traceless; the analysis relates imperfect-state variance to the non-code-space component.Source: §VII · p. 9
  • Recovery sampling adds a probability over selected errors and may have lower sample variance than strict projection.The recovered-space volume improves the conditioning of the marginalization, while strict projection is expected to provide better ultimate recovery quality by removing errors up to weight d − 1 rather than (d − 1)/2.Source: §VII · p. 9
  • Importance sampling can favor lower-weight stabilizer terms, using probabilities proportional to (1−p)^Wχ for single-qubit depolarizing noise.This choice reflects the stated intuition that information in low-weight operators decays more slowly under local noise.Source: §VII · p. 9
  • The recovery estimator samples Pauli operators formed from the recovery operator, observable terms, and stabilizer terms, with error choices sampled according to their probabilities.Sampling frequency over errors may be chosen proportional to the expected physical error probabilities.Source: §VII · pp. 9–10

VIII. DISCUSSION

Studying and optimizing codes under actual device conditions may contribute to progress toward fully fault-tolerant computation.

VIII. DISCUSSION

  • Optimizing quantum error-correcting codes under actual device conditions may contribute to progress toward fully fault-tolerant computation.The discussion notes that modeling true device noise becomes very difficult as system size grows.Source: §VIII · p. 10
  • Knowledge of biased noise sources can vastly increase a code's threshold.Source: §VIII · p. 10
  • The discussion identifies natural-condition code performance and optimization as targets for early quantum-device studies.Source: §VIII · p. 10

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