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Quantum Error Correction on Error-mitigated Physical Qubits

Minjun Jeon, Zhenyu Cai

arXiv:2601.18384v1quant-ph

TL;DR

Near-term quantum devices face noise while QEC and QEM each have resource or implementation limitations. This paper develops a general framework that applies linear QEM to physical qubits before QEC without modifying the decoder, and shows with PEC that leading-order logical errors can be removed, increasing effective code distance by 2 and reducing qubit requirements in simulations.

  • Problem

    Near-term devices have limited physical-qubit budgets, modest code distances, and finite measurement rounds, while applying QEM at the logical layer requires challenging logical-level calibration and circuit modification.

  • Method

    The paper applies linear QEM methods to physical qubits within QEC codes, then applies the existing QEC process without modifying its decoder.

  • Results

    PEC removes leading-order physical errors and increases effective code distance by 2; distance-3 PEC codes match or outperform distance-5 unmitigated codes with 40% and 64% fewer qubits.

  • Takeaways & Limitations

    Physical-level QEM is compatible with QEC and can improve logical performance while reducing qubit overhead, especially at smaller code distances.

Abstract

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We present a general framework for applying linear quantum error mitigation (QEM) techniques directly to physical qubits within a logical qubit to suppress logical errors. By exploiting the linearity of quantum error correction (QEC), we demonstrate that any linear QEM method$\unicode{x2014}$including probabilistic error cancellation (PEC), zero-noise extrapolation (ZNE), and symmetry verification$\unicode{x2014}$can be integrated into the physical layer without requiring modifications to the subsequent QEC decoder. Applying this framework to memory experiments using PEC, we analytically prove and numerically verify that the leading-order contribution to the logical error can be removed, increasing the effective code distance by 2. Our simulations on repetition and rotated surface codes show that a distance-3 code with physical-level PEC achieves logical error rates lower than or similar to a distance-5 unmitigated code while using 40% and 64% fewer qubits, respectively. These results establish physical-level QEM as a widely compatible and resource-efficient strategy for enhancing logical performance in early fault-tolerant architectures.

I. INTRODUCTION

Quantum error correction and quantum error mitigation address noise differently, but limited near-term resources make combining them important. This paper proposes applying linear QEM at the physical layer before QEC, avoiding decoder modifications and improving logical performance.

  • QEC encodes information across physical qubits, whereas QEM combines noisy circuit executions to reduce observable bias without substantial qubit overhead.
  • Limited physical-qubit budgets, modest code distances, and finite measurement rounds make neither QEC nor QEM alone sufficient for many near-term tasks.
  • Applying QEM at the logical layer requires logical-level calibration and circuit modification, which can be slower and more challenging than corresponding physical operations.
  • The proposed framework applies any linear QEM method to physical qubits within a QEC code and requires no decoder modification.
  • PEC on repetition and surface codes removes leading-order physical errors, analytically and numerically increasing the effective code distance by 2.
  • The paper evaluates the framework through PEC memory experiments and numerical simulations of repetition and rotated surface codes.

A. Linear QEM: definition and examples

Linear QEM represents a mitigated estimator as a weighted combination of circuit configurations and their measured observables. This formulation yields an effective mitigated state that may not be a physical quantum state.

  • A linear QEM method has an error-mitigated estimator whose expectation value is linear in the observable of interest.
  • The effective error-mitigated state may have unit trace but fail positivity, so it need not be a physical quantum state.
  • The mitigated expectation value is constructed as a linear combination of outputs from circuit configurations indexed by b.
  • Each configuration contributes an output state, measured observable, and weight β_b, with auxiliary registers handled by the observable’s identity extension.
  • The formalism covers PEC, Richardson ZNE, symmetry verification, and virtual purification, and can be rewritten using process-matrix notation.

B. Examples: Probabilistic Error Cancellation

PEC approximates inverse noise channels using weighted combinations of physically implementable operations, allowing selected error components to be canceled. The same linear construction extends to other QEM methods through different circuit configurations.

  • For a noise channel E, PEC decomposes its inverse into a linear combination of physically implementable basis operations {B_b}.
  • Applying the decomposition to a noisy state recovers the ideal expectation value for the observable of interest.
  • Adjusting the coefficients α_b to β_b enables approximate inversion that targets selected error components rather than perfect error removal.
  • The resulting composite channel F ◦ E acts as an effective error channel, with each branch applying B_b to the noisy state.
  • The framework also supports inserting gates during noisy state preparation to address mid-circuit noise.
  • ZNE varies noise levels, symmetry verification uses symmetry projectors or operators, and virtual purification uses multiple noisy-state copies with cyclic permutations.

C. The QEC Channel

QEC projects a corrupted encoded state into a syndrome subspace, applies the decoder-selected correction, and returns the state to the code space before logical measurement. The framework inserts this unchanged QEC process after physical-level QEM.

  • QEC obtains an error syndrome s through stabilizer checks, projects into its syndrome subspace, and applies the corresponding decoder correction C_s.
  • The complete QEC process R combines syndrome projection and correction to recover a state in the code space.
  • Applying QEC before measuring the logical observable produces an error-corrected expectation-value estimator.
  • Figure 1 contrasts physical-level QEM without QEC against the same implementation with the dashed-boxed R channel added before measurement.

D. QEC on top of Error-mitigated Physical Qubits

The framework applies physical-level QEM before QEC, producing an error-mitigated encoded state that can be processed by the existing recovery map and decoder. Because the relevant maps are linear, the approach extends to multiple noisy QEC rounds without modifying QEC.

  • Physical-level QEM: Physical-level QEM transforms the incoming corrupted encoded state before QEC, after which the recovery map produces an error-corrected-and-mitigated expectation value.The incoming state may lie outside the code space, and the procedure targets noise on physical qubits.
  • Physical-level QEM: If the mitigated state contains less noise that escapes recovery than the original state, its estimator is less biased.
  • Decoder compatibility: No modification to the QEC process or decoder is required despite inserting different physical-level circuit configurations.The decoder remains unaware of the gate insertions used in the QEM branches.
  • PEC implementation: PEC combines circuit configurations by sampling branches, applying QEC from the measured syndrome, measuring the logical observable, and attaching the appropriate branch sign.
  • Multiple QEC rounds: Linear effective error-mitigated processes remain composable, enabling the framework to cover consecutive multiple rounds of noisy QEC.

A. Noise Inversion Process

The noise-inversion construction uses PEC to target the leading physical-error weight that dominates logical errors. Its validity requires a regime where higher-order errors remain less likely, while circuit-level generalization becomes less direct when error locations have different scales.

  • Leading-order errors: For a distance-d code in the code-capacity model, leading logical errors arise from physical errors of weight ω = ⌈d/2⌉ with probability O(p^ω).
  • Inverse construction: PEC constructs an approximate inverse by sampling an identity branch and signed branches that insert the targeted weight-ω error processes.The nonidentity branches are collectively called a superbranch.
  • Noise suppression: The effective error channel suppresses errors of weight ω and above from O(p^ω) to a higher-order contribution.
  • Validity regime: The approximation requires higher-order errors to be less likely, so noise suppression is restricted to physical error rates below p_pole.
  • Circuit-level generalization: The construction generalizes to circuit-level noise by treating error locations across time as a quantum comb rather than a single-time-step error channel.
  • Circuit-level generalization: When error locations have different physical error-rate orders, a single dominant weight ω is ambiguous and the optimal inverse channel requires refinement.

B. Logical Error Rate

The logical-error analysis decomposes PEC into an unchanged identity branch and a branch that deliberately introduces targeted physical errors. Their signed combination cancels the leading logical-error contribution without decoder adjustment.

  • Memory experiment: The memory experiment estimates the logical error rate by starting in |0_L⟩ and measuring the projector onto |1_L⟩ after error mitigation and correction.
  • Branch-conditioned picture: PEC separates the mitigated logical error rate into an unmitigated identity component and a component from deliberately introduced weight-ω physical errors.
  • Branch-conditioned picture: The deliberately introduced errors need not trigger decoder adjustments because their raw logical-error contribution is used to cancel matching damage in the unmitigated branch.
  • Logical-error reduction: The amount of logical-error reduction is defined by the difference between the unmitigated and PEC-treated logical error rates.
  • Leading-order cancellation: PEC explicitly cancels the O(p^ω) contribution for bit-flip and depolarising noise.

IV. NUMERICS AND DISCUSSION

Numerical memory experiments show that physical-level PEC increases the effective error-scaling order by removing leading-order error contributions, while its absolute benefits are largest at small code distances. The simulations also identify lower PEC thresholds and explain deviations from an unmitigated code with distance increased by 2.

  • Simulation setup: The simulations use one-round code-capacity errors followed by stabiliser measurements and MWPM decoding for repetition and rotated surface codes.Repetition codes use bit-flip noise, while rotated surface codes use depolarising noise; the setup extends to any fixed number of rounds.
  • Repetition-code results: For repetition codes with d ∈ {3, 5, 7, 9}, PEC slopes are {3.11, 4.11, 5.12, 6.13} versus {1.98, 2.95, 3.92, 4.89} without PEC.The PEC slopes closely match the predicted increased order of the dominant error probability.
  • Surface-code results: For rotated surface codes with d = {3, 5, 7, 9}, PEC slopes are {2.91, 3.94, 4.94, 6.04} versus {1.98, 2.96, 3.93, 4.90} without PEC.These results indicate the same effective increase in uncorrectable-error order as increasing the code distance by 2.
  • Resource comparisons: At distance 3, PEC gives a ∼5-times lower logical error rate than an unmitigated distance-5 repetition code while using 40% fewer qubits.The comparison is 3 versus 5 qubits.
  • Resource comparisons: At distance 3, a PEC surface code achieves almost the same logical error rate as an unmitigated distance-5 surface code while using 64% fewer qubits.The comparison is 9 versus 25 qubits.
  • Limitations and interpretation: PEC improvements decrease with code distance, and its estimated thresholds are p_th ≈ 0.19 for repetition codes and 0.03 for surface codes versus 0.32 and 0.12 without mitigation.The lower PEC thresholds arise alongside the steeper logical-error curves; the reported values come from linear extrapolation of the plotted small-distance curves.

V. CONCLUSIONS

Physical-level linear QEM is compatible with QEC without modifying the decoder, and PEC suppresses leading-order logical errors. Simulations show substantial qubit savings at small code distances, while several numerical and implementation extensions remain open.

  • No decoder modification is required when linear QEM is applied to physical qubits, because QEC remains agnostic to the physical-layer mitigation technique.The error-mitigated physical layer can present a cleaner effective channel to the QEC code.
  • PEC cancels the leading-order O(p^ω) contribution to the logical error rate, increasing the effective code distance by 2.The cancellation is analytically established and extends to depolarising noise when the inverse targets all weight-ω errors.
  • A distance-3 repetition code with PEC achieves a logical error rate ∼5 times lower than a distance-5 unmitigated code while using 40% fewer qubits.
  • A distance-3 surface code with PEC rivals a distance-5 unmitigated surface code with a 64% reduction in qubit count.
  • Numerical testing beyond the code-capacity setting remains needed, including circuit-level noise, while exact decoder-adapted mitigation procedures remain open directions.For rotated surface codes, component-wise logical error rates are difficult to evaluate fully analytically because decoder-specific error-pattern counts are nontrivial.

Appendix H: More Result Plots

The appendix compares numerical and theoretical logical-error curves across code distances and noise models. PEC steepens the curves, producing lower thresholds than unmitigated codes.

  • Theoretical curves reproduce numerical logical-error curves with high accuracy across all code distances and noise models.The curves use analytic and semi-analytic component-wise logical-error expressions, while simulations use stratified sampling.
  • PEC lifts the logical-error rate by one order to O(...), steepening logical-error curves and lowering the threshold.The cited passage is truncated after O, but explicitly connects the order increase with steeper curves and a lower threshold.
  • Repetition-code fitting slopes increase with PEC from 1.98 to 3.11 at d = 3 and from 4.89 to 6.13 at d = 9.Intermediate distances show the same pattern: d = 5 changes from 2.95 to 4.11, and d = 7 from 3.92 to 5.12.
  • The PEC repetition-code threshold is 0.19, compared with 0.5 for the unmitigated code.The threshold of the PEC code was estimated from the intersection of the two largest distances, d = 7 and 9.
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