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Quantum error mitigation as a universal error-minimization technique: applications from NISQ to FTQC eras

Yasunari Suzuki, Suguru Endo, Keisuke Fujii, Yuuki Tokunaga

arXiv:2010.03887v6quant-ph

TL;DR

Early FTQC is limited by restricted code distances, magic-state counts, and classical decoding capacity, while QEM alone has sampling costs that grow exponentially with error events. The paper combines QEC and QEM in an FTQC architecture, showing reduced physical-qubit requirements across target workloads and greater logical-operation capacity at limited code distance.

  • Problem

    Early FTQC faces restricted code distance and magic-state resources, while applying QEM in logical space involves different operation costs and dominant errors than in NISQ computing.

  • Method

    The paper integrates probabilistic error cancellation and Pauli-frame updates with FTQC to mitigate decoding, magic-state, and approximation errors.

  • Results

    The scheme reduces required physical qubits by approximately 80% for 10^4 logical operations and 45% for 10^10 logical operations.

  • Takeaways & Limitations

    Tuning code distance, distillation levels, and decomposition precision can create practical FTQC regimes where QEM relaxes hardware requirements with constant sampling overheads.

Abstract

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In the early years of fault-tolerant quantum computing (FTQC), it is expected that the available code distance and the number of magic states will be restricted due to the limited scalability of quantum devices and the insufficient computational power of classical decoding units. Here, we integrate quantum error correction and quantum error mitigation into an efficient FTQC architecture that effectively increases the code distance and $T$-gate count at the cost of constant sampling overheads in a wide range of quantum computing regimes. For example, while we need $10^4$ to $10^{10}$ logical operations for demonstrating quantum advantages from optimistic and pessimistic points of view, we show that we can reduce the required number of physical qubits by $80\%$ and $45\%$ in each regime. From another perspective, when the achievable code distance is up to about 11, our scheme allows executing $10^3$ times more logical operations. This scheme will dramatically alleviate the required computational overheads and hasten the arrival of the FTQC era.

I. INTRODUCTION

Early FTQC is constrained by limited qubit, magic-state, and classical decoding resources, motivating a framework that combines QEC with QEM. The proposed architecture mitigates dominant logical errors with adjustable sampling overhead and reduces FTQC resource requirements.

  • I. INTRODUCTION: Combining QEC and QEM enables logical-space mitigation of decoding, magic-state, and approximation errors in early FTQC.The framework integrates probabilistic error cancellation with Pauli-frame updates and addresses errors from restricted code distances, insufficient distillation, and Solovay–Kitaev decomposition.
  • I. INTRODUCTION: The early-FTQC regime lies between hardware requirements for classically intractable tasks and conventional long-term quantum advantage.The figure uses an error-event threshold of 10^-3 and contrasts it with operation at approximately one error event under the proposed technique.
  • I. INTRODUCTION: Adjusting code distance, distillation levels, and Solovay–Kitaev precision can place logical error events near unity, where QEM has practical sampling cost.This tunability relaxes hardware requirements while retaining constant sampling overheads in suitable regimes.
  • I. INTRODUCTION: 80% fewer physical qubits are required for at least 10^4 logical operations, while 45% fewer are required for 10^10 operations under the stated sampling constraint.The estimates assume a mean logical error-event target of 10^-3 and QEM sampling overhead within 10^2 times the unmitigated sample count.
  • I. INTRODUCTION: QEM sampling overhead grows exponentially with physical error events in conventional settings, limiting its usefulness as circuit size increases.Probabilistic error cancellation amplifies variance, requiring Γ_Q times more samples for comparable accuracy.

B. Fault-tolerant quantum computing

FTQC encodes logical qubits in a stabilizer-code subspace using redundant physical qubits and repeated syndrome measurements. Syndrome-derived recovery operations protect the encoded state against physical errors.

  • B. Fault-tolerant quantum computing: FTQC protects logical qubits by encoding them in a code space defined within a larger physical-qubit Hilbert space.The code space is specified by stabilizer operators, with logical Pauli operators acting on the encoded qubits.
  • B. Fault-tolerant quantum computing: Repeated stabilizer measurements produce syndrome values that identify physical errors during computation.One repetition of these measurements is a code cycle, and the syndrome outcomes determine recovery operations.
  • B. Fault-tolerant quantum computing: Recovery operations are estimated from syndromes to restore the encoded state while computation proceeds through logical operations.The stabilizer formalism supplies the code space and logical basis used for fault-tolerant manipulation.

2. Logical operations

Universal FTQC combines transversal operations with magic-state-based non-transversal gates, while QEM targets logical decoding and approximation errors. Logical error probabilities decrease with code distance, but characterization of those probabilities is required for mitigation.

  • 2. Logical operations: No stabilizer code supports a universal transversal gate set, so FTQC uses magic-state injection and gate teleportation for non-transversal operations.T gates consume encoded magic states, whose fidelity can be improved through distillation.
  • 2. Logical operations: The Pauli frame stores recovery operations classically and updates measurement outcomes instead of physically applying those operations.This makes logical Pauli updates effectively noiseless when the classical computer is reliable.
  • 2. Logical operations: The integrated architecture models decoding errors as stochastic logical Pauli noise and applies QEM to errors from error-estimation failures and magic-state distillation.The framework assumes an effectively Markovian logical error map for each logical operation.
  • 2. Logical operations: Logical error probability decreases exponentially with code distance below the physical error threshold, while cycles per logical gate increase at most linearly with distance.These relationships connect code-distance choices to both logical reliability and computational cost.
  • 2. Logical operations: Probabilistic error cancellation requires advance knowledge of logical error probabilities, which can be estimated through gate-set tomography but is not exact.Estimation errors are treated separately from the idealized logical-noise model.

2. Approximation error

The framework mitigates decoding and approximation errors within FTQC by combining probabilistic error cancellation with Pauli-frame updates and physical recovery operations when needed.

  • Decoding-error mitigation: Probabilistic error cancellation represents decoding-error inverses using Pauli operations, enabling error mitigation after decoding without directly applying recovery gates.The Pauli frame and measurement post-processing track the recovery operations and QEM cost.
  • Decoding-error mitigation: Logical Pauli recovery operations are implemented by updating the Pauli frame, while non-Pauli recovery operations are applied physically.The resulting measurement outcomes are post-processed using the Pauli frame, recovery parity, and QEM cost.
  • Cost analysis: QEM costs for decoding errors are exponentially related to code distance and exhibit threshold behavior, matching first-order approximations at sufficiently low physical error rates.The total cost follows the circuit’s logical error events under the stated equal-error-rate assumption.

3. Quantum error mitigation for approximation errors

Approximation errors from Solovay–Kitaev decomposition are mitigated through probabilistic error cancellation using Clifford and Pauli recovery operations, with effectiveness determined by estimation accuracy and sampling overhead.

  • Approximation-error mitigation: Approximation errors are mitigated by probabilistic error cancellation without requiring T-gates for recovery, using Clifford operations and Pauli channels.Pauli recoveries use the Pauli frame, whereas non-Pauli recoveries are physically applied after single-qubit logical operations.
  • Sampling overhead: The sampling overhead has a trade-off relationship with the number of available T-gates.The approximation-error QEM cost depends on the recovery operations used throughout the circuit.
  • Estimation errors: QEM is beneficial when the estimation-error reduction factor r is below one.The comparison uses observable deviations with and without error mitigation under diamond-norm bounds.

2. Efficiency of characterization of decoding errors

The paper evaluates decoding-error mitigation and its characterization cost, showing that QEM can trade sampling overhead for effective code-distance increases while gate-set tomography remains comparatively manageable under stated conditions.

  • Characterization efficiency: Improved gate-set tomography reduces characterization measurements to O(r^-2 ε^-1), making pre-computation costs O(n_q r^-2).The method is designed for decoding errors of Clifford processes and is compatible with the Pauli frame.
  • Effective code-distance increase: QEM effectively increases code distance without increasing physical-qubit count when it reduces the logical error rate by a factor r.Setting r = (p/pth)^x increases the effective code distance by 2x, with exp(O(Ndec pdec)) additional repetitions.
  • Numerical evaluation: Logical error probability decreases exponentially with code distance below the threshold, which is approximately pth = 0.044 in the evaluated model.The analysis uses surface codes with phenomenological single-qubit depolarizing noise and lattice surgery.
  • Numerical evaluation: QEM costs decrease exponentially with code distance and follow the first-order approximation closely at sufficiently small physical error rates.The comparison is shown for d-cycle syndrome measurements across several code distances.

2. Performance analysis

The performance analysis evaluates QEM for decoding and approximation errors, finding bias reduction with increased sampling variance and improved approximation-error mitigation as T-gate availability grows. Resource estimates indicate that QEM can reduce required code distances and physical-qubit counts while extending effective logical-operation capacity.

  • Decoding-error mitigation: The simplified evaluation model omits operation-dependent cycle counts, logical-error probabilities, and correlated Pauli errors from lattice-surgery CNOT gates.The authors state that Clifford-operation cycles and logical-error probabilities depend on the specific operation.
  • Decoding-error mitigation: QEM removed expectation-value bias in 100-qubit random Clifford circuits, although its standard deviation increased; it was effective when mean circuit Pauli errors were below unity.For code distances d = 5 and d = 7, the mean Pauli-error counts were 3.6 and 0.28, respectively.
  • Approximation-error mitigation: QEM cost γSK^-1 and its variance decreased exponentially as the allowed T-gate count increased for Haar-random unitary decompositions.The fitted parameters were β1 = 3.9(5) and β2 = 0.072(1).
  • Approximation-error mitigation: In a 7-qubit SWAP-test simulation, QEM mitigated approximation errors from Solovay–Kitaev decomposition while trading improved accuracy for greater sampling variance.The evaluation varied the allowed T-gate count from 24 to 60 and measured expectation-value means and standard deviations.
  • Practical utility: For NG ∼10^4, QEM reduced the required code distance from about 9 to 4 and the physical-qubit requirement to 21%.The estimate assumes Ne = 10^-3; including distillation and lattice-surgery costs still gives about 10^4 physical qubits for 100 logical qubits.
  • Practical utility: For NG ∼10^10, QEM reduced the required code distance from 19 to 14 and the physical-qubit requirement to 55%, while also increasing available logical gates from the Ne = 10^-3 baseline to 10^8.The paper further discusses effective T-gate increases and applications to probabilistic magic-state generation and entanglement distribution.

V. DISCUSSION

The paper argues that integrating QEM with FTQC can reduce resource requirements in early-FTQC regimes while improving computation accuracy for useful applications. It positions the framework as a way to adapt NISQ-era mitigation techniques to restricted FTQC hardware and compares its scope with related proposals.

  • V. DISCUSSION: The required number of physical qubits can be suppressed by tens of percent using QEM in early-FTQC resource estimates.The discussion reports resource estimates with and without QEM and identifies substantial reductions in physical-qubit requirements.
  • V. DISCUSSION: QEM can improve computation accuracy for useful FTQC applications under realistic assumptions, demonstrated with quantum simulation.The authors contrast this with NISQ settings where algorithmic runtime and accuracy are often less reliably characterized.
  • V. DISCUSSION: FTQC can tune code distances, magic-state distillation levels, and T-gate counts so QEM operates near its most effective error regime.The paper identifies the order-unity mean-error regime as especially favorable for QEM.
  • V. DISCUSSION: The framework is intended for applications including quantum phase estimation and Hamiltonian simulation, where algorithmic errors may also be mitigated by extrapolation.These applications are identified as promising targets alongside the paper’s FTQC error-mitigation framework.
  • V. DISCUSSION: The framework targets intermediate FTQC systems that may be too small to solve large useful problems directly, extending NISQ techniques to this regime.The authors motivate this setting by the limited size of first-generation FTQC and distinguish their proposal from related QEC-QEM approaches.

Appendix A: Pauli transfer matrix

The appendix formulates quantum processes, states, and measurements in the Pauli transfer matrix representation and develops probabilistic error cancellation from inverse-map decompositions. It emphasizes that stochastic Pauli noise can be mitigated using Pauli operations, which are implementable through the FTQC Pauli frame.

  • Appendix A: Pauli transfer matrix: Any quantum map can be represented in the Pauli basis as a real Pauli transfer matrix acting on the corresponding state vector.The Pauli operators form an orthonormal basis, and measurements can be represented as row vectors in the same formalism.
  • Appendix A: Pauli transfer matrix: Composite quantum maps multiply in Pauli transfer matrix form, while probabilistic mixtures map to corresponding linear combinations.These properties provide the algebraic basis for composing operations and representing sampled error-cancellation maps.
  • Appendix A: Pauli transfer matrix: Probabilistic error cancellation represents the inverse of a noise map as a linear combination of implementable operations.For arbitrary channels, Clifford operations and Pauli channels form a complete basis for the required decomposition.
  • Appendix A: Pauli transfer matrix: Stochastic Pauli noise can be canceled using only Pauli operations, because Pauli transfer matrices span the relevant diagonal noise maps.In FTQC, these logical Pauli operations can be realized by updating the Pauli frame rather than acting directly on the quantum device.
  • Appendix A: Pauli transfer matrix: The appendix gives explicit single-qubit Pauli-noise coefficients and first-order approximations for the probabilistic error-cancellation cost.The displayed coefficient expressions depend on the Pauli error probabilities pX, pY, and pZ.

Appendix C: Probabilistic error cancellation with gate set tomography

The appendix explains how gate set tomography supplies the process estimates needed for probabilistic error cancellation and how gauge choices make the procedure compatible with the Pauli frame. Under stochastic Pauli errors, the required mitigation operations remain Pauli operations for gates and state preparation.

  • Appendix C: Probabilistic error cancellation with gate set tomography: Gate set tomography estimates gates, initial states, and measurements despite SPAM errors, using a gauge transformation that preserves sequence expectation values.The gauge is represented through an invertible matrix and affects the estimated gate set without changing predicted gate-sequence expectations.
  • Appendix C: Probabilistic error cancellation with gate set tomography: SPAM errors cannot be experimentally separated into independent preparation and measurement contributions.This prevents direct experimental measurement of the individual A(in) and A(out) components.
  • Appendix C: Probabilistic error cancellation with gate set tomography: The gauge must be chosen carefully because Pauli-frame QEM restricts the available mitigation operations to Pauli operations.The appendix proposes a gauge choice compatible with stochastic Pauli errors and the Pauli frame.
  • Appendix C: Probabilistic error cancellation with gate set tomography: Under stochastic Pauli gate and measurement errors, probabilistic error cancellation requires only Pauli operations and is therefore fully compatible with the Pauli frame.The same compatibility is established for state preparation.
  • Appendix C: Probabilistic error cancellation with gate set tomography: The estimated noisy process is inverted to construct error-mitigated gates, states, measurements, and expectation values for gate sequences.The construction applies the estimated inverse noise process within the gate-set representation.

3. Efficiency of gate set tomography for decoding errors

The appendix analyzes gate set tomography for stochastic Pauli decoding errors and describes the FTQC architecture, latency, and logical-operation model used in the analysis. It provides sampling-scaling results and situates the framework in surface-code and lattice-surgery implementations.

  • 3. Efficiency of gate set tomography for decoding errors: NGST = O((r pL)^-2) samples are required to estimate logical error probabilities with relative accuracy factor r under the analyzed tomography procedure.The same scaling applies to magic-state preparation noise characterization through state tomography.
  • 3. Efficiency of gate set tomography for decoding errors: The improved tomography analysis reduces the required sampling to NGST = O(r^-2 pL^-1) for the stated estimation setting.This result follows from estimating noisy stabilizer-state observables whose deviations scale with the logical error probability.
  • 3. Efficiency of gate set tomography for decoding errors: Surface codes with lattice surgery provide the example FTQC architecture, using patch merging and splitting for logical multi-qubit Pauli measurements.Magic-state injection and gate teleportation supply non-transversal operations such as S and T gates, with each T gate consuming a magic state.
  • 3. Efficiency of gate set tomography for decoding errors: The framework models FTQC through quantum-device states and a Pauli frame, treating decoding errors as logical errors mitigated by probabilistic error cancellation.The logical-operation set includes logical-state preparation, Clifford operations, Pauli-Z measurements, and T-gate teleportation.
  • 3. Efficiency of gate set tomography for decoding errors: Pauli-frame recovery has latency because successive syndrome values are needed, and the latency is at least proportional to the code distance.The analysis defines a step by the slowest logical operation and assumes all logical qubits wait until the next operation is ready.

a. Preparation of logical states

The framework prepares logical |0L⟩ and magic states, while representing recovery operations in a Pauli frame and postponing compatible corrections. It combines software and hardware updates to support logical operations under QEM-compatible FTQC constraints.

  • a. Preparation of logical states: Logical |0L⟩ is prepared by measuring joined qubits in the Pauli-Z basis, correcting the outcome with a Pauli operator, and projecting into the code space with X-stabilizer measurements.Measurement errors can be detected in subsequent stabilizer measurements and incorporated through Pauli-frame updates.
  • a. Preparation of logical states: Magic states are prepared by creating a noisy small-distance state, expanding its code distance fault-tolerantly, and applying magic-state distillation.The resulting state must have fidelity comparable to or below the required logical error rate for gate teleportation.
  • a. Preparation of logical states: Pauli operations can be implemented by updating only the Pauli frame, making the operation instantaneous and noiseless because it is processed classically.A hardware update instead acts on the physical device and can introduce errors, although transversal single-qubit Pauli operations are expected to add negligibly to the physical error rate per cycle.
  • a. Preparation of logical states: Successive Clifford operations and Pauli measurements can proceed before recovery estimation finishes, because Pauli-frame updates and recovery applications can be postponed.This postponement improves logical-operation throughput and avoids exponential growth in latency.

d. Single-qubit logical Pauli measurement

Single-qubit logical measurements combine physical Z-basis readout with Pauli-frame information to recover a fault-tolerant logical outcome. In gate teleportation, delayed decoding affects when adaptive corrections can be applied.

  • d. Single-qubit logical Pauli measurement: A destructive logical Pauli-Z measurement computes the logical outcome as a parity of physical Pauli-Z measurement results, with other bases obtained by basis changes or Clifford operations.The procedure estimates the physical outcome string and applies the corresponding classical parity function.
  • d. Single-qubit logical Pauli measurement: The recovery-corrected logical result is obtained by evaluating the parity function on the measured string combined with the Pauli-frame mask.The mask records which data qubits receive effective X or Y recovery actions.
  • d. Single-qubit logical Pauli measurement: Direct Z-basis readout removes information needed to construct the Z-part of the Pauli frame, but only the X-part is relevant to determining the logical measurement result.The X-part can be estimated from syndrome information and related to a minimum-weight perfect matching problem.
  • d. Single-qubit logical Pauli measurement: In gate teleportation, the delayed mask determines whether the adaptive S operation is applied, creating an additional delay after the measurement outcome is obtained.The relevant decoding latency prevents the measurement-dependent correction from being applied simultaneously with the measurement operation.

a. Logical Pauli operation

The QEM architecture treats residual logical Pauli noise as an inverse noise map implemented through probabilistic Pauli-frame updates. The construction is verified for logical operations, including gate teleportation, under the stated noise model.

  • a. Logical Pauli operation: Decoding failures leave logical Pauli noise that is not canceled by ordinary Pauli-frame updates, so QEM cancels it through probabilistic updates of the Pauli frame.Detected physical errors are revealed with latency and corrected by frame updates, whereas decoding-failure noise remains in expectation values without QEM.
  • a. Logical Pauli operation: The inverse logical noise map is decomposed into signed Pauli operations, sampled according to normalized coefficient magnitudes, and appended to the history of measurement and mitigation outcomes.The sampling probabilities are q_i = |η_i|/γ_Q and the normalization is γ_Q = Σ_i |η_i|.
  • a. Logical Pauli operation: The same probabilistic-cancellation construction is extended to logical Pauli measurements and gate teleportation by incorporating both measurement outcomes and QEM choices into the history.The procedure updates the corresponding outcome and operation probabilities before applying the next logical step.
  • a. Logical Pauli operation: All logical operations considered in the construction are verified to obey the target QEM relation, including the logical Pauli operation and the gate-teleportation process.The verification is inductive for successive steps and includes the expectation-value evaluation for the teleportation procedure.

Appendix F: On the noise model of the decoding errors

The appendix analyzes assumptions behind modeling decoding errors and supports an effective exponential noise model numerically for stochastic Pauli noise. It also frames QEM as applicable to decision-problem formulations of sampling algorithms.

  • Appendix F: On the noise model of the decoding errors: The decoding-noise model assumes that each logical operation can be represented by an effective map acting on the logical code space, enabling QEM to cancel that map.The approximation uses perfect syndrome measurements at selected cycles and assumes a simplified cycle count for logical operations.
  • Appendix F: On the noise model of the decoding errors: The effective-map assumption is equivalent to exponential decay of the diagonal Pauli-transfer-matrix elements with syndrome-measurement cycles.The numerical analysis examines a single logical qubit under a depolarizing noise map with p = 0.01.
  • Appendix F: On the noise model of the decoding errors: Non-Pauli decoding noise can be treated through physical Pauli twirling, because non-stochastic-Pauli noise cannot remain represented solely as a Pauli-frame operator.The appendix notes that physical twirling is required when the decoding noise is not stochastic Pauli noise.
  • Appendix F: On the noise model of the decoding errors: QEM is extended beyond expectation values by converting phase-estimation sampling procedures into decision problems with a quantum process, measurement sampling, and classical post-processing.The framework is applied to ground-state energy estimation and is motivated by similar sampling structure in factoring algorithms.
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