The potential of quantum computers to outperform classical ones in practically useful tasks remains challenging in the near term due to scaling limitations and high error rates of current quantum hardware. While quantum error correction (QEC) offers a clear path towards fault tolerance, overcoming the scalability issues will take time. Early applications will likely rely on QEC combined with quantum error mitigation (QEM). We introduce a QEM scheme against both compilation errors and logical-gate noise that is circuit-, QEC code-, and compiler-agnostic. The scheme builds on quasi-probability methods and uses information about the circuit's gates' compilations to attain an unbiased estimation of noiseless expectation values incurring a constant sample-complexity overhead. Moreover, it features maximal circuit size and code distance both independent of the target precision, in contrast to strategies based on QEC alone. We formulate the mitigation procedure as a linear program, demonstrate its efficacy through numerical simulations, and illustrate it for estimating the Jones polynomials of knots. Our method significantly reduces quantum resource requirements for high-precision estimations, offering a practical route towards fault-tolerant quantum computation with precision-independent overheads for fixed circuit size and code distance.
The potential of quantum computers to outperform classical ones in practically useful tasks remains challenging in the near term due to scaling limitations and high error rates of current quantum hardware. While quantum error correction (QEC) offers a clear path towards fault tolerance, overcoming the scalability issues will take time. Early applications will likely rely on QEC combined with quantum error mitigation (QEM). We introduce a QEM scheme against both compilation errors and logical-gate noise that is circuit-, QEC code-, and compiler-agnostic. The scheme builds on quasi-probability methods and uses information about the circuit's gates' compilations to attain an unbiased estimation of noiseless expectation values incurring a constant sample-complexity overhead. Moreover, it features maximal circuit size and code distance both independent of the target precision, in contrast to strategies based on QEC alone. We formulate the mitigation procedure as a linear program, demonstrate its efficacy through numerical simulations, and illustrate it for estimating the Jones polynomials of knots. Our method significantly reduces quantum resource requirements for high-precision estimations, offering a practical route towards fault-tolerant quantum computation with precision-independent overheads for fixed circuit size and code distance.
Quantum error mitigation (QEM) is typically viewed as a suite of practical techniques for today's noisy intermediate-scale quantum devices, with limited relevance once fault-tolerant quantum computers become available. In this work, we challenge this conventional wisdom by showing that QEM can continue to provide substantial benefits in the era of quantum error correction (QEC), and in an even more efficient manner than it does on current devices. We introduce a framework for logical-level QEM that leverages soft information naturally produced by QEC decoders, requiring no additional data, hardware modifications, or runtime overhead beyond what QEC protocols already provide. Within this framework, we develop and analyze three logical-level QEM techniques: post-selection and runtime abort policies, probabilistic error cancellation, and zero-noise extrapolation. Our techniques reduce logical error rates by more than 100x while discarding fewer than 0.1% of shots; they also provide in situ characterization of logical channels for QEM protocols. As a proof of principle, we benchmark our approach using a surface-code architecture and two state-of-the-art decoders based on tensor-network contraction and minimum-weight perfect matching. We evaluate logical-level QEM on random Clifford circuits and molecular simulation algorithms and find that, compared to previous approaches relying on QEC only or QEC combined with QEM, we can achieve up to 87.4% spacetime overhead savings. Our results demonstrate that logical-level QEM with QEC decoder soft information can reliably improve logical performance, underscoring the efficiency and usefulness of QEM techniques for fault-tolerant quantum computers.
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.
Quantum error mitigation (QEM) is critical for extracting reliable computations from noisy quantum processors, proving itself essential not only in the near term but also as a valuable supplement to fully fault-tolerant systems in the future. Despite the necessity, practical QEM deployment still confronts challenges, including the excessive cost of sampling quantum circuits and reliance on unrealistic assumptions such as gate-independent noise. In this paper, we propose Classical Noise Inversion (CNI), which shifts the QEM from sampling diverse quantum circuits to repeated single-circuit measurement, a drastic cost reduction given that the former incurs far heavier time overhead than the latter on realistic quantum hardware. We target general noise models and establish critical conditions for their classical tractability, under which CNI remains effective. For noise models that violate the condition, we propose partial CNI, which mitigates the classically tractable factor of noise via CNI and mitigates the other part via probabilistic error cancellation tailored to gate-dependent noise. To minimize the sampling overhead, we introduce noise compression, which groups noise components with equivalent effects on measurement outcomes, thereby attaining theoretically optimal error-mitigation overhead. The proposed protocols are particularly efficient for error-propagatable circuits, including adaptive universal quantum computing represented by Clifford+T fault-tolerant circuits, and generalized measurement represented by classical shadows. To demonstrate the practical merits of CNI, we integrate it with thrifty classical shadow, and use analysis and numerical simulations to show its advantages in both efficiency and accuracy over existing approaches.
Early fault-tolerant quantum algorithms are a key step in scalable quantum computing, but are ultimately constrained by residual, uncorrectable logical errors. Logical quantum error mitigation can eliminate this bias, however its standard requirements include prior noise characterization, more complex experiments, and substantial sampling overhead. Although no mitigation technique can simultaneously overcome all three constraints, we demonstrate that the first two---noise learning and need for modified circuits---can be completely bypassed for logical circuits by leveraging only syndrome records. Specifically, we introduce virtual error cancellation: a method that, unlike previous bias-free approaches, operates entirely in classical post-processing on universal quantum circuits. We develop and numerically test end-to-end protocols that reduce the sampling overhead beyond fundamental limitations of existing syndrome-aware mitigation techniques, including syndrome-based postselection. In particular, we propose a scheme that pairs the low-latency decoder used in the experiment with a high-complexity decoder run in post-processing. In numerical simulations of a distance-7 rotated surface code, we show that our full-stack dual-decoder algorithm exhibits an error that diminishes as the number of samples increases, capable of reaching more than three orders of magnitude of suppression compared to the decoded logical error rate in an experimentally achievable setting. These results establish syndrome records as a resource for correcting universal computational estimates after quantum execution.
Quantum computation holds the promise of solving certain complex problems exponentially faster than classical computers. However, the high prevalent noise in current quantum devices impedes the accurate execution of even basic algorithms. This can be remedied by protecting quantum information with a quantum error-correcting code, where the logical information of an algorithmic qubit is spread across multiple physical qubits. Individual quantum errors are then located and corrected by the fault-tolerant measurement of multi-qubit stabilizer operators (parity checks). Unfortunately, error correction and fault tolerance both impose large demands on the qubit overhead: hundreds to thousands of physical qubits per logical qubit.
We reduce the spacetime cost of fault tolerance by redesigning key building blocks of an error-corrected quantum computer. First, we develop a combinatorial proof with flag fault tolerance that exponentially reduces the extra qubits needed to measure a stabilizer of any size, while tolerating one fault. We leverage these proofs to then design state preparation circuits for the Steane and Golay codes with 100% yield. Next, we improve error correction on a planar layout by showing that a distance-four code encoding six logical qubits protects information as well as the distance-five surface code, using one-tenth as many physical qubits. Finally, we optimize the time overhead of logical gates in surface code quantum computers by protecting measurement results with a classical code, cutting computation time by a factor of two to six. Our hardware-agnostic optimizations of fault tolerance overheads thus suggest new routes to advance the timeline of error-free quantum computing.
A long-standing challenge in quantum computing is developing technologies to overcome the inevitable noise in qubits. To enable meaningful applications in the early stages of fault-tolerant quantum computing, devising methods to suppress post-correction logical failures is becoming increasingly crucial. In this work, we propose and experimentally demonstrate the application of zero-noise extrapolation, a practical quantum error mitigation technique, to error correction circuits on superconducting processors. By amplifying the noise on physical qubits, the circuits yield outcomes that exhibit a predictable dependence on noise strength, following a polynomial function determined by the code distance. This property enables the effective application of polynomial extrapolation to mitigate logical errors. Our experiments demonstrate a universal reduction in logical errors across various quantum circuits, including fault-tolerant circuits of repetition and surface codes. We observe a favorable performance in multi-round error correction circuits, indicating that this method remains effective when the circuit depth increases. These results advance the frontier of quantum error suppression technologies, opening a practical way to achieve reliable quantum computing in the early fault-tolerant era. Quantum error mitigation refers to techniques that reduce, rather than correct, errors in quantum computing. Here the authors demonstrate zero-noise extrapolation applied to quantum error correction circuits on superconducting processors, effectively reducing logical errors and advancing early fault-tolerant quantum computing.