arXiv subjects

Stephen D. Bartlett

Publications and source records attributed to Stephen D. Bartlett.

At least 19 recordsLinked to original sources

arXiv2026

Materialised symmetries of 2D translationally invariant codes

There has been significant recent interest in near-term qLDPC codes as high-performance alternatives to surface and color codes. One such class of codes is 2-dimensional translationally invariant (TI) codes, such as bivariate bicycle codes, which share similar properties to topological codes. Fundamental objects in the study of such codes are the materialised symmetries, which can be used for the construction of matching-based decoders. These decoders generalise the minimum-weight perfect matching decoder for toric codes and provide similar asymptotic performance guarantees that heuristic decoders such as BP and its variants lack. Despite this, the mathematical structure of symmetries along with their properties under translation and restriction to finite-sized lattices has not been well-studied. We describe a decomposition of symmetry spaces of 2D CSS TI codes on infinite lattices into translation-invariant subspaces which allow us to write an explicit basis of symmetries in a plane-wave-like form. We then describe how to adapt this infinite lattice basis so that any corresponding rectangular periodic lattice basis can be obtained by restriction to basis elements compatible with the periodic boundaries. This allows us to describe how the symmetry space varies with different rectangular dimensions. We illustrate via examples that it is often straightforward in practice to determine a basis of symmetries on twisted boundaries as well, without needing to recompute the decomposition. We comment on the applications of this framework to matching-based decoders and provide examples of symmetries of such codes as the gross code.

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arXiv2026

Exact efficient simulation of noisy logical magic states using Clifford stabilizers

The preparation of high-fidelity logical magic states is a crucial subroutine for universal fault-tolerant quantum computation (FTQC). Predicting the performance of FTQC and developing improved protocols rely on numerical methods to classically simulate logical magic state preparation in the presence of noise. Clifford logic on Pauli-stabilizer codes with circuit-level Pauli errors can be efficiently simulated using Pauli-stabilizer formalism, but the non-Clifford operations required to prepare logical magic states render generic simulation inefficient. We introduce Clifford-stabilizer simulation, an exact and efficient algorithm based on updating a Clifford-stabilizer group to simulate noisy preparation protocols for a broad class of logical magic states used to implement non-Clifford gates in the third level of the Clifford hierarchy under circuit-level Pauli errors. Clifford-stabilizer simulation applies to a range of operations that commonly appear in preparation protocols for such logical magic states, including Pauli-stabilizer measurements, logical Clifford measurements, and transversal non-Clifford gates. Our algorithm for Clifford-stabilizer simulation maps a non-Clifford circuit with sampled circuit-level Pauli errors to a Clifford circuit that exactly reproduces its measurement outcome distribution, achieving time and space complexities polynomial in relevant protocol parameters. We perform exact simulation of magic state cultivation up to fault distance 7 by Clifford-stabilizer simulation. Our method provides a route to perform exact benchmarking of large-scale logical magic state preparation protocols required for useful FTQC.

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arXiv2026

Ising on the donut: Regimes of topological quantum error correction from statistical mechanics

Utility-scale quantum computers require quantum error correcting codes with large numbers of physical qubits to achieve sufficiently low logical error rates. The performance of quantum error correction (QEC) is generally predicted through large-scale numerical simulations, used to estimate thresholds, finite-size scaling, and exponential suppression of logical errors below threshold. The connection of QEC to models from statistical mechanics provides an alternative tool for analysing QEC performance. However, predicting the behaviour of these models also requires large-scale numerical simulations, as analytic solutions are not generally known. Here we exploit an exact mapping, from a toric code under bit-flip noise that is post-selected on being syndrome free to the exactly-solvable two-dimensional Ising model on a torus, to derive an analytic solution for the logical failure rate across its full domain of physical error rates. In particular, this mapping provides closed-form expressions for the logical failure rate in four distinct regimes: the path-counting, below-threshold (ordered), near-threshold (critical), and above-threshold (disordered) regimes. Our framework places a number of familiar and long-standing numerical observations on firm theoretical ground. It also motivates explicit ansatze for the conventional QEC setting of non-post-selected codes whose statistical mechanics mappings involve random-bond disorder. Specifically, we introduce an effective tension model for the below-threshold regime, and a new scaling ansatz for the near-threshold regime, derived from an analysis of the domain wall energy cost distributions. By bridging statistical mechanics theory and quantum error correction practice, our results offer a new toolkit for designing, benchmarking, and understanding topological codes beyond current computational limits.

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arXiv2026

Duality constrains optimal thresholds in quantum error correction

Error correction thresholds are often treated as the primary figure of merit for comparing quantum error-correcting code families. We show that the optimal error correction threshold for many commonly considered codes is constrained to a single universal value at leading order in a replica limit. Through a statistical mechanical mapping, we demonstrate that duality constrains all zero-rate em-symmetric CSS codes to have the same optimal code capacity threshold. Here, em symmetry means that the X- and Z-type parity-check matrices are equivalent up to row and column permutations. Under this statistical mechanical mapping, em-symmetric CSS codes are self-dual under a generalized Kramers-Wannier duality up to a mixing of logical sectors. For zero-rate code families, this mixing contributes only subextensive corrections, so the thermodynamic bulk free energy is self-dual in the trivial logical sector. This self-duality fixes the clean critical point and constrains the disordered phase boundary. We also show that self-duality is preserved under code concatenation, and that optimal decoding of concatenated codes can be reformulated as a renormalization group flow on a hierarchical lattice. Our results provide a common framework for analyzing topological, concatenated, and more general quantum low-density parity-check code families, including both their optimal code capacity thresholds and their sub-threshold logical error suppression.

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arXiv2026

Error correction on an array of superconducting qubits with defective components

A solid-state quantum-computing architecture will require the fabrication of arrays of many coupled qubits. It is inevitable that this process will produce qubits and couplers with varying performance, with some components underperforming due to imperfect fabrication. Quantum error-correction requires high-performing components and hence these defects must be dealt with, either by adapting the code to exclude the defects, or by informing the decoder to accommodate defects in post-processing. Here we implement and compare strategies to operate distance-5 surface codes on a quantum processor consisting of a square-lattice array of 120 superconducting qubits. We demonstrate a dramatic reduction in the probability of a logical error in a memory experiment by excluding underperforming components, compared with both a standard approach of ignoring defects, and a defect-aware decoding approach. We observe up to 2.8X improvement in logical errors per round when excluding defects compared with the standard defect-ignorant approach (1.62% compared to 4.49%). In contrast, defect-aware decoding gives only modest gains. Defects are also expected to be particularly harmful for measurement-based logical operations. Using a stability experiment we show that excluding defects resurrects measurement-based logic gate performance, observing a 6.3% per-round suppression of failure rate when excluding defects, compared to zero suppression otherwise. Furthermore, we show a further substantial decrease in logical errors when using leakage post-selection in combination with our defect exclusion strategies, resulting in a distance-5 code outperforming the best distance-3 in one basis. Our experiments therefore give a proof-of-principle demonstration of the essential utility of defect exclusion methods in the scale-up of solid-state quantum computing approaches.

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arXiv2026

Scalable quantum error correction tailored for a heavy-hex qubit array

To produce an operable quantum computer that is made with imperfect hardware, we must design and test scalable quantum error correcting codes that are suited for the devices we can build and, in unison, develop decoding strategies that accommodate device-specific noise characteristics. Here, we introduce the \emph{dynamic compass code}, a subsystem code with a novel syndrome extraction cycle, that has a competitive threshold while making efficient use of qubits arranged on a heavy-hex lattice. We use a superconducting qubit array to implement a distance-5 instance of this code, and demonstrate how detailed noise characterisation can boost decoder performance to yield significant improvements in logical error rates. We perform averaged circuit eigenvalue sampling (ACES) to acquire detailed context-dependent error information on all elements of the syndrome extraction process. Furthermore, we leverage soft information produced from measurement devices to augment the decoder with measurement error information and detect leakage errors for exclusion through post-selection. Our noise-informed approach yields up to 38.3\% improvement in the logical error rate of a distance-5 implementation of the dynamic compass code in experiment.

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arXiv2026

Low-valency scalable quantum error correction with a dynamic compass code

The ongoing development of hardware that is capable of reliably executing general quantum algorithms requires quantum error-correcting codes that are both practical for realisation and rapidly reduce logical error rates as they are scaled up. Here we introduce the dynamic compass code, a code that can be implemented with a modest footprint on the heavy-hex lattice while also demonstrating a threshold. The dynamic code is obtained by choosing a novel measurement schedule for the syndrome extraction circuit of the heavy-hex subsystem code. We numerically evaluate its performance and observe that different choices of schedule can provide a trade-off in protection against logical errors in the $X$ vs $Z$ basis. We also demonstrate that this new measurement schedule provides the code with a threshold for stability experiments. We finally show how the dynamic compass code could be used for fault-tolerant logic by illustrating lattice surgery between code patches.

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arXiv2026

How contextuality and antidistinguishability are related

Contextuality is a key characteristic that separates quantum from classical phenomena and an important tool in understanding the potential advantage of quantum computation. However, when assessing the quantum resources available for quantum information processing, there is no formalism to determine whether a set of states can exhibit contextuality and whether such proofs of contextuality indicate anything about the resourcefulness of that set. Introducing a well-motivated notion of what it means for a set of states to be contextual, we establish a relationship between contextuality and antidistinguishability of sets of states. We go beyond the traditional notions of contextuality and antidistinguishability and treat both properties as resources, demonstrating that the degree of contextuality within a set of states has a direct connection to its level of antidistinguishability. If a set of states is contextual, then it must be weakly antidistinguishable and vice-versa. However, critical contextuality emerges as a stronger property than traditional antidistinguishability.

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arXiv2026

Characterizing physical and logical errors in a transversal CNOT via cycle error reconstruction

The development of prototype quantum information processors has progressed to a stage where small instances of logical qubit systems perform better than the best of their physical constituents. Advancing towards fault-tolerant quantum computing will require an understanding of the underlying error mechanisms in logical primitives as they relate to the performance of quantum error correction. In this work we demonstrate the novel capability to characterize the physical error properties relevant to fault-tolerant operations via cycle error reconstruction. We illustrate this diagnostic capability for a transversal CNOT, a prototypical component of quantum logical operations, in a 16-qubit register of a trapped-ion quantum computer. Our error characterization technique offers three key capabilities: (i) identifying context-dependent physical layer errors, enabling their mitigation; (ii) contextualizing component gates in the environment of logical operators, validating the performance differences in terms of characterized component-level physics, and (iii) providing a scalable method for predicting quantum error correction performance using pertinent error terms, differentiating correctable versus uncorrectable physical layer errors. The methods with which our results are obtained have scalable resource requirements that can be extended with moderate overhead to capture overall logical performance in increasingly large and complex systems.

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arXiv2026

Efficient Post-Selection for General Quantum LDPC Codes

Post-selection strategies that discard low-confidence computational results can significantly improve the effective fidelity of quantum error correction at the cost of reduced acceptance rates, which can be particularly useful for offline resource state generation and other moderate-depth fault-tolerant circuits. Prior work has primarily relied on the "logical gap" metric with the minimum-weight perfect matching decoder, but this approach faces fundamental limitations including computational overhead that scales exponentially with the number of logical qubits and poor generalizability to arbitrary codes beyond surface codes. We develop post-selection strategies based on computationally efficient heuristic confidence metrics that leverage error cluster statistics (specifically, aggregated cluster sizes and log-likelihood ratios) from clustering-based decoders, which are applicable to arbitrary quantum low-density parity check (QLDPC) codes. We validate our method through extensive numerical simulations on surface codes, bivariate bicycle codes, and hypergraph product codes, demonstrating orders of magnitude reductions in logical error rates with moderate abort rates. For instance, applying our strategy to the [[144, 12, 12]] bivariate bicycle code achieves approximately three orders of magnitude reduction in the logical error rate with an abort rate of only 1% (19%) at a physical error rate of 0.1% (0.3%). Additionally, we integrate our approach with the sliding-window framework for real-time decoding, featuring early mid-circuit abort decisions that eliminate unnecessary overheads. Notably, its performance matches or even surpasses the original strategy for global decoding, while exhibiting favorable scaling in the number of rounds. Our approach provides a practical foundation for efficient post-selection in fault-tolerant quantum computing with QLDPC codes.

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arXiv2025

Explicit Instances of Quantum Tanner Codes

We construct several explicit instances of quantum Tanner codes, a class of asymptotically good quantum low-density parity check (qLDPC) codes. The codes are constructed using dihedral groups and random pairs of classical codes and exhibit high encoding rates, relative distances, and pseudo-thresholds. Using the BP+OSD decoder, we demonstrate good performance in the phenomenological and circuit-level noise settings, comparable to the surface code with similar distances. Finally, we conduct an analysis of the space-time overhead incurred by these codes.

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arXiv2025

Thresholds for post-selected quantum error correction from statistical mechanics

We identify regimes where post-selection can be used scalably in quantum error correction (QEC) to improve performance. We use statistical mechanical models to analytically quantify the performance and thresholds of post-selected QEC, with a focus on the surface code. Based on the non-equilibrium magnetization of these models, we identify a simple heuristic technique for post-selection that does not require a decoder. Along with performance gains, this heuristic allows us to derive analytic expressions for post-selected conditional logical thresholds and abort thresholds of surface codes. We find that such post-selected QEC is characterised by four distinct thermodynamic phases, and detail the implications of this phase space for practical, scalable quantum computation.

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arXiv2025

Enhancing Decoding Performance using Efficient Error Learning

Lowering the resource overhead needed to achieve fault-tolerant quantum computation is crucial to building scalable quantum computers. We show that adapting conventional maximum likelihood (ML) decoders to a small subset of efficiently learnable physical error characteristics can significantly improve the logical performance of a quantum error-correcting code. Specifically, we leverage error information obtained from efficient characterization methods based on Cycle Error Reconstruction (CER), which yields Pauli error rates on the $n$ qubits of an error-correcting code. Although the total number of Pauli error rates needed to describe a general noise process is exponentially large in $n$, we show that only a few of the largest few Pauli error rates are needed and that a heuristic technique can complete the Pauli error distribution for ML decoding from this restricted dataset. Using these techniques, we demonstrate significant performance improvements for decoding quantum codes under a variety of physically relevant error models. For instance, with CER data that constitute merely $1\%$ of the Pauli error rates in the system, we achieve a $10X$ gain in performance compared to the case where decoding is based solely on the fidelity of the underlying noise process. Our conclusions underscore the promise of recent error characterization methods for improving quantum error correction and lowering overheads.

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arXiv2025

Logical Noise Bias in Magic State Injection

Fault-tolerant architectures aim to reduce the noise of a quantum computation. Despite such architectures being well studied a detailed understanding of how noise is transformed in a fault-tolerant primitive such as magic state injection is currently lacking. We use numerical simulations of logical process tomography on a fault-tolerant gadget that implements a logical $T = Z(π/4)$ gate using magic state injection, to understand how noise characteristics at the physical level are transformed into noise characteristics at the logical level. We show how, in this gadget, a significant phase ($Z$) bias can arise in the logical noise, even with unbiased noise at the physical level. While the magic state injection gadget intrinsically induces biased noise, with extant phase bias being further amplified at the logical level, we identify noisy error correction circuits as a key limiting factor in the circuits studied on the magnitude of this logical noise bias. Our approach provides a framework for assessing the detailed noise characteristics, as well as the overall performance, of fault-tolerant logical primitives.

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arXiv2025

Low-overhead magic state distillation with color codes

Fault-tolerant implementation of non-Clifford gates is a major challenge for achieving universal fault-tolerant quantum computing with quantum error-correcting codes. Magic state distillation is the most well-studied method for this but requires significant resources. Hence, it is crucial to tailor and optimize magic state distillation for specific codes from both logical- and physical-level perspectives. In this work, we perform such optimization for two-dimensional color codes, which are promising due to their higher encoding rates compared to surface codes, transversal implementation of Clifford gates, and efficient lattice surgery. We propose two carefully designed distillation schemes based on the 15-to-1 distillation circuit and lattice surgery, differing in their methods for handling faulty rotations. Our first scheme employs faulty T-measurement, achieving infidelities of $O(p^3)$ for physical noise strength $p$. To achieve lower infidelities, our second scheme integrates distillation with 'cultivation' (a distillation-free approach to fault-tolerantly prepare magic states through transversal Clifford measurements). Our second scheme achieves significantly lower infidelities (e.g., $\sim 2 \times 10^{-16}$ at $p = 10^{-3}$), surpassing the capabilities of both cultivation and single-level distillation. Notably, to reach a given target infidelity, our schemes require approximately two orders of magnitude fewer resources than the previous best magic state distillation schemes for color codes.

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arXiv2025

Characterising the failure mechanisms of error-corrected quantum logic gates

Mid-circuit measurements used in quantum error correction are essential in quantum computer architecture, as they read out syndrome data and drive logic gates. Here, we use a heavy-hex code prepared on a superconducting qubit array to investigate how different noise sources impact error-corrected logic. First, we identify that idling errors occurring during readout periods are highly detrimental to a quantum memory. We demonstrate significant improvements to the memory by designing and implementing a low-depth syndrome extraction circuit. Second, we perform a stability experiment to investigate the type of failures that can occur during logic gates due to readout assignment errors. We find that the error rate of the stability experiment improves with additional stabilizer readout cycles, revealing a trade-off as additional stability comes at the expense of time over which the memory can decay. We corroborate our results using holistic device benchmarking and by comparison to numerical simulations. Finally, by varying different parameters in our simulations we identify the key noise sources that impact the fidelity of fault-tolerant logic gates, with measurement noise playing a dominant role in logical gate performance.

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arXiv2025

Color code decoder with improved scaling for correcting circuit-level noise

Two-dimensional color codes are a promising candidate for fault-tolerant quantum computing, as they have high encoding rates, transversal implementation of logical Clifford gates, and resource-efficient magic state preparation schemes. However, decoding color codes presents a significant challenge due to their structure, where elementary errors violate three checks instead of just two (a key feature in surface code decoding), and the complexity of extracting syndrome is greater. We introduce an efficient color-code decoder that tackles these issues by combining two matching decoders for each color, generalized to handle circuit-level noise by employing detector error models. We provide comprehensive analyses of the decoder, covering its threshold and sub-threshold scaling both for bit-flip noise with ideal measurements and for circuit-level noise. Our simulations reveal that this decoding strategy nearly reaches the best possible scaling of logical failure ($p_\mathrm{fail} \sim p^{d/2}$) for both noise models, where $p$ is the noise strength, in the regime of interest for fault-tolerant quantum computing. While its noise thresholds are comparable with other matching-based decoders for color codes ($8.2\%$ for bit-flip noise and $0.46\%$ for circuit-level noise), the scaling of logical failure rates below threshold significantly outperforms the best matching-based decoders.

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arXiv2024

Mitigating errors in logical qubits

Quantum error correcting codes protect quantum information, allowing for large quantum computations provided that physical error rates are sufficiently low. We combine post-selection with surface code error correction through the use of a parameterized family of exclusive decoders, which are able to abort on decoding instances that are deemed too difficult. We develop new numerical sampling methods to quantify logical failure rates with exclusive decoders as well as the trade-off in terms of the amount of post-selection required. For the most discriminating of exclusive decoders, we demonstrate a threshold of 50\% under depolarizing noise for the surface code (or $32(1)\%$ for the fault-tolerant case with phenomenological measurement errors), and up to a quadratic improvement in logical failure rates below threshold. Furthermore, surprisingly, with a modest exclusion criterion, we identify a regime at low error rates where the exclusion rate decays with code distance, providing a pathway for scalable and time-efficient quantum computing with post-selection. We apply our exclusive decoder to the 15-to-1 magic state distillation protocol, and report a $75\%$ reduction in the number of physical qubits required, and a $60\%$ reduction in the total spacetime volume required, including accounting for repetitions required for post-selection. We also consider other applications, as an error mitigation technique, and in concatenated schemes. Our work highlights the importance of post-selection as a powerful tool in quantum error correction.

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