arXiv subjects

Michael E. Beverland

Publications and source records attributed to Michael E. Beverland.

At least 19 recordsLinked to original sources

arXiv2026

High-performance syndrome extraction circuits for quantum codes

We present a fast and effective framework for analysing and designing syndrome extraction circuits (SECs). Our approach is based on left--right circuits, a general SEC design that maintains low depth and avoids gate interleaving constraints by staggering $X$ and $Z$ stabiliser checks. Initially proposed for specific classes of codes, we generalise this construction to arbitrary CSS codes and optimise the circuit structure to achieve low qubit idling time, large effective distance, and reduced minimum-weight failure mechanisms. A key component of our framework is the formal notion of residual errors and their associated distance metrics, which form lightweight proxies for capturing error propagation and quantifying the potential harm of circuit-level errors. Applying our automated framework to diverse classes of codes, we observe consistent improvements in logical performance of up to an order of magnitude compared to existing single-ancilla SEC designs. Moreover, by directly applying these proxies, we prove that no non-interleaving SEC can achieve circuit distance $12$ for the gross code, and we identify an explicit circuit that we conjecture achieves distance $11$, exceeding previously known constructions.

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arXiv2026

Concatenating Algebraic Codes over High-Rate Quantum LDPC Codes

Different quantum error correction schemes trade off overhead, error suppression, and hardware connectivity. Code concatenation can relax these tradeoffs by using an outer code whose non-local connectivity is supplied by logical operations of an inner code rather than directly by hardware. Prior works showed that this can reduce memory overhead for local low-rate inner codes such as the surface code. Here, we study concatenation over non-local, high-rate inner codes. Such inner codes experience correlated errors among the many logical qubits in a single codeblock. We handle this by treating each block as a single logical Galois qudit, enabling concatenation with algebraic outer codes with excellent parameters and, crucially, list decoders. In particular, we consider a memory system formed by concatenating quantum Reed-Solomon outer codes over the gross code. For fault-tolerant syndrome extraction, we develop a Galois qudit Shor scheme using "time-like" Reed-Solomon protection against measurement errors. Interestingly, a lightweight fault tolerance scheme, that would fail for qubits, works well for large-alphabet qudits, suggesting a very different theory of fault tolerance for such qudits. The whole protocol is optimised via improved bicycle instruction logical error rates, novel compilation strategies, and recent decoder post-selection rules. At uniform $10^{-3}$ physical noise, the concatenated gross code reaches the teraquop regime, which it previously could not access, with a lower space overhead than the $288$-qubit two-gross code, while offering several advantages from the engineering standpoint. Beyond our main case study, we believe the core ideas of Galois qudits, quantum Reed-Solomon outer codes, and list decoding, will prove generically powerful and highly transferable ideas across high-rate quantum architectures.

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arXiv2026

FPGA-tailored algorithms for real-time decoding of quantum LDPC codes

Real-time decoding is crucial for fault-tolerant quantum computing but likely requires specialized hardware such as field-programmable gate arrays (FPGAs), whose parallelism can alter relative algorithmic performance. We analyze FPGA-tailored versions of three decoder classes for quantum low-density parity-check (qLDPC) codes: message passing, ordered statistics, and clustering. For message passing, we analyze the recently introduced Relay decoder and its FPGA implementation; for ordered statistics decoding (OSD), we introduce a filtered variant that concentrates computation on high-likelihood fault locations; and for clustering, we design an FPGA-adapted generalized union-find decoder. We design a systolic algorithm for Gaussian elimination on rank-deficient systems that runs in linear parallel time, enabling fast validity checks and local corrections in clustering and eliminating costly full-rank inversion in filtered-OSD. Despite these improvements, both remain far slower and less accurate than Relay, suggesting message passing is the most viable route to real-time qLDPC decoding.

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arXiv2025

Fail fast: techniques to probe rare events in quantum error correction

The ultimate goal of quantum error correction is to create logical qubits with very low error rates (e.g. 1e-12) and assemble them into large-scale quantum computers capable of performing many (e.g. billions) of logical gates on many (e.g. thousands) of logical qubits. However, it is necessarily difficult to directly assess the performance of such high-quality logical qubits using standard Monte Carlo sampling because logical failure events become very rare. Building on existing approaches to this problem, we develop three complementary techniques to characterize the rare-event regime for general quantum low-density parity-check (qLDPC) codes under circuit noise. (I) We propose a well-motivated, low-parameter ansatz for the failure spectrum (the fraction of fault sets of each size that fail) that empirically fits all the QEC systems we studied and predicts logical error rates at all physical error rates. (II) We find min-weight logical operators of syndrome measurement circuits and exactly compute the number of min-weight failing configurations. (III) We generalize the splitting method to qLDPC codes using multi-seeded Metropolis sampling to improve convergence for systems with many inequivalent logical operators. We apply these tools to distance-6, -12, and -18 bivariate bicycle codes under circuit noise, observing strong low-error-rate performance with the recently proposed Relay decoder but also considerable scope for further improvement.

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arXiv2025

Improved belief propagation is sufficient for real-time decoding of quantum memory

We introduce a new heuristic decoder, Relay-BP, targeting real-time quantum circuit decoding for large-scale quantum computers. Relay-BP achieves high accuracy across circuit-noise decoding problems: significantly outperforming BP+OSD+CS-10 for bivariate-bicycle codes and comparable to min-weight-matching for surface codes. As a lightweight message-passing decoder, Relay-BP is inherently parallel, enabling rapid low-footprint decoding with FPGA or ASIC real-time implementations, similar to standard BP. A core aspect of our decoder is its enhancement of the standard BP algorithm by incorporating disordered memory strengths. This dampens oscillations and breaks symmetries that trap traditional BP algorithms. By dynamically adjusting memory strengths in a relay approach, Relay-BP can consecutively encounter multiple valid corrections to improve decoding accuracy. We observe that a problem-dependent distribution of memory strengths that includes negative values is indispensable for good performance.

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arXiv2025

Tour de gross: A modular quantum computer based on bivariate bicycle codes

We present the bicycle architecture, a modular quantum computing framework based on high-rate, low-overhead quantum LDPC codes identified in prior work. For two specific bivariate bicycle codes with distances 12 and 18, we construct explicit fault-tolerant logical instruction sets and estimate the logical error rate of the instructions under circuit noise. We develop a compilation strategy adapted to the constraints of the bicycle architecture, enabling large-scale universal quantum circuit execution. Integrating these components, we perform end-to-end resource estimates demonstrating that an order of magnitude larger logical circuits can be implemented with a given number of physical qubits on the bicycle architecture than on surface code architectures. We anticipate further improvements through advances in code constructions, circuit designs, and compilation techniques.

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arXiv2025

Decision-tree decoders for general quantum LDPC codes

We introduce Decision Tree Decoders (DTDs), which rely only on the sparsity of the binary check matrix, making them broadly applicable for decoding any quantum low-density parity-check (qLDPC) code and fault-tolerant quantum circuits. DTDs construct corrections incrementally by adding faults one-by-one, forming a path through a Decision Tree (DT). Each DTD algorithm is defined by its strategy for exploring the tree, with well-designed algorithms typically needing to explore only a small portion before finding a correction. We propose two explicit DTD algorithms that can be applied to any qLDPC code: (1) A provable decoder: Guaranteed to find a minimum-weight correction. While it can be slow in the worst case, numerical results show surprisingly fast median-case runtime, exploring only $w$ DT nodes to find a correction for weight-$w$ errors in notable qLDPC codes, such as bivariate bicycle and color codes. This decoder may be useful for ensemble decoding and determining provable code distances, and can be adapted to compute all minimum-weight logical operators of a code. (2) A heuristic decoder: Achieves higher accuracy and faster performance than BP-OSD on the gross code with circuit noise in realistic parameter regimes.

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arXiv2024

Fault tolerance of stabilizer channels

Stabilizer channels are stabilizer circuits that implement logical operations while mapping from an input stabilizer code to an output stabilizer code. They are widely used to implement fault tolerant error correction and logical operations in stabilizer codes such as surface codes and LDPC codes, and more broadly in subsystem, Floquet and space-time codes. We introduce a rigorous and general formalism to analyze the fault tolerance properties of any stabilizer channel under a broad class of noise models. This includes rigorous but easy-to-work-with definitions and algorithms for the fault distance and hook faults for stabilizer channels. The generalized notion of hook faults which we introduce, defined with respect to an arbitrary subset of a circuit's faults rather than a fixed phenomenological noise model, can be leveraged for fault-tolerant circuit design. Additionally, we establish necessary conditions such that channel composition preserves the fault distance. We apply our framework to design and analyze fault tolerant stabilizer channels for surface codes, revealing novel aspects of fault tolerant circuits.

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arXiv2023

Addressing Stopping Failures for Small Set Flip Decoding of Hypergraph Product Codes

For a quantum error correcting code to be used in practice, it needs to be equipped with an efficient decoding algorithm, which identifies corrections given the observed syndrome of errors.Hypergraph product codes are a promising family of constant-rate quantum LDPC codes that have a linear-time decoding algorithm called Small-Set-Flip ($\texttt{SSF}$) (Leverrier, Tillich, Zémor FOCS 2015). The algorithm proceeds by iteratively applying small corrections which reduce the syndrome weight. Together, these small corrections can provably correct large errors for sufficiently large codes with sufficiently large (but constant) stabilizer weight. However, this guarantee does not hold for small codes with low stabilizer weight. In this case, $\texttt{SSF}$ can terminate with stopping failures, meaning it encounters an error for which it is unable to identify a small correction. We find that the structure of errors that cause stopping failures have a simple form for sufficiently small qubit failure rates. We propose a new decoding algorithm called the Projection-Along-a-Line ($\texttt{PAL}$) decoder to supplement $\texttt{SSF}$ after stopping failures. Using $\texttt{SSF}+\texttt{PAL}$ as a combined decoder, we find an order-of-magnitude improvement in the logical error rate.

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arXiv2022

Assessing requirements to scale to practical quantum advantage

While quantum computers promise to solve some scientifically and commercially valuable problems thought intractable for classical machines, delivering on this promise will require a large-scale quantum machine. Understanding the impact of architecture design choices for a scaled quantum stack for specific applications, prior to full realization of the quantum system, is an important open challenge. To this end, we develop a framework for quantum resource estimation, abstracting the layers of the stack, to estimate resources required across these layers for large-scale quantum applications. Using a tool that implements this framework, we assess three scaled quantum applications and find that hundreds of thousands to millions of physical qubits are needed to achieve practical quantum advantage. We identify three qubit parameters, namely size, speed, and controllability, that are critical at scale to rendering these applications practical. A goal of our work is to accelerate progress towards practical quantum advantage by enabling the broader community to explore design choices across the stack, from algorithms to qubits.

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arXiv2021

Bounds on stabilizer measurement circuits and obstructions to local implementations of quantum LDPC codes

In this work we establish lower bounds on the size of Clifford circuits that measure a family of commuting Pauli operators. Our bounds depend on the interplay between a pair of graphs: the Tanner graph of the set of measured Pauli operators, and the connectivity graph which represents the qubit connections required to implement the circuit. For local-expander quantum codes, which are promising for low-overhead quantum error correction, we prove that any syndrome extraction circuit implemented with local Clifford gates in a 2D square patch of $N$ qubits has depth at least $Ω(n/\sqrt{N})$ where $n$ is the code length. Then, we propose two families of quantum circuits saturating this bound. First, we construct 2D local syndrome extraction circuits for quantum LDPC codes with bounded depth using only $O(n^2)$ ancilla qubits. Second, we design a family of 2D local syndrome extraction circuits for hypergraph product codes using $O(n)$ ancilla qubits with depth $O(\sqrt{n})$. Finally, we use circuit noise simulations to compare the performance of a family of hypergraph product codes using this last family of 2D syndrome extraction circuits with a syndrome extraction circuit implemented with fully connected qubits. While there is a threshold of about $10^{-3}$ for a fully connected implementation, we observe no threshold for the 2D local implementation despite simulating error rates of as low as $10^{-6}$. This suggests that quantum LDPC codes are impractical with 2D local quantum hardware. We believe that our proof technique is of independent interest and could find other applications. Our bounds on circuit sizes are derived from a lower bound on the amount of correlations between two subsets of qubits of the circuit and an upper bound on the amount of correlations introduced by each circuit gate, which together provide a lower bound on the circuit size.

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arXiv2021

Constant-overhead quantum error correction with thin planar connectivity

Quantum LDPC codes may provide a path to build low-overhead fault-tolerant quantum computers. However, as general LDPC codes lack geometric constraints, naïve layouts couple many distant qubits with crossing connections which could be hard to build in hardware and could result in performance-degrading crosstalk. We propose a 2D layout for quantum LDPC codes by decomposing their Tanner graphs into a small number of planar layers. Each layer contains long-range connections which do not cross. For any CSS code with a degree-$δ$ Tanner graph, we design stabilizer measurement circuits with depth at most $(2δ+2)$ using at most $\lceil δ/2 \rceil$ layers. We observe a circuit-noise threshold of 0.28\% for a positive-rate code family using 49 physical qubits per logical qubit. For a physical error rate of $10^{-4}$, this family reaches a logical error rate of $10^{-15}$ using fourteen times fewer physical qubits than the surface code.

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arXiv2021

Improved quantum error correction using soft information

The typical model for measurement noise in quantum error correction is to randomly flip the binary measurement outcome. In experiments, measurements yield much richer information - e.g., continuous current values, discrete photon counts - which is then mapped into binary outcomes by discarding some of this information. In this work, we consider methods to incorporate all of this richer information, typically called soft information, into the decoding of quantum error correction codes, and in particular the surface code. We describe how to modify both the Minimum Weight Perfect Matching and Union-Find decoders to leverage soft information, and demonstrate these soft decoders outperform the standard (hard) decoders that can only access the binary measurement outcomes. Moreover, we observe that the soft decoder achieves a threshold 25\% higher than any hard decoder for phenomenological noise with Gaussian soft measurement outcomes. We also introduce a soft measurement error model with amplitude damping, in which measurement time leads to a trade-off between measurement resolution and additional disturbance of the qubits. Under this model we observe that the performance of the surface code is very sensitive to the choice of the measurement time - for a distance-19 surface code, a five-fold increase in measurement time can lead to a thousand-fold increase in logical error rate. Moreover, the measurement time that minimizes the physical error rate is distinct from the one that minimizes the logical performance, pointing to the benefits of jointly optimizing the physical and quantum error correction layers.

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arXiv2021

The cost of universality: A comparative study of the overhead of state distillation and code switching with color codes

Estimating and reducing the overhead of fault tolerance (FT) schemes is a crucial step toward realizing scalable quantum computers. Of particular interest are schemes based on two-dimensional (2D) topological codes such as the surface and color codes which have high thresholds but lack a natural implementation of a non-Clifford gate. In this work, we directly compare two leading FT implementations of the T gate in 2D color codes under circuit noise across a wide range of parameters in regimes of practical interest. We report that implementing the T gate via code switching to a 3D color code does not offer substantial savings over state distillation in terms of either space or space-time overhead. We find a circuit noise threshold of 0.07(1)% for the T gate via code switching, almost an order of magnitude below that achievable by state distillation in the same setting. To arrive at these results, we provide and simulate an optimized code switching procedure, and bound the effect of various conceivable improvements. Many intermediate results in our analysis may be of independent interest. For example, we optimize the 2D color code for circuit noise yielding its largest threshold to date 0.37(1)%, and adapt and optimize the restriction decoder finding a threshold of 0.80(5)% for the 3D color code with perfect measurements under Z noise. Our work provides a much-needed direct comparison of the overhead of state distillation and code switching, and sheds light on the choice of future FT schemes and hardware designs.

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arXiv2020

Optimization of the surface code design for Majorana-based qubits

The surface code is a prominent topological error-correcting code exhibiting high fault-tolerance accuracy thresholds. Conventional schemes for error correction with the surface code place qubits on a planar grid and assume native CNOT gates between the data qubits with nearest-neighbor ancilla qubits. Here, we present surface code error-correction schemes using $\textit{only}$ Pauli measurements on single qubits and on pairs of nearest-neighbor qubits. In particular, we provide several qubit layouts that offer favorable trade-offs between qubit overhead, circuit depth and connectivity degree. We also develop minimized measurement sequences for syndrome extraction, enabling reduced logical error rates and improved fault-tolerance thresholds. Our work applies to topologically protected qubits realized with Majorana zero modes and to similar systems in which multi-qubit Pauli measurements rather than CNOT gates are the native operations.

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arXiv2020

Noncommuting conserved quantities in quantum many-body thermalization

In statistical mechanics, a small system exchanges conserved quantities---heat, particles, electric charge, etc.---with a bath. The small system thermalizes to the canonical ensemble, or the grand canonical ensemble, etc., depending on the conserved quantities. The conserved quantities are represented by operators usually assumed to commute with each other. This assumption was removed within quantum-information-theoretic (QI-theoretic) thermodynamics recently. The small system's long-time state was dubbed ``the non-Abelian thermal state (NATS).'' We propose an experimental protocol for observing a system thermalize to the NATS. We illustrate with a chain of spins, a subset of which form the system of interest. The conserved quantities manifest as spin components. Heisenberg interactions push the conserved quantities between the system and the effective bath, the rest of the chain. We predict long-time expectation values, extending the NATS theory from abstract idealization to finite systems that thermalize with finite couplings for finite times. Numerical simulations support the analytics: The system thermalizes to the NATS, rather than to the canonical prediction. Our proposal can be implemented with ultracold atoms, nitrogen-vacancy centers, trapped ions, quantum dots, and perhaps nuclear magnetic resonance. This work introduces noncommuting conserved quantities from QI-theoretic thermodynamics into quantum many-body physics: atomic, molecular, and optical physics and condensed matter.

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arXiv2018

The role of entropy in topological quantum error correction

The performance of a quantum error-correction process is determined by the likelihood that a random configuration of errors introduced to the system will lead to the corruption of encoded logical information. In this work we compare two different variants of the surface code with a comparable number of qubits: the surface code defined on a square lattice and the same model on a lattice that is rotated by $π/ 4$. This seemingly innocuous change increases the distance of the code by a factor of $\sqrt{2}$.However, as we show, this gain can come at the expense of significantly increasing the number of different failure mechanisms that are likely to occur. We use a number of different methods to explore this tradeoff over a large range of parameter space under an independent and identically distributed noise model. We rigorously analyze the leading order performance for low error rates, where the larger distance code performs best for all system sizes. Using an analytical model and Monte Carlo sampling, we find that this improvement persists for fixed sub-threshold error rates for large system size, but that the improvement vanishes close to threshold. Remarkably, intensive numerics uncover a region of system sizes and sub-threshold error rates where the square lattice surface code marginally outperforms the rotated model.

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arXiv2018

Spectrum estimation of density operators with alkaline-earth atoms

We show that Ramsey spectroscopy of fermionic alkaline-earth atoms in a square-well trap provides an efficient and accurate estimate for the eigenspectrum of a density matrix whose $n$ copies are stored in the nuclear spins of $n$ such atoms. This spectrum estimation is enabled by the high symmetry of the interaction Hamiltonian, dictated, in turn, by the decoupling of the nuclear spin from the electrons and by the shape of the square-well trap. Practical performance of this procedure and its potential applications to quantum computing, quantum simulation, and time-keeping with alkaline-earth atoms are discussed.

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