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arXiv:2609.10666

Walking Floquet code circuits for zero-overhead leakage reduction

Hanna Westerheim, Kaavya Sahay, Shruti Puri

quant-phSubmitted Sep 9, 2026• Announced Sep 11, 2026• 0 citations
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Abstract

Leakage, occurring when a qubit undetectably exits the computational subspace, poses a significant challenge for quantum error correction by inducing correlated errors in space and time. These correlations reduce both the code threshold and the effective code distance. To address this challenge, we introduce two zero-overhead walking circuits for the honeycomb Floquet code (hFC), termed the swirling and sliding circuits, which periodically remove leakage while dynamically protecting logical qubits via a schedule of anticommuting two-body measurements. Unlike previous dynamic circuits for the hFC, ours preserves the distance to Pauli errors. We further study their performance under leakage and find that, for two leakage noise models, they may correct as many leakage errors as Pauli errors, improving on the more widely known walking surface code. Through numerical simulation, we observe that finite-error-rate performance under leakage is strongly influenced by entropic effects, producing a pronounced waterfall regime in which the logical error rate decreases much more rapidly with physical error rate than the expected distance-limited scaling. In this regime, even when asymptotic predictions suggest otherwise, the hFC can outperform the same-distance walking surface code in error rates and sub-threshold logical error scaling.

Comments
10 pages main text (11 figures), 7 pages appendix

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Leakage, occurring when a qubit undetectably exits the computational subspace, poses a significant challenge for quantum error correction by inducing correlated errors in space and time. These correlations reduce both the code threshold and the effective code distance. To address this challenge, we introduce two zero-overhead walking circuits for the honeycomb Floquet code (hFC), termed the swirling and sliding circuits, which periodically remove leakage while dynamically protecting logical qubits via a schedule of anticommuting two-body measurements. Unlike previous dynamic circuits for the hFC, ours preserves the distance to Pauli errors. We further study their performance under leakage and find that, for two leakage noise models, they may correct as many leakage errors as Pauli errors, improving on the more widely known walking surface code. Through numerical simulation, we observe that finite-error-rate performance under leakage is strongly influenced by entropic effects, producing a pronounced waterfall regime in which the logical error rate decreases much more rapidly with physical error rate than the expected distance-limited scaling. In this regime, even when asymptotic predictions suggest otherwise, the hFC can outperform the same-distance walking surface code in error rates and sub-threshold logical error scaling.
In the typical implementation of a quantum error-correcting code, each stabilizer is measured by entangling one or more ancilla qubits with the data qubits and measuring the ancilla qubits to deduce the value of the stabilizer. Recently, the dynamic circuit approach has been introduced, in which stabilizers are measured without ancilla qubits. Here, we demonstrate that dynamic circuits are particularly useful for the Floquet code. Our dynamic circuit increases the timelike distance of the code, automatically removes leakage, and both significantly increases the threshold and lowers the logical error rate compared to the standard ancilla-based circuit. At a physical error rate of ${10}^{\ensuremath{-}3}$, we estimate a nearly $3\ifmmode\times\else\texttimes\fi{}$ reduction in the number of qubits required to reach a ${10}^{\ensuremath{-}12}$ logical error rate.
Quantum error correction (QEC) codes can correct twice as many erasure errors as Pauli errors. Because of this scaling advantage, there is significant interest in developing qubits whose dominant error channel can be converted into erasures via mid-circuit erasure checks. However, such erasure checks come with hardware overhead in practice. End-of-the-line three-state readout, in which one simultaneously measures a qubit's erasure status and computational state, is an alternative to mid-circuit erasure checks that is generally simpler to implement. In this work, we systematically study the conditions required to enable erasure performance---the doubled error-correction capacity---in the surface code with and without mid-circuit erasure checks. We introduce the moonwalking surface code, the time-reversal of the walking surface code, as a zero-overhead circuit with superior handling of leakage and erasure. Specifically, we show that it enables erasure-like logical error rate scaling when combined with three-state measurement if leaked qubits cause two-qubit gates to be skipped and an appropriate decoder is used. Our decoder, based on a branch-and-bound algorithm, specifically incorporates the noise structure of the skip-gate leaked-qubit effect.
A central goal in quantum error correction is to reduce the overhead of fault-tolerant quantum computing by increasing noise thresholds and reducing the number of physical qubits required to sustain a logical qubit. We introduce a potential path towards this goal based on a family of dynamically generated quantum error correcting codes that we call "hyperbolic Floquet codes.'' These codes are defined by a specific sequence of non-commuting two-body measurements arranged periodically in time that stabilize a topological code on a hyperbolic manifold with negative curvature. We focus on a family of lattices for n qubits that, according to our prescription that defines the code, provably achieve a finite encoding rate ( 1 / 8 + 2 / n ) while still requiring only two-body measurements. Similar to hyperbolic surface codes, the distance of the code at each time-step scales at most logarithmically in n . The family of lattices we choose indicates that this scaling is achievable in practice. We develop and benchmark an efficient matching-based decoder that provides evidence of a threshold near 0.1% in a phenomenological noise model and 0.25% in an entangling measurements noise model. Utilizing weight-two check operators and a qubit connectivity of 3, one of our hyperbolic Floquet codes uses 400 physical qubits to encode 52 logical qubits with a code distance of 8, i.e., it is a [ [ 400 , 52 , 8 ] ] code. At small error rates, comparable logical error suppression to this code requires 5x as many physical qubits (1924) when using the honeycomb Floquet code with the same noise model and decoder.
Fault-tolerant quantum memories depend on the syndrome extraction circuit as much as on the underlying code. Ancilla-free or dynamic circuits are an effective way to improve this circuit layer. For the 6.6.6 honeycomb Floquet code, making the circuit dynamic raises the threshold and lowers the qubit overhead, but at the cost of halving the spatial code distance. A dynamic construction for the 4.8.8 lattice layout was conjectured to preserve full distance. I confirm this and give a dynamic measurement circuit for the CSS 4.8.8 Floquet code. To benchmark it, I construct and compare four circuit-level implementations on a torus, including two dynamic variants (with and without mid-circuit resets), the standard ancilla-based circuit, and a pipelined ancilla-based circuit. Under circuit-level depolarising noise, the reset dynamic circuit reaches a per-round threshold of 0.463%0.463\%0.463% (0.490%)(0.490\%)(0.490%) with MWPM (BP+matching), while the no-reset variant reaches the highest threshold of all four circuits at 0.512%0.512\%0.512% (0.574%)(0.574\%)(0.574%). The standard ancilla-based circuit only achieves 0.228%0.228\%0.228% (0.240%)(0.240\%)(0.240%), but the pipelined schedule reaches 0.478%0.478\%0.478% (0.489%)(0.489\%)(0.489%). The reset dynamic circuit also has a faster-growing timelike distance, with 2≤dt/nqec≤32\le d_t/n_{\mathrm{qec}}\le 32≤dt​/nqec​≤3 asymptotically against a tight 3/23/23/2 for the other three, and running it for fewer rounds gives the smallest spacetime volume in the fast-reset regime, while the no-reset variant is smallest in the slow-reset regime. The 4.8.8 dynamic circuits therefore see the expected threshold gain and overhead reduction without the spatial-distance cost, demonstrating the advantage of dynamic syndrome extraction in Floquet codes.
We construct families of Floquet codes derived from color-code tilings of closed hyperbolic surfaces. These codes have weight-two check operators, a finite encoding rate and can be decoded efficiently with minimum-weight perfect matching. We also construct semi-hyperbolic Floquet codes, which have improved distance scaling, and are obtained via a fine-graining procedure. Using a circuit-based noise model that assumes direct two-qubit measurements, we show that semi-hyperbolic Floquet codes can be 48 times more efficient than planar honeycomb codes and therefore over 100 times more efficient than alternative compilations of the surface code to two-qubit measurements, even at physical error rates of 0.3% to 1%. We further demonstrate that semi-hyperbolic Floquet codes can have a teraquop footprint of only 32 physical qubits per logical qubit at a noise strength of 0.1%. For standard circuit-level depolarizing noise at p=0.1% , we find a 30 times improvement over planar honeycomb codes and a 5.6 times improvement over surface codes. Finally, we analyze small instances that are amenable to near-term experiments, including a Floquet code derived from the Bolza surface that encodes four logical qubits into 16 physical qubits. Published by the American Physical Society 2024
Abstract The leakage of quantum information out of the two computational states of a qubit into other energy states represents a major challenge for quantum error correction. During the operation of an error-corrected algorithm, leakage builds over time and spreads through multi-qubit interactions. This leads to correlated errors that degrade the exponential suppression of the logical error with scale, thus challenging the feasibility of quantum error correction as a path towards fault-tolerant quantum computation. Here, we demonstrate a distance-3 surface code and distance-21 bit-flip code on a quantum processor for which leakage is removed from all qubits in each cycle. This shortens the lifetime of leakage and curtails its ability to spread and induce correlated errors. We report a tenfold reduction in the steady-state leakage population of the data qubits encoding the logical state and an average leakage population of less than 1 × 10 −3 throughout the entire device. Our leakage removal process efficiently returns the system back to the computational basis. Adding it to a code circuit would prevent leakage from inducing correlated error across cycles. With this demonstration that leakage can be contained, we have resolved a key challenge for practical quantum error correction at scale.