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arXiv:2603.05391

SpiderCat: Optimal Fault-Tolerant Cat State Preparation

Andrey Boris Khesin, Sarah Meng Li, Boldizsár Poór, Benjamin Rodatz, John van de Wetering, Richie Yeung

quant-phSubmitted Mar 5, 2026• Announced Mar 6, 2026• 1 citation
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Abstract

The ability to fault-tolerantly prepare CAT states, also known as multi-qubit GHZ states, is an important primitive for quantum error correction. It is required for Shor-style syndrome extraction, and can also be used as a subroutine for doing fault-tolerant state preparation of CSS codewords. Existing approaches to fault-tolerant CAT state preparations have been found using computationally expensive heuristics involving SAT solving, reinforcement learning, or exhaustive analysis. In this paper, we constructively find optimal circuits for CAT states in a more scalable way. In particular, we derive formal lower bounds on the number of CNOT gates required for circuits implementing $n$-qubit CAT states that do not spread errors of weight at most $t$ for $1\leq t \leq 5$. We do this by using fault-equivalent rewrites of ZX-diagrams to reduce it to a problem of characterising certain 3-regular simple graphs. We then provide families of such optimal graphs for infinitely many values of $n$ and $t\leq5$. By encoding the construction of optimal graphs as a constraint satisfaction problem we find explicit constructions for circuits that match this lower bound on CNOT count for all $n\leq50$ and $t \leq 5$ and for nearly all pairs $(n,t)$ with $n\leq 100$ and $t\leq 5$ or $n\leq 50$ and $t\leq 7$, significantly extending the regimes that were achievable by previous methods and improving the resource counts for existing constructions. We additionally show how to trade CNOT count against depth, allowing us to construct constant-depth fault-tolerant implementations using $O(n)$ ancilla and $O(n)$ CNOT gates.

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Optimizing Fault-tolerant Cat State Preparation

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Cat states are an important resource for fault-tolerant quantum computing, where they serve as building blocks for a variety of fault-tolerant primitives. Consequently, the ability to prepare high-quality cat states at large fault distances is essential. While optimizations for low fault distances or small numbers of qubits exist, higher fault distances can be achieved via generalized constructions with potentially suboptimal circuit sizes. In this work, we propose a cat state preparation scheme based on preparing two cat states with low-depth circuits, followed by a transversal CNOT and measurement of one of the states. This scheme prepares $w$-qubit cat states fault-tolerantly up to fault distances of $9$ using $\lceil\log_2 w\rceil+1$ depth and at most $3w-2$ CNOTs and $2w$ qubits. We discuss that the combinatorially challenging aspect of this construction is the precise wiring of the transversal CNOT and propose three methods for finding these: two based on Satisfiability Modulo Theory solving and one heuristic search based on a local repair strategy. Numerical evaluations show that our circuits achieve a high fault-distance while requiring fewer resources as generalized constructions.

quant-pharXiv:2601.03343•Physical Review A•Jan 8, 2026•0 citations

SpiderCSS: Scalable Fault-Tolerant CSS State Preparation

Boldizsár Poór, Andrey Boris Khesin, Sarah Meng Li +3 more

Fault-tolerant preparation of logical states is a critical primitive for the realization of large-scale fault-tolerant quantum computing. This paper introduces SpiderCSS, a scalable compilation pipeline that generates highly-optimized, fault-tolerant state preparation circuits for arbitrary Calderbank-Shor-Steane (CSS) codes. Starting from an idealized specification of a CSS state, we use the ZX-calculus and fault-equivalent rewrites to construct a diagram that prepares the given state while preserving the fault tolerance of the initial ideal ZX-diagram. From this diagram, SpiderCSS constructs a fault-tolerant CSS-state preparation circuit using a modular construction that is CNOT-minimal both within each component and also in the way these components are connected together. By only using fault-equivalent rewrites, the final circuit is provably fault-tolerant without needing expensive verification. Monte Carlo simulations across a variety of CSS codes up to distance 15 show that SpiderCSS scales efficiently and constructs circuits with substantially reduced depth and qubit footprint compared to existing scalable and heuristic methods, while simultaneously achieving an average decrease of 33.7% in logical error rate and an increase of 9.5% in acceptance rate.

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Fault-Tolerant Cut-Cat State Syndrome Extraction for Quantum Codes

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Reliable quantum computation requires fault-tolerant protocols to prevent errors from propagating during syndrome extraction in quantum error correction. We present a novel fault-tolerant syndrome extraction technique for CSS codes, which we refer to as the cut-cat state scheme. While each ancilla qubit interacts non-fault-tolerantly with a pair of data qubits, we introduce additional cat stabilizer measurements to identify and correct the resulting hook errors. Our approach maintains the key benefit of cat-based extraction, i.e., parallelized data qubit interactions, while reducing the number of simultaneous qubits required by more than half. Compared to flag-based state-of-the-art protocols, the cut-cat scheme offers a notable advantage in terms of two-qubit gate count as the code distance increases.

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Mirror codes: High-threshold quantum LDPC codes beyond the CSS regime

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The realization of quantum error correction protocols whose logical error rates are suppressed far below physical error rates relies on an intricate combination: the error-correcting code's efficiency, the syndrome extraction circuit's fault tolerance and overhead, the decoder's quality, and the device's constraints, such as physical qubit count and connectivity. This work makes two contributions towards error-corrected quantum devices. First, we introduce mirror codes, a simple yet flexible construction of LDPC stabilizer codes parameterized by a group $G$ and two subsets of $G$ whose total size bounds the check weight. These codes contain all abelian two-block group algebra codes, such as bivariate bicycle (BB) codes. At the same time, they are manifestly not CSS in general, thus deviating substantially from most prior constructions. Fixing a check weight of 6, we find $[[ 60, 4, 10 ]], [[ 36, 6, 6 ]], [[ 48, 8, 6 ]]$, and $[[ 85, 8, 9 ]]$ codes, all of which are not CSS; we also find several weight-7 codes with $kd > n$. Next, we construct syndrome extraction circuits that trade overhead for provable fault tolerance. These circuits use 1-2, 3, and 6 ancillae per check, and respectively are partially fault-tolerant (FT), provably FT on weight-6 CSS codes, and provably FT on \emph{all} weight-6 stabilizer codes. Using our constructions, we perform end-to-end quantum memory experiments on several representative mirror codes under circuit-level noise. We achieve an error pseudothreshold on the order of $0.2\%$, approximately matching that of the $[[ 144, 12, 12 ]]$ BB code under the same model. These findings position mirror codes as a versatile candidate for fault-tolerant quantum memory, especially on smaller-scale devices in the near term.

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Fault-Tolerant Syndrome Extraction and Cat State Preparation with Fewer Qubits

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Leibniz international proceedings in informatics•Jan 1, 2021•2 citations

Synthesis of Fault-tolerant State Preparation Circuits using Steane-type Error Detection

Erik Weilandt, Tom Peham, Robert Wille

Fault-tolerant state preparation is essential for reliable quantum error correction, particularly in Steane-type error correction, which relies on robust ancilla states for syndrome readout. One method of fault-tolerant state preparation is to initialize multiple ancilla states and check them against each other to detect problematic errors. In the worst case, the number of states required for successful initialization grows polynomially with the code distance, but it has been shown that this can be reduced to constant ancilla overhead-in the best case, only four states are required. However, existing techniques for finding low-overhead initialization schemes are limited to codes with large symmetry groups, such as the Golay code. In this work, we propose a general, automated synthesis methodology for Steane-type fault-tolerant state preparation circuits that applies to arbitrary Calderbank-Shor-Steane (CSS) codes and does not rely on code symmetries. We apply the proposed methods to various CSS codes up to a distance of seven and simulate the successful fault-tolerant initialization of logical basis states under circuit-level depolarizing noise. The circuits synthesized using the proposed methodology provide an important step towards experimental realizations of high-fidelity ancilla states for near-term demonstration of fault-tolerant quantum computation.

quant-pharXiv:2601.13313•Jan 21, 2026•0 citations
The ability to fault-tolerantly prepare CAT states, also known as multi-qubit GHZ states, is an important primitive for quantum error correction. It is required for Shor-style syndrome extraction, and can also be used as a subroutine for doing fault-tolerant state preparation of CSS codewords. Existing approaches to fault-tolerant CAT state preparations have been found using computationally expensive heuristics involving SAT solving, reinforcement learning, or exhaustive analysis. In this paper, we constructively find optimal circuits for CAT states in a more scalable way. In particular, we derive formal lower bounds on the number of CNOT gates required for circuits implementing nnn-qubit CAT states that do not spread errors of weight at most ttt for 1≤t≤51\leq t \leq 51≤t≤5. We do this by using fault-equivalent rewrites of ZX-diagrams to reduce it to a problem of characterising certain 3-regular simple graphs. We then provide families of such optimal graphs for infinitely many values of nnn and t≤5t\leq5t≤5. By encoding the construction of optimal graphs as a constraint satisfaction problem we find explicit constructions for circuits that match this lower bound on CNOT count for all n≤50n\leq50n≤50 and t≤5t \leq 5t≤5 and for nearly all pairs (n,t)(n,t)(n,t) with n≤100n\leq 100n≤100 and t≤5t\leq 5t≤5 or n≤50n\leq 50n≤50 and t≤7t\leq 7t≤7, significantly extending the regimes that were achievable by previous methods and improving the resource counts for existing constructions. We additionally show how to trade CNOT count against depth, allowing us to construct constant-depth fault-tolerant implementations using O(n)O(n)O(n) ancilla and O(n)O(n)O(n) CNOT gates.
Cat states are an important resource for fault-tolerant quantum computing, where they serve as building blocks for a variety of fault-tolerant primitives. Consequently, the ability to prepare high-quality cat states at large fault distances is essential. While optimizations for low fault distances or small numbers of qubits exist, higher fault distances can be achieved via generalized constructions with potentially suboptimal circuit sizes. In this work, we propose a cat state preparation scheme based on preparing two cat states with low-depth circuits, followed by a transversal CNOT and measurement of one of the states. This scheme prepares www-qubit cat states fault-tolerantly up to fault distances of 999 using ⌈log⁡2w⌉+1\lceil\log_2 w\rceil+1⌈log2​w⌉+1 depth and at most 3w−23w-23w−2 CNOTs and 2w2w2w qubits. We discuss that the combinatorially challenging aspect of this construction is the precise wiring of the transversal CNOT and propose three methods for finding these: two based on Satisfiability Modulo Theory solving and one heuristic search based on a local repair strategy. Numerical evaluations show that our circuits achieve a high fault-distance while requiring fewer resources as generalized constructions.
Fault-tolerant preparation of logical states is a critical primitive for the realization of large-scale fault-tolerant quantum computing. This paper introduces SpiderCSS, a scalable compilation pipeline that generates highly-optimized, fault-tolerant state preparation circuits for arbitrary Calderbank-Shor-Steane (CSS) codes. Starting from an idealized specification of a CSS state, we use the ZX-calculus and fault-equivalent rewrites to construct a diagram that prepares the given state while preserving the fault tolerance of the initial ideal ZX-diagram. From this diagram, SpiderCSS constructs a fault-tolerant CSS-state preparation circuit using a modular construction that is CNOT-minimal both within each component and also in the way these components are connected together. By only using fault-equivalent rewrites, the final circuit is provably fault-tolerant without needing expensive verification. Monte Carlo simulations across a variety of CSS codes up to distance 15 show that SpiderCSS scales efficiently and constructs circuits with substantially reduced depth and qubit footprint compared to existing scalable and heuristic methods, while simultaneously achieving an average decrease of 33.7% in logical error rate and an increase of 9.5% in acceptance rate.
Reliable quantum computation requires fault-tolerant protocols to prevent errors from propagating during syndrome extraction in quantum error correction. We present a novel fault-tolerant syndrome extraction technique for CSS codes, which we refer to as the cut-cat state scheme. While each ancilla qubit interacts non-fault-tolerantly with a pair of data qubits, we introduce additional cat stabilizer measurements to identify and correct the resulting hook errors. Our approach maintains the key benefit of cat-based extraction, i.e., parallelized data qubit interactions, while reducing the number of simultaneous qubits required by more than half. Compared to flag-based state-of-the-art protocols, the cut-cat scheme offers a notable advantage in terms of two-qubit gate count as the code distance increases.
The realization of quantum error correction protocols whose logical error rates are suppressed far below physical error rates relies on an intricate combination: the error-correcting code's efficiency, the syndrome extraction circuit's fault tolerance and overhead, the decoder's quality, and the device's constraints, such as physical qubit count and connectivity. This work makes two contributions towards error-corrected quantum devices. First, we introduce mirror codes, a simple yet flexible construction of LDPC stabilizer codes parameterized by a group GGG and two subsets of GGG whose total size bounds the check weight. These codes contain all abelian two-block group algebra codes, such as bivariate bicycle (BB) codes. At the same time, they are manifestly not CSS in general, thus deviating substantially from most prior constructions. Fixing a check weight of 6, we find [[60,4,10]],[[36,6,6]],[[48,8,6]][[ 60, 4, 10 ]], [[ 36, 6, 6 ]], [[ 48, 8, 6 ]][[60,4,10]],[[36,6,6]],[[48,8,6]], and [[85,8,9]][[ 85, 8, 9 ]][[85,8,9]] codes, all of which are not CSS; we also find several weight-7 codes with kd>nkd > nkd>n. Next, we construct syndrome extraction circuits that trade overhead for provable fault tolerance. These circuits use 1-2, 3, and 6 ancillae per check, and respectively are partially fault-tolerant (FT), provably FT on weight-6 CSS codes, and provably FT on \emph{all} weight-6 stabilizer codes. Using our constructions, we perform end-to-end quantum memory experiments on several representative mirror codes under circuit-level noise. We achieve an error pseudothreshold on the order of 0.2%0.2\%0.2%, approximately matching that of the [[144,12,12]][[ 144, 12, 12 ]][[144,12,12]] BB code under the same model. These findings position mirror codes as a versatile candidate for fault-tolerant quantum memory, especially on smaller-scale devices in the near term.
We reduce the extra qubits needed for two fault-tolerant quantum computing protocols: error correction, specifically syndrome bit measurement, and cat state preparation. For fault-tolerant syndrome extraction, we show an exponential reduction in qubit overhead over the previous best protocol. For a weight-w stabilizer, we demonstrate that stabilizer measurement tolerating one fault (distance-three) needs at most ⌈ log₂ w ⌉ + 1 ancillas. If qubits reset quickly, four ancillas suffice. We also study the preparation of cat states, simple yet versatile entangled states. We prove that the overhead needed for distance-three fault tolerance is only logarithmic in the cat state size. These results could be useful both for near-term experiments with a few qubits, and for the general study of the asymptotic resource requirements of syndrome measurement and state preparation. For 'a' measured flag bits, there are 2^a possible flag patterns that can identify faults. Hence our results come from solving a combinatorial problem: the construction of maximal-length paths in the a-dimensional hypercube, corresponding to maximal-weight stabilizers or maximal-weight cat states.
Fault-tolerant state preparation is essential for reliable quantum error correction, particularly in Steane-type error correction, which relies on robust ancilla states for syndrome readout. One method of fault-tolerant state preparation is to initialize multiple ancilla states and check them against each other to detect problematic errors. In the worst case, the number of states required for successful initialization grows polynomially with the code distance, but it has been shown that this can be reduced to constant ancilla overhead-in the best case, only four states are required. However, existing techniques for finding low-overhead initialization schemes are limited to codes with large symmetry groups, such as the Golay code. In this work, we propose a general, automated synthesis methodology for Steane-type fault-tolerant state preparation circuits that applies to arbitrary Calderbank-Shor-Steane (CSS) codes and does not rely on code symmetries. We apply the proposed methods to various CSS codes up to a distance of seven and simulate the successful fault-tolerant initialization of logical basis states under circuit-level depolarizing noise. The circuits synthesized using the proposed methodology provide an important step towards experimental realizations of high-fidelity ancilla states for near-term demonstration of fault-tolerant quantum computation.