Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2601.03343 · Physical Review A

Optimizing Fault-tolerant Cat State Preparation

Tom Peham, Erik Weilandt, Robert Wille

quant-phcs.ETSubmitted Jan 6, 2026• Announced Jan 8, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

Comments
17 pages, 8 figures

Related papers

SpiderCat: Optimal Fault-Tolerant Cat State Preparation

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

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.

quant-pharXiv:2603.05391•Mar 6, 2026•1 citation

Automated Synthesis of Fault-Tolerant State Preparation Circuits for Quantum Error Correction Codes

Tom Peham, L. Schmid, Lucas Berent +2 more

A central ingredient in fault-tolerant quantum algorithms is the initialization of a logical state for a given quantum error-correcting code from a set of noisy qubits. A scheme that has demonstrated promising results for small code instances that are realizable on currently available hardware composes a non-fault-tolerant state preparation step with a verification step that checks for spreading errors. Known circuit constructions of this scheme are mostly obtained manually, and no algorithmic techniques for constructing depth- or gate-optimal circuits exist. As a consequence, the current state of the art exploits this scheme only for specific code instances and mostly for the special case of distance 3 codes. In this work, we propose an automated approach for synthesizing fault-tolerant state preparation circuits for arbitrary CSS codes. We utilize methods based on satisfiability solving (SAT) techniques to construct fault-tolerant state preparation circuits consisting of depth- and gate-optimal preparation and verification circuits. We also provide heuristics that can synthesize fault-tolerant state preparation circuits for code instances where no optimal solution can be obtained in an adequate timeframe. Moreover, we give a general construction for non-deterministic state preparation circuits beyond distance 3. Numerical evaluations using $d=3$ and $d=5$ codes confirm that the generated circuits exhibit the desired scaling of the logical error rates. The resulting methods are publicly available as part of the Munich Quantum Toolkit (MQT) at https://github.com/cda-tum/mqt-qecc. Such methods are an important step in providing fault-tolerant circuit constructions that can aid in near-term demonstration of fault-tolerant quantum computing.

arXiv:2408.11894•Aug 21, 2024•0 citations

Efficient Fault-Tolerant Ancilla Preparation for Quantum BCH codes via Cyclic Symmetry

Kohei Yamamoto, Keisuke Fujii

One of the major challenges in realizing fault-tolerant quantum computers (FTQCs) is the requirement for a large number of physical qubits. To address this issue, high-rate quantum error correcting codes, which efficiently embed logical qubits into physical qubits, have recently attracted considerable attention. Among such codes, quantum BCH codes, which offer both high rates and large code distances, are promising yet underexplored candidates. However, no fault-tolerant ancilla preparation method specialized for this class had been established. We employ a two-stage approach (non-fault-tolerant preparation + entanglement distillation) for ancilla preparation. We then propose a framework for designing low-overhead distillation method that strategically leverages the cyclic symmetry of quantum BCH codes to determine which non-fault-tolerant circuits can successfully produce a fault-tolerant state. Numerical simulations on several high-performance quantum BCH codes up to 127 qubits demonstrate that our method achieves lower spatial overhead and logical error rates than conventional distillation circuits. Furthermore, we evaluated the logical error rates under a circuit-level noise model, and obtained performance benchmarks in realistic settings. This efficient state preparation technique is expected to contribute to the early realization of practical FTQCs, particularly on highly connected quantum platforms such as neutral atom systems.

quant-pharXiv:2605.19471•May 20, 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.

quant-pharXiv:2610.03714•Oct 5, 2026•0 citations

Low-overhead Magic State Circuits with Transversal CNOTs

Fazio, Nicholas, Webster, Mark, Zhenyu Cai

With the successful demonstration of transversal CNOTs in many recent experiments, it is the right moment to examine its implications on one of the most critical parts of fault-tolerant computation -- magic state preparation. Using an algorithm that can recompile and simplify a circuit of consecutive multi-qubit phase rotations, we manage to construct fault-tolerant circuits for CCZ, CS and T states with much lower CNOT depths and qubit counts than before and minimal T-depth for the given workspace. These circuits can play crucial roles in fault-tolerant computation with transversal CNOTs, and we hope that the algorithms and methods developed in this paper can be used to further simplify other protocols in similar contexts.

arXiv:2501.10291•Jan 17, 2025•0 citations

Synthesis and Optimization of Encoding Circuits for Fault-Tolerant Quantum Computation

Tom Peham, Matthew Steinberg, Robert Wille +1 more

Preparing arbitrary logical states is a central primitive for universal fault-tolerant quantum computation and the cost of encoded-state preparation contributes directly to the overall resource overhead. This makes the synthesis of efficient general-state encoding circuits an important problem, particularly with respect to two-qubit gate count and circuit depth. Yet the synthesis of such encoders has been studied less extensively than general Clifford circuit synthesis or the preparation of specific logical Pauli-eigenstates. In this work, we develop methods for synthesizing efficient encoders for arbitrary stabilizer codes. We formulate encoder synthesis as a search over stabilizer tableaus and introduce greedy and rollout-based algorithms that exploit the freedom among stabilizer-equivalent realizations of the same encoding isometry. For code families with a modular structure, such as generalized concatenated and holographic codes, we show how large encoders can be assembled from optimized local constituent encoders, and we use SMT-based exact synthesis to obtain optimal local circuits for small instances. We further evaluate the proposed methods on a broad set of stabilizer codes, including holographic and quantum low-density parity-check (qLDPC) codes, and compare them against recent encoder-synthesis methods and existing constructions from the literature, obtaining improvements of up to 43% in two-qubit gate count and up to 70% in depth. Our results support the optimization of encoded-state preparation in several fault-tolerant quantum-computing schemes, and all methods are openly available as part of the Munich Quantum Toolkit.

quant-pharXiv:2605.15266•May 18, 2026•0 citations
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.
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.
A central ingredient in fault-tolerant quantum algorithms is the initialization of a logical state for a given quantum error-correcting code from a set of noisy qubits. A scheme that has demonstrated promising results for small code instances that are realizable on currently available hardware composes a non-fault-tolerant state preparation step with a verification step that checks for spreading errors. Known circuit constructions of this scheme are mostly obtained manually, and no algorithmic techniques for constructing depth- or gate-optimal circuits exist. As a consequence, the current state of the art exploits this scheme only for specific code instances and mostly for the special case of distance 3 codes. In this work, we propose an automated approach for synthesizing fault-tolerant state preparation circuits for arbitrary CSS codes. We utilize methods based on satisfiability solving (SAT) techniques to construct fault-tolerant state preparation circuits consisting of depth- and gate-optimal preparation and verification circuits. We also provide heuristics that can synthesize fault-tolerant state preparation circuits for code instances where no optimal solution can be obtained in an adequate timeframe. Moreover, we give a general construction for non-deterministic state preparation circuits beyond distance 3. Numerical evaluations using d=3d=3d=3 and d=5d=5d=5 codes confirm that the generated circuits exhibit the desired scaling of the logical error rates. The resulting methods are publicly available as part of the Munich Quantum Toolkit (MQT) at https://github.com/cda-tum/mqt-qecc. Such methods are an important step in providing fault-tolerant circuit constructions that can aid in near-term demonstration of fault-tolerant quantum computing.
One of the major challenges in realizing fault-tolerant quantum computers (FTQCs) is the requirement for a large number of physical qubits. To address this issue, high-rate quantum error correcting codes, which efficiently embed logical qubits into physical qubits, have recently attracted considerable attention. Among such codes, quantum BCH codes, which offer both high rates and large code distances, are promising yet underexplored candidates. However, no fault-tolerant ancilla preparation method specialized for this class had been established. We employ a two-stage approach (non-fault-tolerant preparation + entanglement distillation) for ancilla preparation. We then propose a framework for designing low-overhead distillation method that strategically leverages the cyclic symmetry of quantum BCH codes to determine which non-fault-tolerant circuits can successfully produce a fault-tolerant state. Numerical simulations on several high-performance quantum BCH codes up to 127 qubits demonstrate that our method achieves lower spatial overhead and logical error rates than conventional distillation circuits. Furthermore, we evaluated the logical error rates under a circuit-level noise model, and obtained performance benchmarks in realistic settings. This efficient state preparation technique is expected to contribute to the early realization of practical FTQCs, particularly on highly connected quantum platforms such as neutral atom systems.
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.
With the successful demonstration of transversal CNOTs in many recent experiments, it is the right moment to examine its implications on one of the most critical parts of fault-tolerant computation -- magic state preparation. Using an algorithm that can recompile and simplify a circuit of consecutive multi-qubit phase rotations, we manage to construct fault-tolerant circuits for CCZ, CS and T states with much lower CNOT depths and qubit counts than before and minimal T-depth for the given workspace. These circuits can play crucial roles in fault-tolerant computation with transversal CNOTs, and we hope that the algorithms and methods developed in this paper can be used to further simplify other protocols in similar contexts.
Preparing arbitrary logical states is a central primitive for universal fault-tolerant quantum computation and the cost of encoded-state preparation contributes directly to the overall resource overhead. This makes the synthesis of efficient general-state encoding circuits an important problem, particularly with respect to two-qubit gate count and circuit depth. Yet the synthesis of such encoders has been studied less extensively than general Clifford circuit synthesis or the preparation of specific logical Pauli-eigenstates. In this work, we develop methods for synthesizing efficient encoders for arbitrary stabilizer codes. We formulate encoder synthesis as a search over stabilizer tableaus and introduce greedy and rollout-based algorithms that exploit the freedom among stabilizer-equivalent realizations of the same encoding isometry. For code families with a modular structure, such as generalized concatenated and holographic codes, we show how large encoders can be assembled from optimized local constituent encoders, and we use SMT-based exact synthesis to obtain optimal local circuits for small instances. We further evaluate the proposed methods on a broad set of stabilizer codes, including holographic and quantum low-density parity-check (qLDPC) codes, and compare them against recent encoder-synthesis methods and existing constructions from the literature, obtaining improvements of up to 43% in two-qubit gate count and up to 70% in depth. Our results support the optimization of encoded-state preparation in several fault-tolerant quantum-computing schemes, and all methods are openly available as part of the Munich Quantum Toolkit.