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

Tom Peham, L. Schmid, Lucas Berent, Markus Müller, Robert Wille

Published May 14, 2025• 16 citations
Publisher

Abstract

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 circuit with a verification circuit 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 d = 3 codes only. 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) 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 time. Moreover, we give a general construction for nondeterministic state preparation circuits for codes beyond distance 3. Numerical evaluations using d = 3 , d = 5 , and d = 7 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 (MQT) at . Such methods are an important step in providing fault-tolerant circuit constructions that can aid in near-term demonstrations of fault-tolerant quantum computing.

DOI
10.1103/prxquantum.6.020330

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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 circuit with a verification circuit 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 d = 3 codes only. 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) 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 time. Moreover, we give a general construction for nondeterministic state preparation circuits for codes beyond distance 3. Numerical evaluations using d = 3 , d = 5 , and d = 7 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 (MQT) at . Such methods are an important step in providing fault-tolerant circuit constructions that can aid in near-term demonstrations of fault-tolerant quantum computing.
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.
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.
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.
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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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.