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arXiv:2605.15266

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

Tom Peham, Matthew Steinberg, Robert Wille, Sascha Heußen

quant-phcs.ETSubmitted May 14, 2026• Announced May 18, 2026• 0 citations
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Abstract

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.

Comments
33 pages, 11 figures

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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.
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.
Quantum low-density parity-check (LDPC) codes are a promising route to fault-tolerant quantum computation, but their use requires efficient preparation of encoded states. Standard encoder constructions generate circuits through fixed algebraic procedures, yet the resulting circuit can contain substantial redundancy. We formulate LDPC encoder preparation as a circuit-resynthesis problem: given the linear-reversible matrix implemented by the encoder's CNOT block, we seek a lower-cost equivalent circuit that can be routed efficiently on the target hardware and which mitigates noise. We propose a novel optimization approach referred as two-sided Hamming descent and a noise-aware optimization pipeline for this task. Across several families of Calderbank-Shor-Steane (CSS) LDPC encoders, including Bivariate Bicycle, hypergraph-product, and entanglement-assisted codes, the proposed pipeline produces substantially smaller and shallower encoder circuits than the standard constructions and the synthesis baselines considered, cutting gate counts by 53.8% in aggregate across the benchmark and by up to 68% on the Bivariate Bicycle family. The gains remain visible after routing, where the two-qubit depth is reduced by up to 71% and translate into higher-fidelity state preparation under circuit-level noise. On the Bivariate Bicycle family, live-range scheduling further reduces routed preparation failure by up to 13.7% without adding two-qubit gates to the selected circuit. These results indicate that encoder-matrix resynthesis, combined with hardware-calibrated selection and scheduling, is an effective compiler-level tool for preparing quantum LDPC code states.
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.
To achieve large-scale fault-tolerant quantum computation, it may be easier to combine many small sets of qubits than to construct a single large set. For example via quantum error correction with block codes, or distributed quantum processors utilizing shared entanglement. In these regimes, the time or error budget of the overall quantum computation may be dominated by non-local operations. Hence, it is worthwhile to minimize the number of these operations. We consider the case where both non-local and local connectivity may be arbitrarily restricted, and give an asymptotically optimal synthesis method for distributed CNOT and Clifford circuits, based on block-matrix Gaussian elimination. We extend this to all Clifford+RZ circuits by generalizing the Pauli exponential circuit representation; this naturally integrates with existing methods for optimizing T-count. As an application, we show how to implement CNOT circuits in a CSS code encoding n logical qubits in k blocks using O(nk) inter-block transversal CNOTs and intra-block Pauli measurements.
To ensure resilience against the unavoidable noise in quantum computers, quantum information needs to be encoded using an error-correcting code, and circuits must have a particular structure to be fault-tolerant. Compilation of fault-tolerant quantum circuits is thus inherently different from the non-fault-tolerant case. However, automated fault-tolerant compilation methods are widely underexplored, and most known constructions are obtained manually for specific codes only. In this work, we focus on the problem of automatically synthesizing fault-tolerant circuits for the deterministic initialization of an encoded state for a broad class of quantum codes that are realizable on current and near-term hardware. To this end, we utilize methods based on techniques from classical circuit design, such as satisfiability solving, resulting in tools for the synthesis of (optimal) fault-tolerant state preparation circuits for near-term quantum codes. We demonstrate the correct fault-tolerant behavior of the synthesized circuits using circuit-level noise simulations. We provide all routines as open-source software as part of the Munich Quantum Toolkit (MQT) at https://github.com/cdatum/mqt-qecc.
Quantum error correcting codes (QECC) are essential for constructing large-scale quantum computers that deliver faithful results. As strong competitors to the conventional surface code, quantum low-density parity-check (qLDPC) codes are emerging rapidly: they offer high encoding rates while maintaining reasonable physical-qubit connectivity requirements. Despite the existence of numerous code constructions, a notable gap persists between these designs -- some of which remain purely theoretical -- and their circuit-level deployment. In this work, we propose Auto-Stabilizer-Check (ASC), a universal compilation framework that generates depth-optimal syndrome extraction circuits for arbitrary qLDPC codes. ASC leverages the sparsity of parity-check matrices and exploits the commutativity of X and Z stabilizer measurement subroutines to search for optimal compilation schemes. By iteratively invoking an SMT solver, ASC returns a depth-optimal solution if a satisfying assignment is found, and a near-optimal solution in cases of solver timeouts. Notably, ASC provides the first definitive answer to one of IBM's open problems: for all instances of bivariate bicycle (BB) code reported in their work, our compiler certifies that no depth-6 syndrome extraction circuit exists. Furthermore, by integrating ASC with an end-to-end evaluation framework -- one that assesses different compilation settings under a circuit-level noise model -- ASC reduces circuit depth by approximately 50% and achieves an average 7x-8x suppression of the logical error rate for general qLDPC codes, compared with as-soon-as-possible (ASAP) and coloration-based scheduling. ASC thus substantially reduces manual design overhead and demonstrates its strong potential to serve as a key component in accelerating hardware deployment of qLDPC codes.