Automated Synthesis of Fault-Tolerant State Preparation Circuits for Quantum Error Correction Codes (2408.11894v3)

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 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.

• The paper introduces an automated SAT-based synthesis method that constructs fault-tolerant state preparation circuits for CSS quantum error correction codes.
• It decomposes the task into non-fault-tolerant and fault-tolerant verification circuits to optimize gate count and circuit depth.
• The approach achieves improved logical error rate scalings and scalability for higher-distance codes using both heuristic and SAT-solving techniques.

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

The paper "Automated Synthesis of Fault-Tolerant State Preparation Circuits for Quantum Error Correction Codes" addresses a critical aspect of quantum computing: the fault-tolerant preparation of logical states for quantum error correction codes (QECCs). This necessity arises from the fundamental susceptibility of quantum bits (qubits) and operations to various forms of noise and decoherence, which mandate error correction mechanisms to enable the execution of quantum algorithms in a fault-tolerant manner.

Context and Problem Definition

Quantum Error Correcting Codes (QECCs) are pivotal for mitigating errors in quantum computing. The QECCs encode logical qubits using multiple physical qubits to spread information redundantly. A crucial prerequisite for fault-tolerant quantum operations is the initialization of logical states fault-tolerantly. The paper investigates the synthesis of circuits that prepare logical states, particularly focusing on Calderbank-Shor-Steane (CSS) codes due to their structural properties, which simplify the initialization process into a series of CNOT gates operating on physical qubits initialized in $\ket{0}$ or $\ket{+}$ states.

Methodology

The authors propose a methodology to automate the synthesis of fault-tolerant state preparation circuits for arbitrary CSS codes. The novelty of this approach is in utilizing SAT (satisfiability solving) techniques to construct these circuits, aiming for gate and depth optimality. The method is exhaustively detailed with:

1. Problem Decomposition: The synthesis task is broken down into the generation of both non-fault-tolerant state preparation and fault-tolerant verification circuits.
2. SAT-Based Synthesis: The synthesis problem is encoded into a SAT instance, where optimal circuits are derived by encoding Gaussian elimination over $\mathbb{F}_2$ matrices, representing the stabilizer generators of the CSS codes.
3. Heuristic Solutions: Given the complexity of SAT-based methods, the authors also propose heuristics that scale better and still yield efficient circuits. These heuristics draw inspiration from existing methods for linear reversible circuit synthesis and stabilizer state preparation circuits.
4. Generalized Fault-Tolerant State Preparation: For higher-distance codes (e.g., $d > 3$), a generalized scheme incorporating multiple verification stages is proposed. This approach mitigates error accumulation and spread throughout the entire circuit, ensuring correctness and robustness.

Results and Evaluation

The results underscore the strength and practicality of the proposed approach:

• Circuit Metrics: The paper presents a thorough comparative analysis of the synthesized circuits, emphasizing metrics like the number of CNOT gates, circuit depth, and the number of measurements. When optimized for gate count and depth, the proposed method achieves significantly better results than recent reinforcement learning-based approaches, particularly for larger CSS codes.
• Logical Error Rates: Simulations using a standard depolarizing noise model affirm that the synthesized circuits exhibit the expected error suppression. For instance, distance $d=3$ codes exhibit $p_L \sim O(p^2)$, and distance $d=5$ codes exhibit $p_L \sim O(p^3)$ logical error rate scalings, demonstrating effective error mitigation.

Practical Implications and Future Directions

The work's implications are multifaceted, both practical and theoretical:

1. Scalability: The heuristic methods scale better than SAT-solver-based methods and also yield competitive, near-optimal solutions highly relevant for implementation on near-term quantum devices.
2. Flexibility Across Quantum Architectures: The synthesized circuits are versatile and can be adapted for different quantum computing platforms, considering trade-offs between gate overhead and error rates.
3. Algorithmic Foundations: The methodology lays foundational groundwork for future automated design and synthesis tools in quantum computing. These tools can expedite experimental realizations of fault-tolerant quantum computation.

Conclusion

This paper offers an insightful advancement in the automated synthesis of fault-tolerant state preparation circuits, specifically for CSS quantum error correction codes. By bridging the optimization capabilities of SAT solvers with heuristic methodologies, it provides a scalable solution to a profoundly technical problem in quantum computing. The proposed methods are shown to effectively generalize known schemes to codes of higher distances, underscoring their importance for near-term quantum computational demonstrations. Future extensions of this work could focus on integrating these synthesis techniques within broader quantum compiler frameworks, making fault-tolerant state preparation an integral part of quantum algorithm deployment.