Design automation and space-time reduction for surface-code logical operations using a SAT-based EDA kernel compatible with general encodings

Abstract: Fault-tolerant quantum computers (FTQCs) based on surface codes and lattice surgery have been widely studied, and there is strong demand for a framework that can identify logical operations with low space-time cost, verify their functionality and fault tolerance, and demonstrate their optimality within a given search space, much like electronic design automation (EDA) in classical circuit design. In this paper, we propose KOVAL-Q, an EDA kernel that verifies and optimizes surface-code logical operations by formulating them as a satisfiability (SAT) problem. Compared with existing SAT-based frameworks such as LaSsynth, our method can handle logical qubits with more flexible surface-code encodings, both as target configurations and as intermediate states. This extension enables the optimization of advanced layouts, such as fast blocks, and broadens the search space for logical operations. We demonstrate that KOVAL-Q can determine the minimum execution time of fundamental logical operations in given spatial layouts, such as $d$-cycle logical CNOTs and $2d$-cycle patch rotations. Their use reduces the execution time of widely studied FTQC applications by about 10% under a simplified scheduling model. KOVAL-Q consists of three subkernels corresponding to different types of constraints, which facilitates its integration as a submodule into scalable heuristic frameworks. Thus, our proposal provides an essential framework for optimizing and validating core FTQC subroutines.

• The paper introduces a novel SAT-based EDA kernel operating at the stabilizer face level to support arbitrary surface-code encodings.
• The paper demonstrates significant improvements, including a twofold reduction in logical CNOT cycles and a 33% decrease in patch rotation cycles.
• The paper enables modular synthesis and verification for FTQC by integrating LS-SAT, Func-SAT, and FT-SAT to ensure scalable and fault-tolerant operations.

SAT-Based EDA for General Surface-Code Logical Operations: KOVAL-Q

Introduction and Motivation

Surface codes, with their high fault-tolerance threshold and compatibility with nearest-neighbor interactions, are foundational for scalable fault-tolerant quantum computing (FTQC). Realizing practical FTQC demands minimization of space-time overhead in logical gate implementation, particularly for Clifford group operations via lattice surgery. Current electronic design automation (EDA)-inspired quantum frameworks—such as LaSsynth—employ SAT solvers but are restricted to fixed encoding grid geometries, typically single logical qubits per patch. This constraint excludes advanced encoding schemes (e.g., multi-qubit and fast block layouts), limiting the exploration space and thus potentially precluding more efficient logical gate synthesis.

KOVAL-Q Kernel Architecture

KOVAL-Q introduces a SAT-based quantum EDA kernel grounded in a universal topological formalism for surface-code operations, distinguishing itself by encoding operations at the level of stabilizer faces instead of cubes. This not only accommodates arbitrary target patch configurations (including multi-qubit encodings and topologically nontrivial intermediate states) but also addresses functional verification and fault-tolerance within a unified constraint satisfaction paradigm.

The kernel architecture comprises three interacting subkernels:

• LS-SAT: Encodes allowed lattice surgery operations via face and edge connectivity, ensuring topological and operational consistency.
• Func-SAT: Captures logical operation equivalence by tracking stabilizer flow through the 3D code domain, enabling functional verification for arbitrary Clifford operations synthesized as code deformations.
• FT-SAT: Enforces code distance constraints to exclude error chains below threshold, ensuring fault-tolerant implementation.

Each subkernel abstracts away dependence on the code distance $d$ through the 3D exploration domain, ensuring generality and scalability. The kernel generates SAT instances whose complexity is linear in domain size and qubit count, positioning it as a modular component for both global optimization and validation workflows.

Synthesis and Optimization Capabilities

KOVAL-Q overcomes limitations of prior art in several significant aspects:

• General Surface-Code Encodings: Supports arbitrary patch partitioning, multi-qubit codes, and fast block layouts, enabling synthesis routines inaccessible to cube-based frameworks.
• Expanded Intermediate State Space: By operating at the level of stabilizer faces, intermediate states with boundary deformations—critical for time-optimal synthesis—are admitted into the solution space.
• Separation of Synthesis and Verification: The modular subkernel design permits flexible integration as both an overview optimizer and a post-hoc verification engine for heuristic/compiled circuits, serving as a quantum analogue to DRC/STA in classical EDA.

Numerical Results and Performance

KOVAL-Q achieves several notable improvements and new constructions:

• Logical CNOT Optimization: Implements a logical CNOT using a 2-qubit patch in a single code beat ($d$ cycles), representing a twofold reduction in execution time versus previous 2-beat implementations for such layouts.
• Patch Rotation Reduction: Demonstrates patch rotations synthesized in $2d$ cycles, a 33% reduction in cycle count compared to the shortest previously known methods restricted to $3d$ cycles.
• Comprehensive Speedups: Application to practical Clifford+T circuits (e.g., modular adders, PREPARE/SELECT blocks, Trotter simulation) yields execution time reductions of 6–14% on realistic FTQC benchmarks, and up to 22% on random Clifford circuits under simplified scheduling assumptions.

These constructions are outside the reach of LaSsynth and previous SAT-based approaches due to their more restrictive encoding spaces. The formulation generates larger SAT instances but the trade-off is justified by the expansion of solution space and improved optimality.

Theoretical and Practical Implications

Theoretical Significance:

• KOVAL-Q establishes a universal SAT-based encoding for FTQC design automation, extending the fundamental toolkit for quantum EDA beyond what fixed-shape, cube-based methods are capable of modeling.
• The introduction of face-based constraints and topological search increases the expressive power for representing code deformations, supporting not only surface codes but extensible to other CSS codes (e.g., color codes).
• Provides a rigorous SAT-based foundation for unified synthesis, verification, and optimization, opening avenues to co-design with physical architectures relying on novel patch geometries or dynamically changing layouts.

Practical Impact:

• The reduction in space-time overhead for frequent logical operations (CNOT, rotations, moves) translates directly into lower quantum hardware resource requirements and improved computational throughput in realistic FTQC pipelines.
• KOVAL-Q can be integrated as a backend optimization or validation kernel in large-scale heuristic FTQC compilers, augmenting or superseding classical path-finding/routing strategies for critical subroutines.
• The modular subkernels facilitate adaptation to future advancements or other code families without wholesale redesign.

Future Directions

Future developments may include:

• Integration with large-scale FTQC compilers: Partitioning kernels for subcircuit-level optimization within end-to-end synthesis pipelines.
• SAT Instance Size Reduction: Algorithmic advances in constraint encoding or symmetry-based instance pruning to further improve scalability.
• Extension to Non-Abelian Topological Codes: Adapting the stabilizer-face formalism for non-CSS or more general topological stabilizer code families.
• Hardware-Aware Co-Design: Incorporation of realistic noise models and hardware constraints directly into the SAT domain for hardware-algorithm co-optimization.

Conclusion

KOVAL-Q establishes a formal, modular SAT-based kernel for EDA in FTQC, enabling optimization and verification of logical surface-code operations on arbitrary encodings. By operating at the level of stabilizer faces, it expands the synthesis and verification space, delivering measurable runtime reductions for fundamental operations in FTQC. Its theoretical foundation and practical performance indicate significant promise as a core component in the emerging stack of quantum EDA methodologies and FTQC compiler infrastructures.

Reference: "Design automation and space-time reduction for surface-code logical operations using a SAT-based EDA kernel compatible with general encodings" (2604.12560).