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Optimal Compilation of Syndrome Extraction Circuits for General Quantum LDPC Codes

Kai Zhang, Dingchao Gao, Zhaohui Yang, Runshi Zhou, Fangming Liu, Zhengfeng Ji, Jianxin Chen

Abstract

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.

Optimal Compilation of Syndrome Extraction Circuits for General Quantum LDPC Codes

Abstract

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.
Paper Structure (23 sections, 1 theorem, 10 equations, 6 figures, 3 tables, 1 algorithm)

This paper contains 23 sections, 1 theorem, 10 equations, 6 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Stabilizer Formalism
  4. Quantum Low-Density Parity-Check Codes
  5. Syndrome extraction circuit
  6. General Rules of Syndrome Extraction Circuit for qLDPC Codes
  7. General rules
  8. Lower bound
  9. X-Z commutativity
  10. Formulated constraints
  11. Implementation
  12. SMT-based scheduler
  13. Definition of variables
  14. Optimization Goal
  15. Problem Constraints
  16. ...and 8 more sections

Key Result

Proposition 1

For any pair of stabilizer checks, if the parity of the inversion number in the $\mathrm{CX}$-$\mathrm{CZ}$ order across their shared data qubits is even, then all corresponding syndrome extraction circuits are equivalent.

Figures (6)

  • Figure 1: Illustration of Hadamard test and stabilizer measurement of a given stabilizer generator $g = XZXZ$ as an example.
  • Figure 2: Examples of equivalent and non-equivalent stabilizer measurement circuits for stabilizer checks $g_1 = XXXX$ and $g_2 = ZZZZ$.
  • Figure 3: Workflow of ASC with the evaluation pipeline.
  • Figure 4: Logical error rate comparison.
  • Figure 5: Improvement of fault-tolerance through the automated circuit filter, with a fixed random seed for noise simulation.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Definition 1: Pauli Group
  • Definition 2: Stabilizer Group
  • Definition 3: Normalizer
  • Proposition 1: X-Z commutativity