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SAQC: A SAT-Aware Compilation Framework for QAOA-Based Quantum Optimization

Till Schnittka, Julie Maria Raju, Abhoy Kole, Rolf Drechsler

arXiv:2609.05737v2quant-phcs.ET

TL;DR

Existing SAT-to-QAOA workflows translate formulas into Hamiltonians before generic compilation, losing Boolean structure that could inform compilation. SAQC preserves that structure in SIR and applies SAT-aware optimizations; experiments report reductions in circuit resources and compilation time compared with conventional Hamiltonian-based workflows.

  • Problem

    Existing SAT-to-QAOA workflows translate formulas into Hamiltonians before generic compilation, so the compiler no longer has explicit Boolean semantics to exploit.

    Source: §I · p. 1

  • Method

    SAQC preserves CNF and eCNF structure in SIR, applies clause rewriting and dependency analysis, and supports penalty-Hamiltonian generation and gate synthesis.

    Source: §I · p. 1

  • Results

    Up to 35.83% lower circuit depth than Qiskit on larger instances with 15 ancillas; overall, SAQC reduces depth, two-qubit gate count, and compilation time versus conventional Hamiltonian-based workflows.

    Source: A. Comparison with Qiskit · p. 5, §V · p. 6

  • Takeaways & Limitations

    SAQC provides a unified flow for penalty-Hamiltonian generation and direct clause-to-ansatz synthesis while remaining compatible with existing quantum compilation frameworks.

    Source: §V · p. 6

Abstract

from arXiv · show

The Quantum Approximate Optimization Algorithm (QAOA) is a promising variational approach for solving Boolean satisfiability (SAT) problems. Existing SAT-to-QAOA workflows first translate Boolean formulas into penalty Hamiltonians before applying generic quantum compilation, thereby discarding SAT-specific information such as clause structure, logical operators, and variable dependencies. Consequently, subsequent compiler optimizations cannot exploit the underlying Boolean formulation. This paper presents SAQC, a SAT-aware compilation framework that introduces a SAT Intermediate Representation (SIR) to preserve Boolean structure during compilation. The proposed framework supports both CNF and XOR-extended CNF (eCNF) formulations and performs SAT-aware optimizations including clause rewriting, dependency analysis, and clause scheduling prior to quantum lowering. Building on the optimized SIR, SAQC provides a unified framework for both penalty Hamiltonian generation and direct clause-to-ansatz synthesis, together with ansatz optimizations based on relative-phase decomposition, ancilla-assisted synthesis, and dynamic uncomputation. Experimental results demonstrate significant reductions in circuit depth, two-qubit gate count, and compilation time compared with conventional Hamiltonian-based workflows while remaining compatible with existing quantum compilation frameworks.

I. INTRODUCTION

Existing SAT-to-QAOA workflows discard Boolean structure when translating formulas into Hamiltonians, limiting compiler access to SAT-specific information. SAQC preserves that structure in an intermediate representation and uses it for SAT-aware compilation.

I. INTRODUCTION

  • SAT’s exponential worst-case complexity motivates exploring alternative computational paradigms despite strong performance by modern classical solvers on many practical instances.SAT also underlies EDA tasks including formal verification, equivalence checking, automatic test generation, logic synthesis, and optimization.Source: §I · p. 1

A. Quantum Approximate Optimization Algorithm

  • QAOA is a variational approach to combinatorial optimization, and SAT formulations map Boolean formulas to cost Hamiltonians whose ground states represent satisfying assignments.The introduction situates this work amid QAOA research on Hamiltonian construction, parameter optimization, mixer design, and theoretical performance.Source: §I · p. 1, A. Quantum Approximate Optimization… · p. 2

I. INTRODUCTION

  • Hamiltonian-based SAT-QAOA workflows lose explicit Boolean semantics, leaving generic gate-level compilers unable to exploit SAT-specific structure.The source identifies clause structure, logical operators, and variable dependencies as information that is no longer explicit after translation.Source: §I · p. 1
  • SAQC introduces SIR to preserve Boolean semantics and supports CNF and eCNF with clause rewriting, dependency analysis, Hamiltonian generation, gate synthesis, and SAT-aware scheduling.The framework also evaluates scalability and the quality of generated Hamiltonians and QAOA ansätze.Source: §I · p. 1

A. Quantum Approximate Optimization Algorithm

  • QAOA alternates problem-dependent and mixer evolutions for p rounds, then uses classical optimization to minimize the variational state’s expected cost.Repeated measurements after optimization produce candidate solutions with high probability of approximating the optimum.Source: A. Quantum Approximate Optimization… · p. 2

B. Boolean Satisfiability(SAT)

The section defines SAT and CNF, describes clause penalties for QAOA cost Hamiltonians, and notes that eCNF represents XOR constraints explicitly.

B. Boolean Satisfiability(SAT)

  • SAT asks whether some assignment to Boolean variables satisfies a formula; in CNF, the formula is a conjunction of clauses, each a disjunction of literals.A literal is a variable or its negation; a clause is satisfied when at least one literal is true, and the formula when all clauses are satisfied.Source: B. Boolean Satisfiability(SAT) · p. 2
  • For QAOA, each SAT clause receives a penalty Hamiltonian that is zero for satisfying assignments and positive otherwise, and these penalties form the cost Hamiltonian.For an OR clause, violation occurs when all its variables are false; negated literals use a corresponding projector.Source: B. Boolean Satisfiability(SAT) · p. 2
  • eCNF represents XOR constraints explicitly rather than expanding them into CNF clauses, often compacting parity-constrained and cryptographic SAT instances.Source: B. Boolean Satisfiability(SAT) · p. 2

C. Related Work and Motivation

Prior QAOA work emphasizes Hamiltonian construction and algorithm design, while compiler support for QAOA is comparatively limited; SAQC compiles SAT structure before Hamiltonian synthesis using an SIR.

  • Prior work studied Hamiltonian encodings for constrained optimization and focused QAOA research on Hamiltonian construction, variational optimization, mixers, and performance.Source: C. Related Work and Motivation · p. 2
  • Existing support includes Hamiltonian-derived circuit optimization in 2QAN and gate-level synthesis or hardware-aware transpilation in general-purpose frameworks.Source: C. Related Work and Motivation · p. 2
  • SAQC compiles before Hamiltonian synthesis, preserving CNF and eCNF Boolean structure for SAT-aware analysis, optimization, and generation of Hamiltonians and QAOA ansätze.Source: C. Related Work and Motivation · p. 3

III. PROPOSED SAT-AWARE COMPILATION FRAMEWORK

SAQC converts CNF and eCNF inputs into a shared SIR that preserves Boolean structure, then applies SAT-aware passes and lowers the representation to Hamiltonians and QAOA ansätze.

III. PROPOSED SAT-AWARE COMPILATION FRAMEWORK

  • The compilation flow builds SIR from DIMACS or qDIMACS, applies clause rewriting, dependency analysis, and scheduling, then lowers to Hamiltonians and ansätze with further ansatz optimization.Source: §III · p. 3

A. A SAT Intermediate Representation

B. SAT-aware Optimization Passes

SAQC applies clause rewriting, dependency analysis, and scheduling before lowering; rewriting recursively breaks large clauses into smaller subexpressions, reducing clause size and synthesis complexity.

B. SAT-aware Optimization Passes

1) Clause Rewriting:

  • Large eCNF conjunctions can yield high-order Pauli interactions and large multi-controlled Toffoli gates, so rewriting introduces auxiliary variables to decompose them into smaller subexpressions.Source: 1) Clause Rewriting · p. 3
  • Each rewriting step approximately halves the maximum clause size, yielding lower-order Hamiltonian terms and smaller multi-controlled Toffoli gates.Source: 1) Clause Rewriting · p. 4

2) Dependency Analysis:

SAQC represents clause dependencies as a graph, schedules independent clauses together to reduce ansatz depth, and lowers SIR through Hamiltonian-generation or direct ansatz-generation paths.

2) Dependency Analysis:

  • The dependency graph uses clauses as vertices and connects clauses sharing a literal; clauses without shared literals are independent and may run in parallel.Source: 2) Dependency Analysis · p. 4

3) Clause Scheduling:

  • Scheduling independent clauses consecutively reduces ansatz depth; a greedy heuristic uses parameter k to approximate the maximum-independent-set scheduling problem.Source: 3) Clause Scheduling · p. 4

C. Lowering the SIR

  • Optimized SIR has two lowering paths: penalty Hamiltonian generation and direct clause-to-ansatz generation.Source: C. Lowering the SIR · p. 4

1) Penalty Hamiltonian Generation:

SAQC lowers SIR clauses into penalty Hamiltonians using rules for their logical operators, while clause rewriting and scheduling support more compact, shallower circuits.

2) Direct Clause-to-Ansatz Generation:

SAQC synthesizes QAOA cost layers directly from optimized SIR clauses, then reduces the cost of multi-controlled gates with relative-phase decomposition.

2) Direct Clause-to-Ansatz Generation:

D. Ansatz Optimizations

1) Relative-phase Decomposition:

  • Relative-phase Toffoli variants replace standard multi-controlled Toffolis; inverse operations cancel introduced phases while reducing CX and T gate counts.Source: 1) Relative-phase Decomposition · p. 5

2) Ancilla-assisted Decomposition:

SAQC's ancilla-assisted and dynamic-uncomputation methods reduce circuit costs, with experiments reporting lower depth, CX counts, and compilation time than the Qiskit workflow.

2) Ancilla-assisted Decomposition:

3) Dynamic Uncomputation:

  • Dynamic uncomputation measures the ancilla in the Hadamard basis and uses classically controlled CZ and Z corrections, replacing much of the inverse circuit with native operations.Source: 3) Dynamic Uncomputation · p. 5

IV. EXPERIMENTAL EVALUATION

  • Experiments used eCNF benchmarks, five-run averages for randomized tests, a native gate set, and clause-scheduling parameter k = 1200 unless stated otherwise.Source: §IV · p. 5

A. Comparison with Qiskit

B. Impact of Individual Optimizations

The optimization ablations show how clause scheduling and ancilla availability affect circuit depth; the conclusion reports resource reductions and summarizes SAQC’s unified CNF/eCNF compilation flow.

B. Impact of Individual Optimizations

V. CONCLUSION

  • SAQC preserves Boolean semantics in a SAT Intermediate Representation before quantum lowering.Source: §V · p. 6
  • From that representation, SAQC supports penalty-Hamiltonian generation and direct clause-to-ansatz synthesis for CNF and eCNF, with SAT-aware and ansatz-level optimizations.Source: §V · p. 6
  • SAQC significantly reduces circuit depth, two-qubit gate count, and compilation time compared with conventional Hamiltonian-based workflows.Source: §V · p. 6

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