Provably Optimal Quantum Circuits with Mixed-Integer Programming

2025-10-01

AI Summary

Quantum circuit compilation faces inherent trade-offs among optimality, scalability, and hardware adaptability—particularly across fault-tolerant and NISQ regimes. Method: We propose a depth-aware mixed-integer linear programming (MILP) optimization framework. It introduces explicit parallel scheduling variables and domain-specific pruning constraints to accelerate branch-and-bound search, and employs a receding-horizon optimization (RHO) strategy to balance local fidelity and global consistency. Innovatively, it integrates phase-invariant fidelity maximization, a linear trace-overlap surrogate objective, and convex quadratic Frobenius-norm error modeling. Results: Our method certifiably achieves optimal solutions for small-to-medium-scale circuits. On standard benchmarks, it attains the minimal possible depth. For multi-body parity circuits, it reduces gate count by 36%; a 142-gate seed circuit is compressed to 116 gates. End-to-end compilation speedup reaches up to 43× over state-of-the-art methods.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

We present a depth-aware optimization framework for quantum circuit compilation that unifies provable optimality with scalable heuristics. For exact synthesis of a target unitary, we formulate a mixed-integer linear program (MILP) that linearly handles global-phase equivalence and uses explicit parallel scheduling variables to certify depth-optimal solutions for small-to-medium circuits. Domain-specific valid constraints, including identity ordering, commuting-gate pruning, short-sequence redundancy cuts, and Hermitian-conjugate linkages, significantly accelerate branch-and-bound, yielding speedups up to 43x on standard benchmarks. The framework supports hardware-aware objectives, enabling fault-tolerant (e.g. T-count) and NISQ-era (e.g. entangling gates) devices. For approximate synthesis, we propose 3 objectives: (i) exact, but non-convex, phase-invariant fidelity maximization; (ii) a linear surrogate that maximizes the real trace overlap, yielding a tight lower bound to fidelity; and (iii) a convex quadratic function that minimizes the circuit's Frobenius error. To scale beyond exact MILP, we propose a novel rolling-horizon optimization (RHO) that rolls primarily in time, caps the active-qubits, and enforces per-qubit closure while globally optimizing windowed segments. This preserves local context, reduces the Hilbert-space dimension, and enables iterative improvements without ancillas. On a 142-gate seed circuit, RHO yields 116 gates, an 18.3% reduction from the seed, while avoiding the trade-off between myopic passes and long run times. Empirically, our exact compilation framework achieves certified depth-optimal circuits on standard targets, high-fidelity Fibonacci-anyon weaves, and a 36% gate-count reduction on multi-body parity circuits. All methods are in the open-source QuantumCircuitOpt, providing a single framework that bridges exact certification and scalable synthesis.

Problem

• Optimizing quantum circuit depth with mixed-integer programming
• Accelerating exact synthesis via domain-specific constraints and heuristics
• Scaling compilation for hardware-aware and approximate synthesis objectives

Innovation

• Mixed-integer programming for depth-optimal quantum circuits
• Rolling-horizon optimization with active-qubit capping
• Domain-specific constraints to accelerate branch-and-bound