Provably Optimal Quantum Circuits with Mixed-Integer Programming

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.