High-Performance Exact Synthesis of Two-Qubit Quantum Circuits

Exact synthesis provides unconditional optimality and canonical structure, but is often limited to small, carefully scoped regimes. We present an exact synthesis framework for two-qubit circuits over the Clifford+$T$ gate set that optimizes $T$-count exactly. Our approach exhausts a bounded search space, exploits algebraic canonicalization to avoid redundancy, and constructs a lookup table of optimal implementations that turns synthesis into a query. Algorithmically, we combine meet-in-the-middle ideas with provable pruning rules and problem-specific arithmetic designed for modern hardware. The result is an exact, reusable synthesis engine with substantially improved practical performance.

💡 Research Summary

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This paper tackles the problem of exact synthesis for two‑qubit quantum circuits in the Clifford+T gate set, with the explicit goal of minimizing the number of T‑gates, which dominate the resource cost in fault‑tolerant quantum computing. Exact synthesis differs from heuristic optimization in that it seeks provably optimal circuits for a given cost metric, rather than merely good approximations. While exact synthesis is known to be computationally hard in the worst case (NP‑hard, co‑NQP‑hard), the authors argue that many practical compilation pipelines repeatedly encounter small sub‑circuits whose optimal implementations can be pre‑computed and reused. Their contribution is a complete framework that exhaustively enumerates all two‑qubit Clifford+T circuits up to a prescribed T‑count, canonicalizes each circuit to eliminate redundancy, and stores a single optimal representative per equivalence class in a lookup table (LUT). Subsequent synthesis queries become constant‑time table look‑ups, dramatically accelerating compilation.

The technical core rests on three pillars: (1) a representation of two‑qubit unitaries in the real orthogonal group SO(6) via the well‑known two‑to‑one homomorphism Φ: SU(4) → SO(6). This replaces 4×4 complex matrices with 6×6 real matrices, allowing fast arithmetic and avoiding complex‑number overhead. (2) An equivalence relation that identifies circuits differing only by a global phase and left/right multiplication by Clifford operators. By quotienting out these symmetries, the search space shrinks dramatically. (3) A generating set G consisting of conjugates of the elementary T⊗I gate by signed permutation matrices (elements of the Clifford group). Lemma 3.3 proves that G generates the entire Clifford+T group J₂, so any circuit can be expressed as a product of G‑elements.

The algorithm proceeds in two stages. The outer stage performs a breadth‑first search over increasing T‑gate depth k, generating all possible products of G‑elements of length k, mapping each product to its SO(6) matrix, and reducing it to a canonical form (e.g., by sorting rows/columns according to a deterministic rule). If a newly generated matrix falls into an already‑seen equivalence class, it is discarded; otherwise it is stored as the optimal representative for that class. The inner stage implements a meet‑in‑the‑middle (MITM) strategy: for a target T‑count K, the algorithm splits K into k₁ + k₂, enumerates all depth‑k₁ and depth‑k₂ circuits, and checks whether the product of a depth‑k₁ matrix and the inverse of a depth‑k₂ matrix yields the target unitary (up to Clifford equivalence). Because the search space is effectively square‑rooted, the overall complexity drops from O(|G|^K) to O(|G|^{K/2}), a substantial gain.

To achieve practical performance, the authors implement a high‑performance backend that exploits modern CPU features: SIMD vectorization for matrix multiplication, multi‑threaded enumeration of independent branches, and custom data structures for the ring ℤ