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High-Performance Exact Synthesis of Two-Qubit Quantum Circuits

Andrew N. Glaudell, Michael Jarret, Swan Klein, Samuel S. Mendelson, T. C. Mooney, Mingzhen Tian

TL;DR

The paper tackles the intractable problem of exact two-qubit Clifford+$T$ synthesis by constructing a pay-once, query-for-life database of $T$-optimal circuits. It combines bounded exhaustive enumeration with a canonicalization scheme in an $ ext{SO}(6)$ representation and a meet-in-the-middle strategy to produce provably minimal $T$-count results, then implements a high-performance, HPC-aware engine that uses problem-specific arithmetic and parallelization. Key contributions include correctness-proven pruning rules, a compact involutive generating set for representatives, and a highly optimized inner kernel that replaces generic linear algebra with constant-time, gate-set-aware operations. The resulting LUT and synthesis engine offer a practical backend for compilers, enabling fast lookup-based optimization and serving as ground truth for evaluating heuristic methods, with potential extensions to larger gate sets and alternative cost metrics through the same outer enumeration–inner arithmetic framework.

Abstract

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.

High-Performance Exact Synthesis of Two-Qubit Quantum Circuits

TL;DR

The paper tackles the intractable problem of exact two-qubit Clifford+ synthesis by constructing a pay-once, query-for-life database of -optimal circuits. It combines bounded exhaustive enumeration with a canonicalization scheme in an representation and a meet-in-the-middle strategy to produce provably minimal -count results, then implements a high-performance, HPC-aware engine that uses problem-specific arithmetic and parallelization. Key contributions include correctness-proven pruning rules, a compact involutive generating set for representatives, and a highly optimized inner kernel that replaces generic linear algebra with constant-time, gate-set-aware operations. The resulting LUT and synthesis engine offer a practical backend for compilers, enabling fast lookup-based optimization and serving as ground truth for evaluating heuristic methods, with potential extensions to larger gate sets and alternative cost metrics through the same outer enumeration–inner arithmetic framework.

Abstract

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+ gate set that optimizes -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.
Paper Structure (46 sections, 10 theorems, 26 equations, 2 figures, 1 table, 2 algorithms)

This paper contains 46 sections, 10 theorems, 26 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Preliminaries
  4. Clifford+T synthesis
  5. Theory
  6. Notation
  7. Equivalence classes
  8. The SO(6) representation
  9. An involutive generating set for representatives
  10. Optimal Exact Synthesis Algorithm
  11. Implementation and Optimization
  12. Design constraints and the inner loop
  13. Fast group action: T-steps as linear maps
  14. Why generic multiplication is the wrong abstraction.
  15. Compile-time specialization and constant-time dispatch.
  16. ...and 31 more sections

Key Result

Theorem 3.1

There exists a two-to-one cover $\Phi:\mathrm{SU}(4)\to \operatorname{SO}(6)$. For $U\in\mathrm{SU}(4)$ we write $\overline{U}=\Phi(U)$. In particular, $\overline{U}=\overline{-U}$ and $\overline{iU}=\overline{-iU}=-\overline{U}$.

Figures (2)

  • Figure 1: Time to build a complete lookup table of all unique representatives (up to equivalence) for a particular T-count.
  • Figure 2: Time to perform MITM search with no pre-built database for a representative. The horizontal axis is the size of an optimal $T$-count circuit for that representative. We plot against the depth-optimized and not$T$-depth optimized circuits found by Amy2013. $T$-depth optimized circuits were too slow to be plotted on the same scale.

Theorems & Definitions (19)

  • Theorem 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • proof
  • Theorem 3.6
  • proof
  • ...and 9 more