Multi-qubit circuit synthesis and Hermitian lattices

TL;DR

This paper introduces new optimal and heuristic algorithms for the exact synthesis of multi-qubit quantum circuits, leveraging Hermitian lattice theory and advanced algebraic techniques to improve efficiency and provide new insights.

Contribution

The paper presents novel algorithms for multi-qubit circuit synthesis using Hermite and Smith Normal Forms, and introduces a new data structure and heuristics for efficient search.

Findings

Optimal algorithms find Clifford and T circuits with entries in specific rings.

Best-first search runtime is proportional to the T-count for certain gate sets.

Techniques rely on Hermite and Smith Normal Forms and Hermitian lattice theory.

Abstract

We present new optimal and heuristic algorithms for exact synthesis of multi-qubit unitaries and isometries. For example, our algorithms find Clifford and T circuits for unitaries with entries in $\mathbb{Z}[i,1/\sqrt{2}]$. The optimal algorithms are the A* search instantiated with a new data structure for graph vertices and new consistent heuristic functions. We also prove that for some gate sets, best-first search synthesis relying on the same heuristic is efficient. For example, for two-qubit Clifford and T circuits, our best-first search runtime is proportional to the T-count of the unitary. Our algorithms rely on Hermite and Smith Normal Forms of matrices with entries in a ring of integers of a number field, and we leverage the theory of and algorithms for Hermitian lattices over number fields to prove efficiency. These new techniques are of independent interest for future work on multi-qubit exact circuit synthesis and related questions.