Near-optimal quantum circuit construction via Cartan decomposition

TL;DR

This paper introduces a method using Cartan decomposition to efficiently synthesize quantum circuits with near-optimal CNOT gate counts, applicable to any unitary operation.

Contribution

It presents a recursive algorithm for quantum circuit synthesis based on Lie algebra Cartan decomposition, achieving near-optimal CNOT gate counts.

Findings

Explicit circuit representations for Lie algebra generators.

Recursive algorithm for circuit expansion to individual qubits.

Asymptotic CNOT cost of 21/16 * 4^n for n qubits.

Abstract

We show the applicability of the Cartan decomposition of Lie algebras to quantum circuits. This approach can be used to synthesize circuits that can efficiently implement any desired unitary operation. Our method finds explicit quantum circuit representations of the algebraic generators of the relevant Lie algebras allowing the direct implementation of a Cartan decomposition on a quantum computer. The construction is recursive and allows us to expand any circuit down to generators and rotation matrices on individual qubits, where through our recursive algorithm we find that the generators themselves can be expressed with controlled-not (CNOT) and SWAP gates explicitly. Our approach is independent of the standard CNOT implementation and can be easily adapted to other cross-qubit circuit elements. In addition to its versatility, we also achieve near-optimal counts when working with CNOT gates, achieving an asymptotic cnot cost of $\frac{21}{16}4^n$ for $n$ qubits.