Optimal simulation of three-qubit gates

TL;DR

This paper establishes the minimal number of two-qubit gates needed to simulate three-qubit gates, providing exact counts for the controlled controlled and Fredkin gates, advancing quantum circuit optimization.

Contribution

It characterizes the two-qubit gate cost for simulating three-qubit controlled gates and proves the optimality of five gates for the Fredkin gate, resolving longstanding open problems.

Findings

Four two-qubit gates are sufficient for controlled controlled gates with determinant 1.

Five two-qubit gates are necessary and sufficient for implementing the Fredkin gate.

The results settle the optimal gate count for key three-qubit gates in quantum computing.

Abstract

In this paper, we study the optimal simulation of three-qubit unitary by using two-qubit gates. First, we give a lower bound on the two-qubit gates cost of simulating a multi-qubit gate. Secondly, we completely characterize the two-qubit gate cost of simulating a three-qubit controlled controlled gate by generalizing our result on the cost of Toffoli gate. The function of controlled controlled gate is simply a three-qubit controlled unitary gate and can be intuitively explained as follows: the gate will output the states of the two control qubit directly, and apply the given one-qubit unitary $u$ on the target qubit only if both the states of the control are $\ket{1}$. Previously, it is only known that five two-qubit gates is sufficient for implementing such a gate [Sleator and Weinfurter, Phys. Rev. Lett. 74, 4087 (1995)]. Our result shows that if the determinant of $u$ is 1, four two-qubit gates is achievable optimal. Otherwise, five is optimal. Thirdly, we show that five two-qubit gates are necessary and sufficient for implementing the Fredkin gate(the controlled swap gate), which settles the open problem introduced in [Smolin and DiVincenzo, Phys. Rev. A, 53, 2855 (1996)]. The Fredkin gate is one of the most important quantum logic gates because it is universal alone for classical reversible computation, and thus with little help, universal for quantum computation. Before our work, a five two-qubit gates decomposition of the Fredkin gate was already known, and numerical evidence of showing five is optimal is found.