Efficient Approximation of Diagonal Unitaries over the Clifford+T Basis

TL;DR

This paper introduces an efficient algorithm for approximating diagonal unitaries over the Clifford+T basis, minimizing T-gate count and entanglement cost, with advantages over previous methods especially for large qubit systems.

Contribution

The paper presents a novel decomposition algorithm that reduces T-gate and entanglement costs for diagonal unitaries, outperforming existing techniques in specific parameter regimes.

Findings

The T-count is bounded by a function of phases, precision, and entanglement cost.

The algorithm achieves lower entanglement cost than previous methods in certain parameter regions.

It can exponentially reduce T-gates in the number of qubits when phases are sparse.

Abstract

We present an algorithm for the approximate decomposition of diagonal operators, focusing specifically on decompositions over the Clifford+$T$ basis, that minimize the number of phase-rotation gates in the synthesized approximation circuit. The equivalent $T$-count of the synthesized circuit is bounded by $k \, C_0 \log_2(1/\varepsilon) + E(n,k)$, where $k$ is the number of distinct phases in the diagonal $n$-qubit unitary, $\varepsilon$ is the desired precision, $C_0$ is a quality factor of the implementation method ($1<C_0<4$), and $E(n,k)$ is the total entanglement cost (in $T$ gates). We determine an optimal decision boundary in $(k,n,\varepsilon)$-space where our decomposition algorithm achieves lower entanglement cost than previous state-of-the-art techniques. Our method outperforms state-of-the-art techniques for a practical range of $\varepsilon$ values and diagonal operators and can reduce the number of $T$ gates exponentially in $n$ when $k << 2^n$.