Block Encodings of Discrete Subgroups on Quantum Computer

TL;DR

This paper presents a novel block encoding technique for representing discrete subgroups on quantum computers, enabling the implementation of primitive gates for complex groups like $ ext{BI}$ and $ ext{V}$, with resource estimates and experimental benchmarks.

Contribution

The paper introduces a new block encoding method for discrete subgroups on quantum computers, including the first implementation for the $ ext{BI}$ group, and details gate constructions and resource estimations.

Findings

Primitive gates for $ ext{BT}$ and $ ext{BI}$ groups constructed.

Benchmark fidelities of 40% and 4% on quantum hardware.

Resource estimates for extracting gluon viscosity.

Abstract

We introduce a block encoding method for mapping discrete subgroups to qubits on a quantum computer. This method is applicable to general discrete groups, including crystal-like subgroups such as $\mathbb{BI}$ of $SU(2)$ and $\mathbb{V}$ of $SU(3)$. We detail the construction of primitive gates -- the inversion gate, the group multiplication gate, the trace gate, and the group Fourier gate -- utilizing this encoding method for $\mathbb{BT}$ and for the first time $\mathbb{BI}$ group. We also provide resource estimations to extract the gluon viscosity. The inversion gates for $\mathbb{BT}$ and $\mathbb{BI}$ are benchmarked on the $\texttt{Baiwang}$ quantum computer with estimated fidelities of $40^{+5}_{-4}\%$ and $4^{+5}_{-3}\%$ respectively.