Efficient quantum circuits for high-dimensional representations of SU(n) and Ramanujan quantum expanders

TL;DR

This paper introduces efficient quantum circuits for high-dimensional $SU(n)$ representations using the Jordan-Schwinger approach, enabling applications like Ramanujan quantum expanders and fast-forwarding quantum system evolution.

Contribution

The authors develop polynomial-size quantum circuits for $SU(n)$ irreps based on harmonic oscillator representations, advancing quantum representation implementation.

Findings

Quantum circuits are polynomial in $ ext{log}(N)$ and $ ext{log}(1/\e)$.

Circuits enable explicit construction of Ramanujan quantum expanders.

Applications include fast-forwarding quantum system evolution.

Abstract

We present efficient quantum circuits that implement high-dimensional unitary irreducible representations (irreps) of $SU(n)$, where $n \ge 2$ is constant. For dimension $N$ and error $\epsilon$, the number of quantum gates in our circuits is polynomial in $\log(N)$ and $\log(1/\epsilon)$. Our construction relies on the Jordan-Schwinger representation, which allows us to realize irreps of $SU(n)$ in the Hilbert space of $n$ quantum harmonic oscillators. Together with a recent efficient quantum Hermite transform, which allows us to map the computational basis states to the eigenstates of the quantum harmonic oscillator, this allows us to implement these irreps efficiently. Our quantum circuits can be used to construct explicit Ramanujan quantum expanders, a longstanding open problem. They can also be used to fast-forward the evolution of certain quantum systems.