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 $ε$, the number of quantum gates in our circuits is polynomial in $\log(N)$ and $\log(1/ε)$. 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.

翻译：我们提出了实现SU(n)（其中n≥2为常数）高维酉不可约表示的高效量子电路。对于维度N和误差ε，我们电路中的量子门数量在log(N)和log(1/ε)上呈多项式级。我们的构建依赖于Jordan-Schwinger表示，该表示允许我们在n个量子谐振子的希尔伯特空间中实现SU(n)的不可约表示。结合近期提出的高效量子埃尔米特变换（该变换可将计算基态映射到量子谐振子的本征态），我们得以高效实现这些不可约表示。我们的量子电路可用于构建显式的拉马努金量子扩展器，这是一个长期存在的开放问题。它们也可用于加速特定量子系统的演化过程。