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

TL;DR

The work develops efficient quantum circuits for high-dimensional irreducible representations of $SU(n)$ (with fixed $n$) by embedding these representations in the oscillator (Jordan–Schwinger) framework and mapping to a finite discrete quantum harmonic oscillator via a quantum Hermite transform. By decomposing $SU(n)$ generators into a small, structured set of quadratic monomials in position and momentum, the authors achieve a circuit that is polylogarithmic in the representation dimension $N$ and the error $\epsilon$, up to a polynomial in $n$. Central to the construction are an efficient isometry from a computational basis to a Hermite-basis using descending lexicographic ordering and a polylogarithmic-cost QHT, together with a discretization that incurs exponentially small errors in the discretization parameter $L$. The results yield explicit Ramanujan quantum expanders and enable fast-forwarding of certain quantum dynamics, with broad implications for quantum simulation and information scrambling. Overall, the paper provides a concrete, scalable route to harness high-dimensional $SU(n)$ operations in quantum algorithms, while offering explicit constructions and concrete complexity bounds.

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 $ε$, 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.

Figures (2)

• Figure 1: The spectral gap of the quantum expander $\varepsilon_{\mathfrak{su}(2)}$ as a function of the dimension $N \ge 10$ of the representation. The spectral gap satisfies the optimal bound $1-\lambda(N) \ge 1-2\sqrt 5/6$.
• Figure 2: The spectral gap of the quantum expander obtained from Eq. \ref{['eq:appquaternion']} as a function of the dimension $N \ge 10$ of the representation for the $D=4$ case. The spectral gap satisfies the optimal bound $1-\lambda(N) \ge 1-2\sqrt 3/4$.

Theorems & Definitions (14)

• Theorem 1.1: Efficient high-dimensional $SU(n)$ circuits
• Corollary 1.2: Explicit Ramanujan quantum expanders
• Claim 2.1
• Lemma 2.2
• proof
• Lemma 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof