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

2026-02-16

AI Summary

This work resolves a long-standing open problem by efficiently realizing high-dimensional irreducible representations of SU(n) and explicitly constructing Ramanujan quantum expanders. Leveraging the Jordan–Schwinger map, the authors embed SU(n) irreducible representations into the Hilbert space of n quantum harmonic oscillators and combine this with an efficient quantum Hermite transform to achieve, for the first time, fast quantum circuit synthesis for high-dimensional representations. The resulting quantum circuits exhibit gate complexity polynomial in log(N) and log(1/ε), providing the first explicit and efficient construction of Ramanujan quantum expanders. This advance also enables accelerated simulation of time evolution for certain quantum systems.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Security and Privacy: Applications of cryptography

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.

Problem

• quantum circuits
• SU(n) representations
• Ramanujan quantum expanders
• high-dimensional irreps

Innovation

• quantum circuits
• SU(n) representations
• Jordan-Schwinger map
• quantum Hermite transform
• Ramanujan quantum expanders