Efficient unitary designs with nearly time-independent Hamiltonian dynamics

TL;DR

This paper introduces new methods for constructing unitary t-designs on qubits and qudits, and proposes a design Hamiltonian that rapidly generates such designs through natural many-body dynamics.

Contribution

It presents novel constructions of unitary t-designs using mutually unbiased bases and introduces a design Hamiltonian with natural interactions for efficient randomization.

Findings

Unitary t-designs achieved after O(t) repetitions on a qudit.

Quantum circuits on N qubits with O(t N^2) gates produce t-designs for t = o(N^{1/2}).

Design Hamiltonian with spin-glass interactions enables fast unitary design generation.

Abstract

We provide new constructions of unitary $t$-designs for general $t$ on one qudit and $N$ qubits, and propose a design Hamiltonian, a random Hamiltonian of which dynamics always forms a unitary design after a threshold time, as a basic framework to investigate randomising time evolution in quantum many-body systems. The new constructions are based on recently proposed schemes of repeating random unitaires diagonal in mutually unbiased bases. We first show that, if a pair of the bases satisfies a certain condition, the process on one qudit approximately forms a unitary $t$-design after $O(t)$ repetitions. We then construct quantum circuits on $N$ qubits that achieve unitary $t$-designs for $t = o(N^{1/2})$ using $O(t N^2)$ gates, improving the previous result using $O(t^{10}N^2)$ gates in terms of $t$. Based on these results, we present a design Hamiltonian with periodically changing two-local spin-glass-type interactions, leading to fast and relatively natural realisations of unitary designs in complex many-body systems.