Quantum Simulation of the Sachdev-Ye-Kitaev Model by Asymmetric Qubitization

TL;DR

This paper presents a highly efficient quantum simulation method for the Sachdev-Ye-Kitaev model, achieving significant improvements in gate complexity and precision over previous algorithms by using a novel asymmetric qubitization technique.

Contribution

Introduces an asymmetric qubitization approach for simulating the SYK model with exponential improvements in efficiency and precision compared to prior methods.

Findings

Achieves gate complexity of O(N^{7/2} t + N^{5/2} t polylog(N/ε))

Provides exponential improvement in 1/ε over previous algorithms

Demonstrates efficient encoding of Hamiltonian using asymmetric projections

Abstract

We show that one can quantum simulate the dynamics of a Sachdev-Ye-Kitaev model with $N$ Majorana modes for time $t$ to precision $\epsilon$ with gate complexity $O(N^{7/2} t + N^{5/2} t \,{\rm polylog}(N/ \epsilon))$. In addition to scaling sublinearly in the number of Hamiltonian terms, this gate complexity represents an exponential improvement in $1/\epsilon$ and large polynomial improvement in $N$ and $t$ over prior state-of-the-art algorithms which scale as $O(N^{10} t^2 / \epsilon)$. Our approach involves a variant of the qubitization technique in which we encode the Hamiltonian $H$ as an asymmetric projection of a signal oracle $U$ onto two different signal states prepared by state oracles, $A\left\vert{0}\right\rangle \mapsto \left\vert{A}\right\rangle$ and $B \left\vert{0}\right\rangle \mapsto \left\vert{B}\right\rangle$, such that $H = \left\langle{B}\right\vert U\left\vert{A}\right\rangle$. Our strategy for applying this method to the Sachdev-Ye-Kitaev model involves realizing $B$ using only Hadamard gates and realizing $A$ as a random quantum circuit.