Trotter error and gate complexity of the SYK and sparse SYK models

TL;DR

This work develops rigorous bounds for Trotter error and gate counts in quantum simulations of the SYK model and its sparse variant using Lie–Trotter–Suzuki product formulas. By extending Chen–Brandão’s random-Hamiltonian toolkit to fermionic Majorana systems and leveraging random-matrix polynomial techniques, the authors derive explicit first-order and higher-order error bounds that depend on locality $k$, number of Majorana modes $n$, and randomness parameters, and translate these into near-optimal gate complexities. They show distinctive scaling depending on even vs. odd $k$, with higher-order formulas dramatically reducing gate counts, and they demonstrate substantial reductions when simulating a fixed input state. The sparse-SYK analysis yields favorable average gate complexities, $\overline{G}_{l\gg 2}=\mathcal{O}(n^2 t)$, and near-optimal fixed-input-state improvements, generalizing to broad Gaussian sparse Hamiltonians. Together, these results offer practical guidance for efficiently simulating strongly interacting fermionic systems on quantum hardware and highlight avenues for further tightening of error bounds and extending to broader random-Gaussian models.

Abstract

The Sachdev-Ye-Kitaev (SYK) model is a prominent model of strongly interacting fermions that serves as a toy model of quantum gravity and black hole physics. In this work, we study the Trotter error and gate complexity of the quantum simulation of the SYK model using Lie-Trotter-Suzuki formulas. Building on recent results by Chen and Brandao (arXiv:2111.05324), we derive bounds on the first- and higher-order Trotter error of the SYK model, and subsequently find near-optimal gate complexities for simulating these models using Lie-Trotter-Suzuki formulas. For the $k$-local SYK model on $n$ Majorana fermions, our gate complexity estimates for the first-order Lie-Trotter-Suzuki formula scales with $O(n^{k+\frac{5}{2}}t^2)$ for even $k$ and $O(n^{k+3}t^2)$ for odd $k$, and the gate complexity of simulations using higher-order formulas scales with $O(n^{k+\frac{1}{2}}t)$ for even $k$ and $O(n^{k+1}t)$ for odd $k$. Given that the SYK model has $Θ(n^k)$ terms, these estimates are close to optimal. These gate complexities can be further improved when simulating the time-evolution of an arbitrary fixed input state $|ψ\rangle$, leading to a $O(n^2)$-reduction in gate complexity for first-order formulas and $O(\sqrt{n})$-reduction for higher-order formulas. We also apply our techniques to the sparse SYK model, a simplified variant of the SYK model obtained by deleting all but a $Θ(n)$ fraction of the terms in a uniformly i.i.d. manner. We compute the average (over the random term removal) gate complexity for simulating this model using higher-order formulas to be $O(n^2 t)$, a bound that also holds for a general class of sparse Gaussian random Hamiltonians. Similar to the full SYK model, we obtain a $O(\sqrt{n})$-reduction simulating the time-evolution of an arbitrary fixed input state $|ψ\rangle$.

Figures (3)

• Figure 1: Scaling behavior of $\Delta_l$ in $n\in [6,500]$ for $l=2,4,6,8,10$ at fixed $k=4$, $t=10$, $r=100000$, $p=2$. When plotting these, we omitted the calculation of the prefactor $\mathcal{D}(l)$ because it gets too large for us to compute. Therefore, this plot does not give the correct prefactors but still captures the correct scaling behavior in $n$.
• Figure 2: Scaling comparison between the chromatic number $\chi_A$ of the anti-commutation graph of the SYK model $A$ and $Q(n,k)$ for even $n\in [6,38]$ and $k=4$. The chromatic number is estimated using a greedy coloring algorithm.
• Figure :

Theorems & Definitions (34)

• Theorem : Simplified version of Theorem \ref{['thm:FOE']}
• Theorem : Simplified version of Theorem \ref{['thm:SYKRPG']}
• Theorem : Simplified version of Theorem \ref{['thm:SSYKb']}
• Lemma 2.0: Concentration inequality for operator norm
• Lemma 2.0: Concentration inequality for fixed input state
• Theorem 2.1
• Theorem 3.1
• Lemma 3.1
• proof : Proof (of Theorem \ref{['thm:FOE']})
• Corollary 3.1.1