Precision of quantum simulation of all-to-all coupling in a local architecture

TL;DR

This paper introduces a 2D local circuit design for implementing all-to-all quantum interactions with quantifiable error, requiring feasible control precision for current hardware sizes, and compares its efficiency to existing methods like minor embedding.

Contribution

The authors develop an analytic framework for a 2D local circuit implementing all-to-all couplings with controlled error, extending the applicability of perturbative gadgets to practical quantum hardware.

Findings

Constant relative error requires energy scale scaling as n^6.

For 40 qubits, 5 digits of control precision suffice at ε=0.1.

Our method outperforms minor embedding in chain length efficiency.

Abstract

We present a simple 2d local circuit that implements all-to-all interactions via perturbative gadgets. We find an analytic relation between the values $J_{ij}$ of the desired interaction and the parameters of the 2d circuit, as well as the expression for the error in the quantum spectrum. For the relative error to be a constant $\epsilon$, one requires an energy scale growing as $n^6$ in the number of qubits, or equivalently a control precision up to $ n^{-6}$. Our proof is based on the Schrieffer-Wolff transformation and generalizes to any hardware. In the architectures available today, $5$ digits of control precision are sufficient for $n=40,~ \epsilon =0.1$. Comparing our construction, known as paramagnetic trees, to ferromagnetic chains used in minor embedding, we find that at chain length $>3$ the performance of minor embedding degrades exponentially with the length of the chain, while our construction experiences only a polynomial decrease.