Observing Algebraic Variety of Lee-Yang Zeros in Asymmetrical Systems via a Quantum Probe

TL;DR

This paper introduces a quantum probe method to experimentally determine Lee-Yang zeros in asymmetrical classical systems, revealing their algebraic variety in a higher-dimensional complex plane, and demonstrates this with a three-qubit NMR system.

Contribution

It presents a novel qubit-based protocol for extracting Lee-Yang zeros in asymmetrical systems without requiring control over system qubits, expanding quantum simulation capabilities.

Findings

Successfully extracted Lee-Yang zeros in a three-qubit NMR experiment.

Mapped the algebraic variety of Lee-Yang zeros using amoeba and coamoeba projections.

Demonstrated the method's applicability to complex, asymmetrical classical systems.

Abstract

Lee-Yang (LY) zeros, points on the complex plane of physical parameters where the partition function goes to zero, have found diverse applications across multiple disciplines like statistical physics, protein folding, percolation, complex networks etc. However, experimental extraction of the complete set of LY zeros for general asymmetrical classical systems remains a crucial challenge to put those applications into practice. Here, we propose a qubit-based method to simulate an asymmetrical classical Ising system, enabling the exploration of LY zeros at arbitrary values of physical parameters like temperature, internal couplings etc. Without assuming system symmetry, the full set of LY zeros forms an algebraic variety in a higher-dimensional complex plane. To determine this variety, we pro ject it into sets representing magnitudes (amoeba ) and phases (coamoeba ) of LY zeros. Our approach uses a probe qubit to initialize the system and to extract LY zeros without assuming any control over the system qubits. This is particularly important as controlling system qubits can get intractable with the increasing complexity of the system. Initializing the system at an amoeba point, coamoeba points are sampled by measuring probe qubit dynamics. Iterative sampling yields the entire algebraic variety. Experimental demonstration of the protocol is achieved through a three-qubit NMR register. This work expands the horizon of quantum simulation to domains where identifying LY zeros in general classical systems is pivotal. Moreover, by extracting abstract mathematical objects like amoeba and coamoeba for a given polynomial, our study integrates pure mathematical concepts into the realm of quantum simulations.