Observing algebraic variety of Lee-Yang zeros in asymmetrical systems via a quantum probe

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, and complex networks. 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 and internal couplings. 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 project 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 paper 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 paper integrates pure mathematical concepts into the realm of quantum simulations.