Detecting the purely imaginary Fisher zeros of an Ising spin system on a quantum computer

TL;DR

The paper presents an ancilla-free quantum-computer protocol to detect purely imaginary Fisher zeros of the Ising model by exploiting the unitary evolution corresponding to imaginary temperature, $Z(\alpha)=2^N\langle {\bf 0}|e^{-i\alpha H_{XX}}|{\bf 0}\rangle$. By mapping the Ising Hamiltonian to a commuting XX-like form via Hadamard conjugation, the evolution factors into simple gates, enabling direct measurement of $|Z(\alpha)|^2$ without extra qubits. The authors implement the protocol on ibm-lagos for a 3-spin chain, a 3-spin triangle, and a 7-spin cluster that reflects the device connectivity, observing zeros that align with theoretical predictions within hardware-imposed errors. This work demonstrates the feasibility of probing partition-function zeros on near-term quantum hardware and suggests routes to reconstruct thermodynamic information from Fisher zeros in finite systems. The study highlights the impact of gate and readout noise on zero detection and provides a platform for exploring finite-size signatures of phase behavior in quantum simulators.

Abstract

We propose a protocol for studying the purely imaginary Fisher zeros of the Ising model on a quantum computer. Our protocol is based on the direct relation between the partition function for purely imaginary temperature and the evolution operator of the Ising model. In this case, the inverse temperature is equal to the time of evolution. This protocol allows one to measure the zeros only those localized on the imaginary axes. We test this protocol on the ibm-lagos quantum computer in the cases of a 3-spin chain and a triangle cluster in a purely imaginary magnetic field, as well as a 7-spin cluster in which the interaction between spins reproduces the architecture of the quantum computer.

Figures (7)

• Figure 1: The protocol designed to measure the Fisher zeros of the spin-1/2 Ising model defined by Hamiltonian (\ref{['isingham']}). Here $\textsf{H}$ is the Hadamard operator.
• Figure 2: The structure of the ibm-lagos quantum computer.
• Figure 3: The structure of the 3-spin chain and triangle cluster. The interaction between spins is defined by value $J$.
• Figure 4: The protocol which allows one to measure the Fisher zeros of the three spin-1/2 system defined by Hamiltonian (\ref{['threespinsham']}).
• Figure 5: The plot depicts the square of the modulus of the partition function (\ref{['isingpartfunc3spins']}) for the 3-spin chain as a function of the $\alpha$. The two cases presented include $h=0$ (a) and $h=J$ (b), where $J$ denotes the coupling constant. The solid lines represent the theoretical predictions, while the dots correspond to the results obtained on the ibm-lagos quantum computer. Fisher zeros are identified at specific points, such as $J\alpha=2\pi$ and $6\pi$ for the case with $h=0$ (a), and $J\alpha=\pi$, $2\pi$, $3\pi$, $5\pi$, $6\pi$, $7\pi$ for the case with $h=J$ (b). Here we divided the partition function by 8.