Simulation of the 1d XY model on a quantum computer

TL;DR

This paper presents a comprehensive method for simulating the 1-D XY model on quantum computers, including Hamiltonian diagonalization, energy spectrum access, and time evolution, enabling studies of quantum phase transitions and benchmarking quantum algorithms.

Contribution

The paper introduces a novel quantum circuit design for exact time evolution of the 1-D XY model, advancing simulation capabilities on quantum computers.

Findings

Successfully diagonalized the XY Hamiltonian for small spin chains.

Computed ground and excited state energies, observing quantum phase transitions.

Simulated time evolution demonstrating the algorithm's scalability and performance.

Abstract

The field of quantum computing has grown fast in recent years, both in theoretical advancements and the practical construction of quantum computers. These computers were initially proposed, among other reasons, to efficiently simulate and comprehend the complexities of quantum physics. In this paper, we present a comprehensive scheme for the exact simulation of the 1-D XY model on a quantum computer. We successfully diagonalize the proposed Hamiltonian, enabling access to the complete energy spectrum. Furthermore, we propose a novel approach to design a quantum circuit to perform exact time evolution. Among all the possibilities this opens, we compute the ground and excited state energies for the symmetric XY model with spin chains of $n=4$ and $n=8$ spins. Further, we calculate the expected value of transverse magnetization for the ground state in the transverse Ising model. Both studies allow the observation of a quantum phase transition from an antiferromagnetic to a paramagnetic state. Additionally, we have simulated the time evolution of the state all spins up in the transverse Ising model. The scalability and high performance of our algorithm make it an ideal candidate for benchmarking purposes, while also laying the foundation for simulating other integrable models on quantum computers.