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arXiv:2112.04625 · Symmetry

Determining Ground-State Phase Diagrams on Quantum Computers via a Generalized Application of Adiabatic State Preparation

Akhil Francis, Ephrata Zelleke, Ziyue Zhang, Alexander F. Kemper, James K. Freericks

Published Apr 13, 2022• 6 citations
arXivHTMLPDF

Abstract

Quantum phase transitions materialize as level crossings in the ground-state energy when the parameters of the Hamiltonian are varied. The resulting ground-state phase diagrams are straightforward to determine by exact diagonalization on classical computers, but are challenging on quantum computers because of the accuracy needed and the near degeneracy of the competing states close to the level crossings. On the other hand, classical computers are limited to small system sizes, which quantum computers may help overcome. In this work, we use a local adiabatic ramp for state preparation to allow us to directly compute ground-state phase diagrams on a quantum computer via time evolution. This methodology is illustrated by examining the ground states of the XY model with a magnetic field in the z-direction in one dimension. We are able to calculate an accurate phase diagram on both two- and three-site systems using IBM quantum machines.

DOI
10.3390/sym14040809

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Quantum phase transitions materialize as level crossings in the ground-state energy when the parameters of the Hamiltonian are varied. The resulting ground-state phase diagrams are straightforward to determine by exact diagonalization on classical computers, but are challenging on quantum computers because of the accuracy needed and the near degeneracy of the competing states close to the level crossings. On the other hand, classical computers are limited to small system sizes, which quantum computers may help overcome. In this work, we use a local adiabatic ramp for state preparation to allow us to directly compute ground-state phase diagrams on a quantum computer via time evolution. This methodology is illustrated by examining the ground states of the XY model with a magnetic field in the z-direction in one dimension. We are able to calculate an accurate phase diagram on both two- and three-site systems using IBM quantum machines.
Background: Solving nuclear many-body problems with an ab initio approach is widely recognized as a computationally challenging problem. Quantum computers offer a promising path to address this challenge. There are urgent needs to develop quantum algorithms for this purpose. Objective: In this work, we explore the application of the quantum algorithm of adiabatic state preparation with quantum phase estimation in ab initio nuclear structure theory. We focus on solving the low-lying spectra (including both the ground and excited states) of simple nuclear systems. Ideas: The efficiency of this algorithm is hindered by the emergence of small energy gaps (level crossings) during the adiabatic evolution. In order to improve the efficiency, we introduce techniques to avoid level crossings: 1) by suitable design of the reference Hamiltonian; 2) by insertions of perturbation terms to modify the adiabatic path. Results: We illustrate this algorithm by solving the deuteron ground state energy and the spectrum of the deuteron bounded in a harmonic oscillator trap implementing the IBM Qiskit quantum simulator. The quantum results agree well the classical results obtained by matrix diagonalization. Outlook: With our improvements to the efficiency, this algorithm provides a promising tool for investigating the low-lying spectra of complex nuclei on future quantum computers.
Abstract Mapping out phase diagrams of quantum systems using classical simulations can be challenging or intractable due to the computational resources required to simulate even small quantum systems far away from the thermodynamic limit. We investigate using quantum computers and the variational quantum eigensolver (VQE) for this task. In contrast to the task of preparing the exact ground state using VQE, sketching phase diagrams might require less quantum resources and accuracy, because low fidelity approximations to the ground state may be enough to correctly identify different phases. We used classical numerical simulations of low-depth VQE circuits to compute order parameters for four well-studied spin and fermion models which represent a mix of 1D and 2D, and exactly-solvable and classically hard systems. We find that it is possible to predict the location of phase transitions up to reasonable accuracy using states produced by VQE even when their overlap with the true ground state is small. Further, we introduce a model-agnostic predictor of phase transitions based on the speed with which the VQE energy improves with respect to the circuit depth, and find that in some cases this is also able to predict phase transitions.
Quantum computing allows for the manipulation of highly correlated states whose properties quickly go beyond the capacity of any classical method to calculate. Thus one natural problem which could lend itself to quantum advantage is the study of ground-states of condensed matter models, and the transitions between them. However, current levels of hardware noise can require extensive application of error-mitigation techniques to achieve reliable computations. In this work, we use several IBM devices to explore a finite-size spin model with multiple “phaselike” regions characterized by distinct ground-state configurations. Using preoptimized Variational Quantum Eigensolver (VQE) solutions, we demonstrate that in contrast to calculating the energy, where zero-noise extrapolation is required in order to obtain qualitatively accurate yet still unreliable results, calculations of the energy derivative, two-site spin correlation functions, and the fidelity susceptibility yield accurate behavior across multiple regions, even with minimal or no application of error-mitigation approaches. Taken together, these sets of observables could be used to identify level crossings in a simple, noise-robust manner which is agnostic to the method of ground state preparation. This work shows promising potential for near-term application to identifying quantum phase transitions, including avoided crossings and nonadiabatic conical intersections in electronic structure calculations. Published by the American Physical Society 2024
One of the main applications of future quantum computers will be the simulation of quantum models. While the evolution of a quantum state under a Hamiltonian is straightforward (if sometimes expensive), using quantum computers to determine the ground state phase diagram of a quantum model and the properties of its phases is more involved. Using the Hubbard model as a prototypical example, we here show all the steps necessary to determine its phase diagram and ground state properties on a quantum computer. In particular, we discuss strategies for efficiently determining and preparing the ground state of the Hubbard model starting from various mean-field states with broken symmetry. We present an efficient procedure to prepare arbitrary Slater determinants as initial states and present the complete set of quantum circuits needed to evolve from these to the ground state of the Hubbard model. We show that, using efficient nesting of the various terms each time step in the evolution can be performed with just O(N)\mathcal{O}(N)O(N) gates and O(log⁡N)\mathcal{O}(\log N)O(logN) circuit depth. We give explicit circuits to measure arbitrary local observables and static and dynamic correlation functions, both in the time and frequency domain. We further present efficient non-destructive approaches to measurement that avoid the need to re-prepare the ground state after each measurement and that quadratically reduce the measurement error.
Many physically interesting models show a quantum phase transition when a single parameter is varied through a critical point, where the ground state and the first excited state become degenerate. When this parameter appears as a coupling constant, these models can be understood as straight-line interpolations between different Hamiltonians HI{H}_{\mathrm{I}}HI​ and HF{H}_{\mathrm{F}}HF​. For finite-size realizations however, there will usually be a finite energy gap between ground and first excited state. By slowly changing the coupling constant through the point with the minimum energy gap one thereby has an adiabatic algorithm that prepares the ground state of HF{H}_{\mathrm{F}}HF​ from the ground state of HI{H}_{\mathrm{I}}HI​. The adiabatic theorem implies that in order to obtain a good preparation fidelity the runtime $\ensuremath{\tau}$ should scale with the inverse energy gap and thereby also with the system size. In addition, for open quantum systems not only nonadiabatic but also thermal excitations are likely to occur. It is shown that---using only local Hamiltonians---for the one-dimensional quantum Ising model and the cluster model in a transverse field the conventional straight-line path can be replaced by a series of straight-line interpolations, along which the fundamental energy gap is always greater than a constant independent on the system size. The results are of interest for adiabatic quantum computation since strong similarities between adiabatic quantum algorithms and quantum phase transitions exist.
We argue the feasibility of studying the phase structure of a quantum physical system on quantum devices via adiabatic preparation of states. We introduce a method and successfully test it in application to the Schwinger model in the presence of a topological $\ensuremath{\theta}$ term. We explore the first-order-phase-transition and the no-transition regions of the corresponding phase diagram. The core idea of the method is to separately evolve the ground and the first excited states with a time-dependent Hamiltonian, the time dependence of which interpolates between different values of $\ensuremath{\theta}$. Despite our approach being a direct application of the adiabatic theorem, in some cases we are able to demonstrate its advantages in comparison to a different method from the literature that also employs adiabatic state preparation.