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arXiv:2303.11200 · Physical Review Letters

Parent Hamiltonian Reconstruction via Inverse Quantum Annealing

Davide Rattacaso, Gianluca Passarelli, Angelo Russomanno, Procolo Lucignano, Giuseppe E. Santoro, Rosario Fazio

Published Apr 16, 2024• 2 citations
arXivHTMLPDF

Abstract

Finding a local Hamiltonian H[over ^] that has a given many-body wave function |ψ⟩ as its ground state, i.e., a parent Hamiltonian, is a challenge of fundamental importance in quantum technologies. Here we introduce a numerical method, inspired by quantum annealing, that efficiently performs this task through an artificial inverse dynamics: a slow deformation of the states |ψ(λ(t))⟩, starting from a simple state |ψ_{0}⟩ with a known H[over ^]_{0}, generates an adiabatic evolution of the corresponding Hamiltonian. We name this approach inverse quantum annealing. The method, implemented through a projection onto a set of local operators, only requires the knowledge of local expectation values, and, for long annealing times, leads to an approximate parent Hamiltonian whose degree of locality depends on the correlations built up by the states |ψ(λ)⟩. We illustrate the method on two paradigmatic models: the Kitaev fermionic chain and a quantum Ising chain in longitudinal and transverse fields.

DOI
10.1103/physrevlett.132.160401

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Finding a local Hamiltonian H[over ^] that has a given many-body wave function |ψ⟩ as its ground state, i.e., a parent Hamiltonian, is a challenge of fundamental importance in quantum technologies. Here we introduce a numerical method, inspired by quantum annealing, that efficiently performs this task through an artificial inverse dynamics: a slow deformation of the states |ψ(λ(t))⟩, starting from a simple state |ψ_{0}⟩ with a known H[over ^]_{0}, generates an adiabatic evolution of the corresponding Hamiltonian. We name this approach inverse quantum annealing. The method, implemented through a projection onto a set of local operators, only requires the knowledge of local expectation values, and, for long annealing times, leads to an approximate parent Hamiltonian whose degree of locality depends on the correlations built up by the states |ψ(λ)⟩. We illustrate the method on two paradigmatic models: the Kitaev fermionic chain and a quantum Ising chain in longitudinal and transverse fields.
Traditional computational methods for studying quantum many-body systems are "forward methods," which take quantum models, i.e., Hamiltonians, as input and produce ground states as output.However, such forward methods often limit one's perspective to a small fraction of the space of possible Hamiltonians.We introduce an alternative computational "inverse method," the eigenstate-to-Hamiltonian construction (EHC), that allows us to better understand the vast space of quantum models describing strongly correlated systems.EHC takes as input a wave function jψ T i and produces as output Hamiltonians for which jψ T i is an eigenstate.This is accomplished by computing the quantum covariance matrix, a quantum mechanical generalization of a classical covariance matrix.EHC is widely applicable to a number of models and, in this work, we consider seven different examples.Using the EHC method, we construct a parent Hamiltonian with a new type of antiferromagnetic ground state, a parent Hamiltonian with two different targeted degenerate ground states, and large classes of parent Hamiltonians with the same ground states as well-known quantum models, such as the Majumdar-Ghosh model, the XX chain, the Heisenberg chain, the Kitaev chain, and a 2D BdG model.EHC gives an alternative inverse approach for studying quantum many-body phenomena.
We study the inverse problem of constructing an appropriate Hamiltonian from a physically reasonable set of orthogonal wave functions for a quantum spin system. Usually, we are given a local Hamiltonian and our goal is to characterize the relevant wave functions and energies (the spectrum) of the system. Here, we take the opposite approach; starting from a reasonable collection of orthogonal wave functions, we try to characterize the associated parent Hamiltonians, to see how the wave functions and the energy values affect the structure of the parent Hamiltonian. Specifically, we obtain (quasi) local Hamiltonians by a complete set of (multilayer) product states and a local mapping of the energy values to the wave functions. On the other hand, a complete set of tree wave functions (having a tree structure) results to nonlocal Hamiltonians and operators which flip simultaneously all the spins in a single branch of the tree graph. We observe that even for a given set of basis states, the energy spectrum can significantly change the nature of interactions in the Hamiltonian. These effects can be exploited in a quantum engineering problem optimizing an objective functional of the Hamiltonian.
Given a generic time-dependent many-body quantum state, we determine the associated parent Hamiltonian. This procedure may require, in general, interactions of any sort. Enforcing the requirement of a fixed set of engineerable Hamiltonians, we find the optimal Hamiltonian once a set of realistic elementary interactions is defined. We provide three examples of this approach. We first apply the optimization protocol to the ground states of the one-dimensional Ising model and a ferromagnetic ppp-spin model but with time-dependent coefficients. We also consider a time-dependent state that interpolates between a product state and the ground state of a ppp-spin model. We determine the time-dependent optimal parent Hamiltonian for these states and analyze the capability of this Hamiltonian of generating the state evolution. Finally, we discuss the connections of our approach to shortcuts to adiabaticity.
We review here some recent work in the field of quantum annealing, alias adiabatic quantum computation. The idea of quantum annealing is to perform optimization by a quantum adiabatic evolution which tracks the ground state of a suitable time-dependent Hamiltonian, where 'ℏ' is slowly switched off. We illustrate several applications of quantum annealing strategies, starting from textbook toy-models—double-well potentials and other one-dimensional examples, with and without disorder. These examples display in a clear way the crucial differences between classical and quantum annealing. We then discuss applications of quantum annealing to challenging hard optimization problems, such as the random Ising model, the travelling salesman problem and Boolean satisfiability problems. The techniques used to implement quantum annealing are either deterministic Schrödinger's evolutions, for the toy models, or path-integral Monte Carlo and Green's function Monte Carlo approaches, for the hard optimization problems. The crucial role played by disorder and the associated non-trivial Landau–Zener tunnelling phenomena is discussed and emphasized.
Computing the ground state properties of quantum systems is an important potential application of quantum computing. The success probability of quantum phase estimation approaches to ground state problems is proportional to the overlap of the input state with the ground state. Local ansatz state approaches suffer from the orthogonality catastrophe, whereas adiabatic state preparation (ASP) can prepare good approximations with a cost growing with the inverse square of the minimum spectral gap along the adiabatic path. When the initial and target Hamiltonians lie in quantum phases separated by a first order phase transition, the minimum spectral gap along the adiabatic path becomes exponentially small as a function of system size. We pursue a solution to this problem based on choosing an initial Hamiltonian for adiabatic state preparation whose ground state lies in the same quantum phase as the target ground state. We develop a protocol for universal adiabatic warm starts with universal parent Hamiltonians (UPHAWS) that can initialize ASP in any state whose preparation circuit is known. We use the Feynman--Kitaev clock Hamiltonian as a universal parent Hamiltonian, for preparation circuits with and without mid circuit measurement. We benchmark the framework on a Z2\mathbb{Z}_2Z2​-symmetric matrix product state (MPS) family interpolating to a target GHZ Hamiltonian, and on the linear H6H_6H6​ chain under symmetric bond stretching. For the H6H_6H6​ system a bond-dimension-444 matrix product state warm-start increases the minimum gap on the adiabatic path by a factor of two relative to the Hartree-Fock initialization. To perform these classical benchmark simulations we develop a momentum-space truncation of the adiabatic Hamiltonian that may be of independent interest.
Abstract Quantum annealing solves combinatorial optimization problems by finding the energetic ground states of an embedded Hamiltonian. However, quantum annealing dynamics under the embedded Hamiltonian may violate the principles of adiabatic evolution and generate excitations that correspond to errors in the computed solution. Here we empirically benchmark the probability of chain breaks and identify sweet spots for solving a suite of embedded Hamiltonians. We further correlate the physical location of chain breaks in the quantum annealing hardware with the underlying embedding technique and use these localized rates in a tailored post-processing strategies. Our results demonstrate how to use characterization of the quantum annealing hardware to tune the embedded Hamiltonian and remove computational errors.