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Progress of Theoretical and Experimental Physics

Quantum Entanglement to Extract a True Ground State from an Approximate Ground State

Kazuto Oshima

Published Nov 1, 2024• 1 citation
Publisher

Abstract

Abstract We propose a procedure to extract a true ground state from an approximate ground state by using quantum entanglement. We use plural ancilla qubits with hierarchical structure, intending to gradually improve the precision of the approximation. We demonstrate the procedure for quantum systems with a few qubits that derive from the (1+1)-dimensional Schwinger model by classically emulated digital quantum simulation. Although we use for simplicity an approximate ground state prepared by adiabatic quantum computation, our procedure is applicable to any approximate ground state that is a superposition of a true ground state and excited states. Our procedure is applicable for an $N$-qubits Hamiltonian with a nondegenerate ground state.

DOI
10.1093/ptep/ptae164

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Abstract We propose a procedure to extract a true ground state from an approximate ground state by using quantum entanglement. We use plural ancilla qubits with hierarchical structure, intending to gradually improve the precision of the approximation. We demonstrate the procedure for quantum systems with a few qubits that derive from the (1+1)-dimensional Schwinger model by classically emulated digital quantum simulation. Although we use for simplicity an approximate ground state prepared by adiabatic quantum computation, our procedure is applicable to any approximate ground state that is a superposition of a true ground state and excited states. Our procedure is applicable for an NNN-qubits Hamiltonian with a nondegenerate ground state.
Preparing the ground state of a system is an important task in physics. We propose a quantum algorithm for preparing the ground state of a physical system that can be simulated on a quantum computer. The system is coupled to an ancillary qubit, by introducing a resonance mechanism between the ancilla qubit and the system, and combined with measurements performed on the ancilla qubit, the system can be evolved to monotonically converge to its ground state through an iterative procedure. We have simulated the application of this algorithm for the Afflect-Kennedy-Lieb-Tasaki model, whose ground state can be used as resource state in one-way quantum computation.
We propose a simple procedure to produce energy eigenstates of a Hamiltonian with discrete eigenvalues. We use ancilla qubits and quantum entanglement to separate an energy eigenstate from the other energy eigenstates. We exhibit a few examples derived from the (1 + 1)‐dimensional massless Schwinger model. Our procedure in principle will be applicable for a Hamiltonian with a finite‐dimensional Hilbert space. Choosing an initial state properly, we can in principle produce any energy eigenstate of the Hamiltonian.
A variety of quantum computing algorithms exist for the preparation of approximate Hamiltonian ground states. A natural and important question is how these ground-state approximations can be further improved using adiabatic state preparation. Here, we present a heuristic method to carry out adiabatic state preparation starting from a generic initial wavefunction. Given a quantum circuit that prepares the initial wavefunction, and a target Hamiltonian for which one wishes to prepare the ground state, we present an algorithm to construct an adiabatic path between these two states. This method works by approximating a parent Hamiltonian for the initial wavefunction, and the quality of this approximation can be can be checked prior to running the ASP algorithm. We apply this technique to simulate the ground state of water and the lowest-lying multireference singlet state of methylene, using various initial wavefunctions as the starting point of the adiabatic path.
We propose a general-purpose quantum algorithm for preparing ground states of quantum Hamiltonians from a given trial state. The algorithm is based on techniques recently developed in the context of solving the quantum linear system problem. We show that, compared to algorithms based on phase estimation, the runtime of our algorithm is exponentially better as a function of the allowed error, and at least quadratically better as a function of the overlap with the trial state. We also show that our algorithm requires fewer ancilla qubits than existing algorithms, making it attractive for early applications of small quantum computers. Additionally, it can be used to determine an unknown ground energy faster than with phase estimation if a very high precision is required.