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IET Quantum Communication

A Novel Procedure for Generating Energy Eigenstates From Physical States by Classically Emulated Quantum Simulation: The Hydrogen Molecule as an Example

Kazuto Oshima

Published Jan 1, 2025• 0 citations
Publisher

Abstract

ABSTRACT We propose a novel procedure for generating an arbitrary energy eigenstate from physical states. To validate it, we demonstrate the generation of an energy eigenstate of the molecule from a superposition of energy eigenstates by classically emulated quantum simulation. We use a Hamiltonian represented by Pauli matrices and concatenated ancilla qubits. Starting from an adequate initial state of physical qubits, we generate a corresponding energy eigenstate by twirling operations, which evolve the system under the Hamiltonian controlled by the ancilla qubits.

DOI
10.1049/qtc2.70021

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•Aug 25, 2023•0 citations
ABSTRACT We propose a novel procedure for generating an arbitrary energy eigenstate from physical states. To validate it, we demonstrate the generation of an energy eigenstate of the molecule from a superposition of energy eigenstates by classically emulated quantum simulation. We use a Hamiltonian represented by Pauli matrices and concatenated ancilla qubits. Starting from an adequate initial state of physical qubits, we generate a corresponding energy eigenstate by twirling operations, which evolve the system under the Hamiltonian controlled by the ancilla qubits.
We construct superposition states of energy eigenstates of the hydrogen molecule using classically emulated digital quantum simulation. We generate the ground state and excited states of the system via the twirling operation method, and construct superposition states of the ground state and an excited state of the system by applying a controlled excitation unitary transformation on the ground state with one ancillary qubit as the control. To verify the correctness of the resulting superposition state, we calculate matrix elements of several physical observables.
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.
We present a quantum algorithm that provides a general approach for obtaining the energy spectrum of a physical system without making a guess on its eigenstates. In this algorithm, a probe qubit is coupled to a quantum register R which consists of one ancilla qubit and an n-qubit register that represents the system. R is prepared in a general reference state, and a general excitation operator that acts on R is constructed. The probe exhibits a dynamical response only when it is resonant with a transition from the reference state to an excited state of R which contains the eigenstates of the system. By varying the probe's frequency, the energy spectrum and the eigenstates of the system can be obtained.
Abstract In this article, the hydrogen molecular ground-state energies using our algorithm based on quantum variational principle are calculated. They are calculated through a simulator since the system of the present study (i.e., the hydrogen molecule) is relatively small and hence the ground-state energies for this molecule are efficiently classically simulable using a simulator. Complete details of this algorithm are elucidated. For this, a full description on the fermions–qubits and the molecular Hamiltonian–qubit Hamiltonian transformations, is given. The authors search for qubit system parameters ( θ 0 {\theta }_{0} and θ 1 {\theta }_{1} ) that yield the minimum energies for the system and also study the ground state energies as a function of the molecular bond length. Proposed circuit is humble and does not include many parameters compared with that of Kandala et al., the authors control only two parameters ( θ 0 {\theta }_{0} and θ 1 {\theta }_{1} ).
It is difficult to calculate the energy levels and eigenstates of a large physical system on a classical computer because of the exponentially growing size of the Hilbert space. In this work, we experimentally demonstrate a quantum algorithm which could solve this problem via simulated resonant transitions. Using a four-qubit quantum simulator in which two qubits are used as ancillas for control and measurement, we obtain the energy spectrum of a 2-qubit low-energy effective Hamiltonian of the water molecule. The simulated transitions allow the state of the quantum simulator to transform and access large regions of the Hilbert space, including states that have no overlap with the initial state. Furthermore, we make use of this algorithm to efficiently prepare specific eigenstates on the simulator according to the measured eigenenergies.
Abstract In this chapter, we present quantum algorithms to approximate Hamiltonian eigenstates. We start discussing the quantum phase estimation algorithms, an important subroutine that allows us to approximate eigenpairs of a Hamiltonian operator given an initial guess and a quantum circuit implementing the time evolution operator. We then present the adiabatic state preparation, a technique to transform the ground state of a Hamiltonian into the ground state of another Hamiltonian. We describe two important heuristic methods, the quantum approximate optimization algorithm, and the variational quantum eigensolver. The second part of the chapter discusses concrete applications in the field of quantum chemistry. After introducing the Born-Oppenheimer Hamiltonian in second quantization and the Hartree-Fock method, we illustrate how a chemical problem can be mapped onto a quantum computer. We conclude the chapter with the simulation of the hydrogen molecule in a minimal basis with the iterative phase estimation algorithm.