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arXiv:2309.04933 · Quantum Engineering

Twirling Operations to Produce Energy Eigenstates of a Hamiltonian by Classically Emulated Quantum Simulation

Kazuto Oshima

Published Jan 1, 2024• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1155/2024/3854366

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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.
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.
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.
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 quantum algorithm to obtain the lowest eigenstate of any Hamiltonian simulated by a quantum computer. The proposed algorithm begins with an arbitrary initial state of the simulated system. A finite series of transforms is iteratively applied to the initial state assisted with an ancillary qubit. The fraction of the lowest eigenstate in the initial state is then amplified up to ≃1\simeq 1≃1. We prove that our algorithm can faithfully work for any arbitrary Hamiltonian in the theoretical analysis. Numerical analyses are also carried out. We firstly provide a numerical proof-of-principle demonstration with a simple Hamiltonian in order to compare our scheme with the so-called Demon-like algorithmic (DLAC), recently proposed in [Nature Photonics 8, 113 (2014)]. The result shows a good agreement with our theoretical analysis, exhibiting the comparable behavior to the best cooling with the DLAC method. We then consider a random Hamiltonian model for further analysis of our algorithm. By numerical simulations, we show that the total number ncn_cnc​ of iterations is proportional to ≃O(D−1ϵ−0.19)\simeq {\cal O}(D^{-1}\epsilon^{-0.19})≃O(D−1ϵ−0.19), where DDD is the difference between the two lowest eigenvalues, and ϵ\epsilonϵ is an error defined as the probability that the finally obtained system state is in an unexpected (i.e. not the lowest) eigenstate.
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.