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arXiv:2002.12621 · Journal of the Physical Society of Japan

Excited State Search Using Quantum Annealing

Yuya Seki, Yuichiro Matsuzaki, Shiro Kawabata

Published Apr 7, 2021• 6 citations
arXivHTMLPDF

Abstract

Quantum annealing (QA) is one of the ways to search the ground state of the problem Hamiltonian. Here, we propose the QA scheme to search arbitrary excited states of the problem Hamiltonian. In our scheme, an $n$-th excited state of the trivial Hamiltonian is initially prepared and is adiabatically changed into an $n$-th excited state of the target Hamiltonian. Although our scheme is general such that we can search any excited states, we especially discuss the first excited state search in this paper. As a comparison, we consider a non-adiabatic scheme to find the first excited state with non-adiabatic transitions from the ground state. By solving the Lindblad master equation, we evaluate the performance of each scheme under the influence of decoherence. Our conclusion is that the adiabatic scheme show better performance than the non-adiabatic scheme as long as the coherence time of qubits is sufficiently long. These results are important for applications in the area of quantum chemistry, quantum simulation, and post-quantum cryptography.

DOI
10.7566/jpsj.90.054002

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Quantum annealing (QA) is one of the ways to search the ground state of the problem Hamiltonian. Here, we propose the QA scheme to search arbitrary excited states of the problem Hamiltonian. In our scheme, an nnn-th excited state of the trivial Hamiltonian is initially prepared and is adiabatically changed into an nnn-th excited state of the target Hamiltonian. Although our scheme is general such that we can search any excited states, we especially discuss the first excited state search in this paper. As a comparison, we consider a non-adiabatic scheme to find the first excited state with non-adiabatic transitions from the ground state. By solving the Lindblad master equation, we evaluate the performance of each scheme under the influence of decoherence. Our conclusion is that the adiabatic scheme show better performance than the non-adiabatic scheme as long as the coherence time of qubits is sufficiently long. These results are important for applications in the area of quantum chemistry, quantum simulation, and post-quantum cryptography.
Quantum annealing is a way to prepare an eigenstate of the problem Hamiltonian. Starting from an eigenstate of a trivial Hamiltonian, we slowly change the Hamiltonian to the problem Hamiltonian, and the system remains in the eigenstate of the Hamiltonian as long as the so-called adiabatic condition is satisfied. By using devices provided by D-Wave Systems Inc., there were experimental demonstrations to prepare a ground state of the problem Hamiltonian. However, up to date, there are no demonstrations to prepare the excited state of the problem Hamiltonian with quantum annealing. Here, we demonstrate the excited-state search by using the D-wave processor. The key idea is to use the reverse quantum annealing with a hot start where the initial state is the excited state of the trivial Hamiltonian. During the reverse quantum annealing, we control not only the transverse field but also the longitudinal field and slowly change the Hamiltonian to the problem Hamiltonian so that we can obtain the desired excited state. As an example of the exited state search, we adopt a two-qubit Ising model as the problem Hamiltonian and succeed to prepare the excited state. Also, we solve the shortest vector problem where the solution is embedded into the first excited state of the Ising Hamiltonian. Our results pave the way for new applications of quantum annealers to use the excited states.
Quantum chemistry is one of the important applications of quantum information technology. Especially, an estimation of the energy gap between a ground state and excited state of a target Hamiltonian corresponding to a molecule is essential. In the previous approach, an energy of the ground state and that of the excited state are estimated separately, and the energy gap can be calculated from the subtraction between them. Here, we propose a direct estimation of the energy gap between the ground state and excited state of the target Hamiltonian with quantum annealing. The key idea is to combine a Ramsey type measurement with the quantum annealing. This provides an oscillating signal with a frequency of the energy gap, and a Fourier transform of the signal let us know the energy gap. Based on typical parameters of superconducting qubits, we numerically investigate the performance of our scheme when we estimate an energy gap between the ground state and first excited state of the Hamiltonian. We show robustness against non-adiabatic transitions between the ground state and first-excited state. Our results pave a new way to estimate the energy gap of the Hamiltonian for quantum chemistry.
Quantum annealing (QA) is a promising approach for not only solving combinatorial optimization problems but also simulating quantum many-body systems such as those in condensed matter physics. However, non-adiabatic transitions constitute a key challenge in QA. The choice of the drive Hamiltonian is known to affect the performance of QA because of the possible suppression of non-adiabatic transitions. Here, we propose the use of a drive Hamiltonian that preserves the symmetry of the problem Hamiltonian for more efficient QA. Owing to our choice of the drive Hamiltonian, the solution is searched in an appropriate symmetric subspace during QA. As non-adiabatic transitions occur only inside the specific subspace, our approach can potentially suppress unwanted non-adiabatic transitions. To evaluate the performance of our scheme, we employ the XY model as the drive Hamiltonian in order to find the ground state of problem Hamiltonians that commute with the total magnetization along the zzz axis. We find that our scheme outperforms the conventional scheme in terms of the fidelity between the target ground state and the states after QA.
Quantum chemistry is one of the important applications of quantum information technology.Especially, an estimation of the energy gap between a ground state and excited state of a target Hamiltonian corresponding to a molecule is essential.In the previous approach, an energy of the ground state and that of the excited state are estimated separately, and the energy gap can be calculated from the subtraction between them.Here, we propose a direct estimation of the energy gap between the ground state and excited state of the target Hamiltonian with quantum annealing.The key idea is to combine a Ramsey type measurement with the quantum annealing.This provides an oscillating signal with a frequency of the energy gap, and a Fourier transform of the signal let us know the energy gap.Based on typical parameters of superconducting qubits, we numerically investigate the performance of our scheme when we estimate an energy gap between the ground state and first excited state of the Hamiltonian.We show robustness against non-adiabatic transitions between the ground state and first-excited state.Our results pave a new way to estimate the energy gap of the Hamiltonian for quantum chemistry.
Abstract The shortest vector problem (SVP) is one of the lattice problems and is the mathematical basis for lattice-based cryptography, which is expected to be post-quantum cryptography. The SVP can be mapped onto the Ising problem, which in principle can be solved by quantum annealing (QA). However, one issue in solving the SVP using QA is that the solution of the SVP corresponds to the first excited state of the problem Hamiltonian. Therefore, QA, which searches for ground states, cannot provide a solution with high probability. In this paper, we propose to adopt an excited-state search of the QA to solve the shortest vector problem. We numerically show that the excited-state search provides a solution with a higher probability than the ground-state search.
Abstract In quantum chemistry, it is important to estimate an energy gap between a ground state and an excited state of molecular Hamiltonians. In previous researches, it was necessary to measure the energy of the ground state and that of the excited state separately, and the energy gap was estimated from the subtraction between them. Here, we show a novel scheme to estimate such an energy gap in a more direct manner. We use a concept of a Ramsey type measurement in the quantum annealing (QA) for such a direct estimation of the energy gap. In our scheme, the measured signal oscillates with a frequency of the energy gap. To study the performance of our scheme, we perform numerical simulations. The results show our scheme is robust against non-adiabatic transitions between the ground state and first excited state. Our scheme paves the way for the practical application of QA in quantum chemistry.