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arXiv:1510.00820 · Physical Review A

Quantum algorithm for obtaining the eigenstates of a physical system

Hefeng Wang

Published May 25, 2016• 20 citations
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Abstract

We propose a quantum algorithm for solving the following problem: given the Hamiltonian of a physical system and one of its eigenvalues, how do we obtain the corresponding eigenstate? The algorithm is based on the resonance phenomenon. For a probe qubit coupled to a quantum system, the system exhibits resonance dynamics when the frequency of the probe qubit matches a transition frequency in the system. Therefore the system can be guided to evolve to the eigenstate with a known eigenvalue by inducing the resonance between the probe qubit and a designed transition in the system. This algorithm can also be used to obtain the energy spectrum of a physical system and can achieve even quadratic speedup over the phase estimation algorithm.

DOI
10.1103/physreva.93.052334

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We propose a quantum algorithm for solving the following problem: given the Hamiltonian of a physical system and one of its eigenvalues, how do we obtain the corresponding eigenstate? The algorithm is based on the resonance phenomenon. For a probe qubit coupled to a quantum system, the system exhibits resonance dynamics when the frequency of the probe qubit matches a transition frequency in the system. Therefore the system can be guided to evolve to the eigenstate with a known eigenvalue by inducing the resonance between the probe qubit and a designed transition in the system. This algorithm can also be used to obtain the energy spectrum of a physical system and can achieve even quadratic speedup over the phase estimation algorithm.
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.
We present a polynomial-time quantum algorithm for obtaining the energy spectrum of a physical system, i.e., the differences between the eigenvalues of the system's Hamiltonian, provided that the spectrum of interest contains at most a polynomially increasing number of energy levels. A probe qubit is coupled to a quantum register that represents the system of interest such that the probe exhibits a dynamical response only when it is resonant with a transition in the system. By varying the probe's frequency and the system-probe coupling operator, any desired part of the energy spectrum can be obtained. The algorithm can also be used to deterministically prepare any energy eigenstate. As an example, we have simulated running the algorithm and obtained the energy spectrum of the water molecule.
In the problem of ground-state energy estimation, one aims to estimate the smallest eigenvalue of a Hamiltonian, often given a guiding state, with some promised overlap γγγ with the ground space. The main approach to this problem is to simulate its evolution, and estimate the smallest (or equivalently, largest) eigenphase of the resulting unitary UUU. We give a quantum algorithm that estimates the largest eigenphase of a unitary UUU in this guided setting using a factor of log⁡1γ\log\frac{1}γlogγ1​ fewer queries to UUU than the previous best approach. The result matches an existing lower bound, and answers an open question from Mande and de Wolf. The algorithm is based on transducers, which often allow composition of quantum algorithms without overhead from error reduction.