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arXiv:1206.4747 · Physical Review A

Quantum algorithm for solving some discrete mathematical problems by probing their energy spectra

Hefeng Wang, Heng Fan, Fuli Li

Published Jan 6, 2014• 3 citations
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Abstract

When a probe qubit is coupled to a quantum register that represents a physical system, the probe qubit will exhibit a dynamical response only when it is resonant with a transition in the system. Using this principle, we propose a quantum algorithm for solving discrete mathematical problems based on the circuit model. Our algorithm has favorable scaling properties in solving some discrete mathematical problems.

DOI
10.1103/physreva.89.012306

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When a probe qubit is coupled to a quantum register that represents a physical system, the probe qubit will exhibit a dynamical response only when it is resonant with a transition in the system. Using this principle, we propose a quantum algorithm for solving discrete mathematical problems based on the circuit model. Our algorithm has favorable scaling properties in solving some discrete mathematical problems.
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.
Linear matrix systems represent a fundamental class of problems in mathematics, physics, and engineering, and their efficient solution is crucial for various applications. Traditional methods often rely on computationally intensive algorithms, limiting scalability. This paper explores the potential of leveraging the inherent non-linear properties of linear matrices to design novel quantum algorithms for solving linear systems. We propose a quantum algorithm predicated on the principle that by mapping a linear system into a non-linear representation via a specific matrix transformation, we can achieve speedups that surpass classical approaches. The core mechanism involves constructing a quantum circuit that systematically explores this non-linear space, leveraging quantum phenomena like superposition and entanglement to enhance the search process. The paper details the design of a quantum algorithm for solving linear systems, emphasizing the benefits of this approach and its potential to unlock new possibilities in quantum computation.