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Quantum algorithms for the generalized eigenvalue problem

Jin‐Min Liang, Shu‐Qian Shen, Ming Li +1 more

arXiv:2112.02554Quantum Information ProcessingDec 28, 202116 citations

Quantum algorithm for measuring the eigenvalues of UÄU-1 for a black-box unitary transformation U

Dominik Janzing, T. Beth

Estimating the eigenvalues of a unitary transformation U by standard phase estimation requires the implementation of controlled-U-gates which are not available if U is only given as a black box. We show that a simple trick allows to measure eigenvalues of U\otimesU^\deggar even in this case. Running the algorithm several times allows therefore to estimate the autocorrelation function of the density of eigenstates of U. This can be applied to find periodicities in the energy spectrum of a quantum system with unknown Hamiltonian if it can be coupled to a quantum computer.Estimating the eigenvalues of a unitary transformation U by standard phase estimation requires the implementation of controlled-U-gates which are not available if U is only given as a black box. We show that a simple trick allows to measure eigenvalues of U\otimesU^\deggar even in this case. Running the algorithm several times allows therefore to estimate the autocorrelation function of the density of eigenstates of U. This can be applied to find periodicities in the energy spectrum of a quantum system with unknown Hamiltonian if it can be coupled to a quantum computer.

Quantum Information and ComputationApr 1, 20021 citation

Adaptive identification of low-degree polynomials in quantum singular value transformation: application to nonlinear quantum properties estimation

Jumpei Kato, Akira Tanji, Hiroyuki Harada +3 more

Estimating properties of unknown quantum states via quantum singular value transformation (QSVT) often requires high-degree polynomials to handle small eigenvalues of density matrices. Specifically, the existing approaches determine the polynomial degree by relying on overly conservative worst-case bounds based on the minimum non-zero eigenvalue or the rank of the density matrices. In this work, we propose a spectral cutoff method that truncates the negligible eigenvalue tail depending on the task, the target accuracy, and the state, which enables the use of significantly lower-degree polynomials. To implement this, we develop a two-stage algorithm to estimate nonlinear properties, particularly von Neumann entropy and R{é}nyi entropy. In the first stage, we execute a search algorithm to identify the spectral cutoff directly from the unknown quantum state. In the second stage, we estimate the nonlinear properties utilizing QSVT with the degree of polynomial adaptively determined by the cutoff. This two-stage algorithm significantly improves the overall estimation cost compared to known bounds, even without knowing the minimum eigenvalue or the rank.Estimating properties of unknown quantum states via quantum singular value transformation (QSVT) often requires high-degree polynomials to handle small eigenvalues of density matrices. Specifically, the existing approaches determine the polynomial degree by relying on overly conservative worst-case bounds based on the minimum non-zero eigenvalue or the rank of the density matrices. In this work, we propose a spectral cutoff method that truncates the negligible eigenvalue tail depending on the task, the target accuracy, and the state, which enables the use of significantly lower-degree polynomials. To implement this, we develop a two-stage algorithm to estimate nonlinear properties, particularly von Neumann entropy and R{é}nyi entropy. In the first stage, we execute a search algorithm to identify the spectral cutoff directly from the unknown quantum state. In the second stage, we estimate the nonlinear properties utilizing QSVT with the degree of polynomial adaptively determined by the cutoff. This two-stage algorithm significantly improves the overall estimation cost compared to known bounds, even without knowing the minimum eigenvalue or the rank.

quant-pharXiv:2606.10994Jun 10, 20260 citations

Quantum techniques for eigenvalue problems

Dean Lee

arXiv:2307.03889The European Physical Journal ANov 20, 20234 citations

Quantum Gates Implementation for Reinforcement Learning Decision- Making Process

Aarash Maroufian

Abstract This paper introduces a novel method to improve the decision-making process of reinforcement learning agents via quantum information technology methods. In this approach, states |s〉 of the system are replaced by quantum states (eigenfunctions) of the system |ψ〉, in which a system can be in a superposition of states, and rewards of each step are calculated based on the calculated eigenvalues of the previous step. The agent decides the next step of the system based on the result of the quantum gate’s effect on the available options. Due to the high-speed, escalated performance of quantum algorithms, this method will improve the performance of reinforcement learning agents in unknown environments.Abstract This paper introduces a novel method to improve the decision-making process of reinforcement learning agents via quantum information technology methods. In this approach, states |s〉 of the system are replaced by quantum states (eigenfunctions) of the system |ψ〉, in which a system can be in a superposition of states, and rewards of each step are calculated based on the calculated eigenvalues of the previous step. The agent decides the next step of the system based on the result of the quantum gate’s effect on the available options. Due to the high-speed, escalated performance of quantum algorithms, this method will improve the performance of reinforcement learning agents in unknown environments.

Oct 27, 20220 citations

An Eigenvalue Problem for Quantum Computing

Selman Hershfield

Bulletin of the American Physical SocietyMar 6, 20070 citations

Calculating unknown eigenvalues with a quantum algorithm

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