Finding eigenvectors with a quantum variational algorithm

Abstract

This paper presents a hybrid variational quantum algorithm that finds a random eigenvector of a unitary matrix with a known quantum circuit. The algorithm is based on the SWAP test on trial states generated by a parametrized quantum circuit. The eigenvector is described by a compact set of classical parameters that can be used to reproduce the found approximation to the eigenstate on demand. This variational eigenvector finder can be adapted to solve the generalized eigenvalue problem, to find the eigenvectors of normal matrices and to perform quantum principal component analysis (QPCA) on unknown input mixed states. These algorithms can all be run with low depth quantum circuits, suitable for an efficient implementation on noisy intermediate state quantum computers (NISQC) and, with some restrictions, on linear optical systems. Limitations and potential applications are discussed.

arXiv comment18 pages, 7 figures. Comments welcome. v2.0 Revised version with a new section (comparison to alternative quantum algorithms) and small corrections

References in corpus (54)

• Quantum Computing in the NISQ era and beyond
• Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
• A variational eigenvalue solver on a quantum processor
• The Security of Practical Quantum Key Distribution
• Quantum algorithm for solving linear systems of equations
• Variational Quantum Algorithms
• Hardware-efficient Variational Quantum Eigensolver for Small Molecules and Quantum Magnets
• The theory of variational hybrid quantum-classical algorithms
• Noisy intermediate-scale quantum (NISQ) algorithms
• Quantum machine learning in feature Hilbert spaces
• Quantum computational chemistry
• Quantum Circuit Learning
• Secure quantum key distribution with realistic devices
• Quantum principal component analysis
• Quantum fingerprinting
• Parameterized quantum circuits as machine learning models
• Universal Linear Optics
• Quantum algorithms: an overview
• The Variational Quantum Eigensolver: a review of methods and best practices
• Circuit-centric quantum classifiers
• A quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors
• Quantum algorithms for quantum chemistry and quantum materials science
• Simulating Hamiltonian dynamics with a truncated Taylor series
• Connecting ansatz expressibility to gradient magnitudes and barren plateaus
• Self-Verifying Variational Quantum Simulation of the Lattice Schwinger Model
• Quantum circuits with many photons on a programmable nanophotonic chip
• Hybrid quantum-classical algorithms and quantum error mitigation
• Variational Quantum Computation of Excited States
• Quantum-assisted quantum compiling
• ``Plug and play'' systems for quantum cryptography
• Optical Simulation of Quantum Logic
• Learning the quantum algorithm for state overlap
• Exponential improvement in precision for simulating sparse Hamiltonians
• Variational Quantum State Diagonalization
• The SWAP test and the Hong-Ou-Mandel effect are equivalent
• Variational Quantum State Eigensolver
• Quantum principal component analysis only achieves an exponential speedup because of its state preparation assumptions
• Methodology for replacing indirect measurements with direct measurements
• Hybrid quantum-classical algorithms in the noisy intermediate-scale quantum era and beyond
• A 12-mode Universal Photonic Processor for Quantum Information Processing
• Crossing a topological phase transition with a quantum computer
• Quantum circuits cannot control unknown operations
• Gigahertz measurement-device-independent quantum key distribution using directly modulated lasers
• Simple factorization of unitary transformations
• Hamiltonian Simulation Algorithms for Near-Term Quantum Hardware
• Dimensional Expressivity Analysis of Parametric Quantum Circuits
• Reconfigurable continuously-coupled 3D photonic circuit for Boson Sampling experiments
• Further Compactifying Linear Optical Unitaries
• Generation of Universal Linear Optics by Any Beamsplitter
• Experimental approximation of the Jones polynomial with DQC1
• Cost function embedding and dataset encoding for machine learning with parameterized quantum circuits
• Detecting drift of quantum sources: not the de Finetti theorem
• Certified variational quantum algorithms for eigenstate preparation
• Quantum-classical algorithms for skewed linear systems with optimized Hadamard test