Quantum algorithms for estimating quantum entropies

Abstract

The von Neumann and quantum Rényi entropies characterize fundamental properties of quantum systems and lead to theoretical and practical applications in many fields. Quantum algorithms for estimating quantum entropies, using a quantum query model that prepares the purification of the input state, have been established in the literature. {However, constructing such a model is almost as hard as state tomography.} In this paper, we propose quantum algorithms to estimate the von Neumann and quantum $\alpha$-Rényi entropies of an $n$-qubit quantum state $\rho$ using independent copies of the input state. We also show how to efficiently construct the quantum circuits for {quantum entropy estimation} using primitive single/two-qubit gates. We prove that the number of required copies scales polynomially in $1/\epsilon$ and $1/\Lambda$, where $\epsilon$ denotes the additive precision and $\Lambda$ denotes the lower bound on all non-zero eigenvalues. Notably, our method outperforms previous methods in the aspect of practicality since it does not require any quantum query oracles, which are usually necessary for previous methods. Furthermore, we conduct experiments to show the efficacy of our algorithms to single-qubit states and study the noise robustness. We also discuss the applications to some quantum states of practical interest as well as some meaningful tasks such as quantum Gibbs state preparation and entanglement estimation.

arXiv comment36 pages, 8 figures

References in corpus (31)

• Quantum entanglement
• Concentrating Partial Entanglement by Local Operations
• Simulated Quantum Computation of Molecular Energies
• Quantum principal component analysis
• Quantum entanglement in condensed matter systems
• Theory of variational quantum simulation
• Universal Quantum Estimator
• Direct estimations of linear and non-linear functionals of a quantum state
• Measuring Renyi Entanglement Entropy with Quantum Monte Carlo
• The Gravity Dual of Renyi Entropy
• Towards Quantum Machine Learning with Tensor Networks
• Fast Reset and Suppressing Spontaneous Emission of a Superconducting Qubit
• Demonstrating a Driven Reset Protocol of a Superconducting Qubit
• Hamiltonian Learning and Certification Using Quantum Resources
• Fast and Unconditional All-Microwave Reset of a Superconducting Qubit
• Initialization by measurement of a two-qubit superconducting circuit
• Quantum Simulations of Classical Annealing Processes
• Learning a local Hamiltonian from local measurements
• Variational Quantum Eigensolver with Fewer Qubits
• Variational Thermal Quantum Simulation via Thermofield Double States
• Sample-efficient learning of quantum many-body systems
• Holographic quantum algorithms for simulating correlated spin systems
• Pulsed reset protocol for fixed-frequency superconducting qubits
• Entanglement spectroscopy with a depth-two quantum circuit
• Evaluation of Entanglement Entropy in High Energy Elastic Scattering
• Quantum algorithm for estimating Renyi entropies of quantum states
• Jacob Bekenstein's Universes: From Black Holes to Modified Theories of Gravity
• Qubit-efficient entanglement spectroscopy using qubit resets
• Measuring Quantum Entropy
• Coherent Feedback Improved Qubit Initialization in the Dispersive Regime
• Spectral thresholding quantum tomography for low rank states

Cited by in corpus (14)

• New Quantum Algorithms for Computing Quantum Entropies and Distances
• Multivariate trace estimation in constant quantum depth
• Quantum Phase Processing and its Applications in Estimating Phase and Entropies
• Quantum Neural Estimation of Entropies
• Quantum Algorithms for Inverse Participation Ratio Estimation in multi-qubit and multi-qudit systems
• Efficient Quantum Algorithm for Filtering Product States
• A Quantum Algorithm Framework for Discrete Probability Distributions with Applications to Rényi Entropy Estimation
• The topology of data hides in quantum thermal states
• Time-Efficient Quantum Entropy Estimator via Samplizer
• Parallel Quantum Signal Processing Via Polynomial Factorization
• Virtual Quantum Markov Chains
• Disentangling quantum neural networks for unified estimation of quantum entropies and distance measures
• Performance Guarantees for Quantum Neural Estimation of Entropies
• Qubit frugal entanglement determination with the deep multi-scale entanglement renormalization ansatz