Variational Quantum Algorithms for Trace Distance and Fidelity Estimation

Abstract

Estimating the difference between quantum data is crucial in quantum computing. However, as typical characterizations of quantum data similarity, the trace distance and quantum fidelity are believed to be exponentially-hard to evaluate in general. In this work, we introduce hybrid quantum-classical algorithms for these two distance measures on near-term quantum devices where no assumption of input state is required. First, we introduce the Variational Trace Distance Estimation (VTDE) algorithm. We in particular provide the technique to extract the desired spectrum information of any Hermitian matrix by local measurement. A novel variational algorithm for trace distance estimation is then derived from this technique, with the assistance of a single ancillary qubit. Notably, VTDE could avoid the barren plateau issue with logarithmic depth circuits due to a local cost function. Second, we introduce the Variational Fidelity Estimation (VFE) algorithm. We combine Uhlmann's theorem and the freedom in purification to translate the estimation task into an optimization problem over a unitary on an ancillary system with fixed purified inputs. We then provide a purification subroutine to complete the translation. Both algorithms are verified by numerical simulations and experimental implementations, exhibiting high accuracy for randomly generated mixed states.

arXiv comment15 pages, v2 updated some results; v3 published version

References in corpus (60)

• Quantum Computing in the NISQ era and beyond
• A computable measure of entanglement
• Quantum Machine Learning
• A variational eigenvalue solver on a quantum processor
• Variational Quantum Algorithms
• Hardware-efficient Variational Quantum Eigensolver for Small Molecules and Quantum Magnets
• Barren plateaus in quantum neural network training landscapes
• The theory of variational hybrid quantum-classical algorithms
• Quantum Coherence as a Resource
• Quantum Circuit Learning
• Quantum Error Correction for Quantum Memories
• An introduction to quantum machine learning
• Quantum Resource Theories
• Quantum fingerprinting
• The Role of Relative Entropy in Quantum Information Theory
• Measuring entanglement entropy through the interference of quantum many-body twins
• Parameterized quantum circuits as machine learning models
• Evaluating analytic gradients on quantum hardware
• An adaptive variational algorithm for exact molecular simulations on a quantum computer
• Circuit-centric quantum classifiers
• Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms
• On quantum Renyi entropies: a new generalization and some properties
• Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Renyi relative entropy
• The Resource Theory of Stabilizer Computation
• Min- and Max- Relative Entropies and a New Entanglement Monotone
• Theory of variational quantum simulation
• Universal Quantum Estimator
• Direct estimations of linear and non-linear functionals of a quantum state
• An initialization strategy for addressing barren plateaus in parametrized quantum circuits
• Layerwise learning for quantum neural networks
• Hierarchical quantum classifiers
• Subspace-search variational quantum eigensolver for excited states
• A universal quantum circuit for two-qubit transformations with three CNOT gates
• The SWAP test and the Hong-Ou-Mandel effect are equivalent
• Sequential minimal optimization for quantum-classical hybrid algorithms
• Variational algorithms for linear algebra
• Quantum generative adversarial learning in a superconducting quantum circuit
• A quantum Fredkin gate
• Large gradients via correlation in random parameterized quantum circuits
• Variational Thermal Quantum Simulation via Thermofield Double States
• Variational Quantum Fidelity Estimation
• Measuring the Renyi entropy of a two-site Fermi-Hubbard model on a trapped ion quantum computer
• Efficiently computable bounds for magic state distillation
• Experimental investigation of initial system-environment correlations via trace distance evolution
• Variational Quantum Singular Value Decomposition
• Quantum Singular Value Decomposer
• Cost of quantum entanglement simplified
• Quantifying Bell non-locality with the trace distance
• A Variational Quantum Algorithm for Preparing Quantum Gibbs States
• Variational Hamiltonian Diagonalization for Dynamical Quantum Simulation
• Subsystem Trace Distance in Quantum Field Theory
• The Fidelity of Recovery is Multiplicative
• Universal and Operational Benchmarking of Quantum Memories
• Practical distributed quantum information processing with LOCCNet
• Irreversibility of Asymptotic Entanglement Manipulation Under Quantum Operations Completely Preserving Positivity of Partial Transpose
• Sequential decoding of a general classical-quantum channel
• Learning unknown pure quantum states
• Simpler semidefinite programs for completely bounded norms
• Variational quantum Gibbs state preparation with a truncated Taylor series
• What The Trace Distance Security Criterion in Quantum Key Distribution Does And Does Not Guarantee

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