Quantum natural gradient generalised to noisy and non-unitary circuits

Abstract

Variational quantum algorithms are promising tools whose efficacy depends on their optimisation method. For noise-free unitary circuits, the quantum generalisation of natural gradient descent has been introduced and shown to be equivalent to imaginary time evolution: the approach is effective due to a metric tensor reconciling the classical parameter space to the device's Hilbert space. Here we generalise quantum natural gradient to consider arbitrary quantum states (both mixed and pure) via completely positive maps; thus our circuits can incorporate both imperfect unitary gates and fundamentally non-unitary operations such as measurements. We employ the quantum Fisher information (QFI) as the core metric in the space of density operators. A modification of the Error Suppression by Derangements (ESD) and Virtual Distillation (VD) techniques enables an accurate and experimentally-efficient approximation of the QFI via the Hilbert-Schmidt metric tensor using prior results on the dominant eigenvector of noisy quantum states. Our rigorous proof also establishes the fundamental observation that the geometry of typical noisy quantum states is (approximately) identical in either the Hilbert-Schmidt metric or as characterised by the QFI. In numerical simulations of noisy quantum circuits we demonstrate the practicality of our approach and confirm it can significantly outperform other variational techniques.

25 pages, 7 figures -- added new proofs and added new numerical simulations

References in corpus (27)

• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• Many body localization and thermalization in quantum statistical mechanics
• Quantum computational chemistry
• Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits
• Quantum Fisher information matrix and multiparameter estimation
• Self-Verifying Variational Quantum Simulation of the Lattice Schwinger Model
• Quantum Natural Gradient
• Theory of variational quantum simulation
• Simulating chemistry using quantum computers
• Virtual Distillation for Quantum Error Mitigation
• Stochastic gradient descent for hybrid quantum-classical optimization
• Tangent-space methods for uniform matrix product states
• Avoiding local minima in variational quantum eigensolvers with the natural gradient optimizer
• Fluxonium: an alternative qubit platform for high-fidelity operations
• Classical and Quantum Fisher Information in the Geometrical Formulation of Quantum Mechanics
• Exponential Error Suppression for Near-Term Quantum Devices
• Simple Mitigation of Global Depolarizing Errors in Quantum Simulations
• Variational-State Quantum Metrology
• Geometry of variational methods: dynamics of closed quantum systems
• Multicore Quantum Computing
• Demonstration of Adiabatic Variational Quantum Computing with a Superconducting Quantum Coprocessor
• Resource Estimation for Quantum Variational Simulations of the Hubbard Model
• The Dominant Eigenvector of a Noisy Quantum State
• Measurement cost of metric-aware variational quantum algorithms
• Quantum Analytic Descent
• QuESTlink -- Mathematica embiggened by a hardware-optimised quantum emulator
• Training variational quantum circuits with CoVaR: covariance root finding with classical shadows

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• Barren Plateaus in Variational Quantum Computing
• Variational Quantum Algorithms for Computational Fluid Dynamics
• Proposal for Observing Yang-Lee Criticality in Rydberg Atomic Arrays
• Quantum Error Mitigated Classical Shadows
• Adaptive variational quantum minimally entangled typical thermal states for finite temperature simulations
• Quantum Simulations of SO(5) Many-Fermion Systems using Qudits
• Probabilistic Interpolation of Quantum Rotation Angles
• Optimizing Variational Quantum Algorithms with qBang: Efficiently Interweaving Metric and Momentum to Navigate Flat Energy Landscapes
• Algorithmic Shadow Spectroscopy
• Can shallow quantum circuits scramble local noise into global white noise?
• Measurement-induced entanglement phase transitions in variational quantum circuits
• Quantum Natural Gradient optimizer on noisy platforms: QAOA as a case study
• Random Natural Gradient
• Sparse Probabilistic Synthesis of Quantum Operations
• Yang-Lee edge singularity and quantum criticality in non-Hermitian PXP model
• Quantum Natural Gradient with Geodesic Corrections for Small Shallow Quantum Circuits
• QKAN: quantum Kolmogorov-Arnold networks with applications in machine learning and multivariate state preparation
• Optimized numerical gradient and Hessian estimation for variational quantum algorithms
• Quantum natural gradient without monotonicity
• Evolved Quantum Boltzmann Machines
• Biased Estimator Channels for Classical Shadows
• Exponential distillation of dominant eigenproperties
• Variational quantum algorithm based on Lagrange polynomial encoding to solve differential equations
• Modified Conjugate Quantum Natural Gradient
• Quantum natural gradient with thermal-state initialization
• Robustness of optimized numerical estimation schemes for noisy variational quantum algorithms
• Weighted Approximate Quantum Natural Gradient for Variational Quantum Eigensolver
• Adaptive time Compressed QITE (ACQ) and its geometrical interpretation
• Information geometry of nonmonotonic quantum natural gradient
• Exploring Entanglement and Parameter Sensitivity in QAOA through Quantum Fisher Information
• Quantum simulation of massive Thirring and Gross--Neveu models for arbitrary number of flavors