Abstract
We consider metric-aware quantum algorithms that use a quantum computer to efficiently estimate both a matrix and a vector object. For example, the recently introduced quantum natural gradient approach uses the Fisher matrix as a metric tensor to correct the gradient vector for the codependence of the circuit parameters. We rigorously characterize and upper bound the number of measurements required to determine an iteration step to a fixed precision, and propose a general approach for optimally distributing samples between matrix and vector entries. Finally, we establish that the number of circuit repetitions needed for estimating the quantum Fisher information matrix is asymptotically negligible for an increasing number of iterations and qubits.Received 18 May 2020Accepted 28 June 2021DOI:https://doi.org/10.1103/PRXQuantum.2.030324Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum algorithmsQuantum measurementsQuantum simulationQuantum InformationGeneral Physics
Highlights
This is of particular interest in the context of noisy, intermediate-scale quantum devices [25], because complex ansatz states can be prepared with shallow circuits [26,27,28,29]
We consider variational quantum algorithms that typically aim to prepare a parameterized quantum state ρ(θ ) := (θ )ρ0 where we model via a mapping (θ ) that acts on the computational zero state ρ0 of N qubits and depends continuously on the parameters θi with i ∈ {1, 2, . . . , ν}
In the following we focus on one prominent algorithm, the recently introduced quantum natural gradient approach [37,44] that is equivalent to imaginary time evolution when quantum circuits are noiseless and unitary [37,39]
Summary
With quantum computers rising as realistic technologies, attention has turned to how such machines could perform as variational tools [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24] This results in a hybrid model with an iterative loop: a classical processor determines how to update the parameters describing a family of quantum states (parameterized ansatz states), while a quantum coprocessor generates and performs measurements on that state (via an ansatz circuit).
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