Abstract
A quantum generalization of Natural Gradient Descent is presented as part of a general-purpose optimization framework for variational quantum circuits. The optimization dynamics is interpreted as moving in the steepest descent direction with respect to the Quantum Information Geometry, corresponding to the real part of the Quantum Geometric Tensor (QGT), also known as the Fubini-Study metric tensor. An efficient algorithm is presented for computing a block-diagonal approximation to the Fubini-Study metric tensor for parametrized quantum circuits, which may be of independent interest.
Highlights
Variational optimization of parametrized quantum circuits is an integral component for many hybrid quantum-classical algorithms, which are arguably the most promising applications of Noisy Intermediate-Scale Quantum (NISQ) computers [29]
In the quantum setting these estimates are obtained by repeated measurements of some Hermitian observables for a quantum state which depends on the variational parameters
Unlike Adam, which involves some kind of historical averaging, the preconditioning matrix suggested by quantum information geometry does not depend on the specific choice of loss function (Hermitian observable)
Summary
Variational optimization of parametrized quantum circuits is an integral component for many hybrid quantum-classical algorithms, which are arguably the most promising applications of Noisy Intermediate-Scale Quantum (NISQ) computers [29]. One motivation for exploiting first-order gradients is theoretical: in the convex case, the expected error in the objective function using the best known zeroth-order stochastic optimization algorithm scales polynomially with the dimension d of the parameter space, whereas Stochastic Gradient Descent (SGD) converges independently of d. The intuitive argument put forth favoring the l1 geometry is that some quantum state ansätze can be physically interpreted as a sequence of pulses of Hamiltonian evolution, starting from a fixed reference state In this particular parametrization, each variational parameter can be interpreted as the duration of the corresponding pulse. We present quantum circuit construction which computes a block-diagonal approximation to the Quantum Geometric Tensor and show that a simple diagonal preconditioning scheme outperforms vanilla gradient descent in terms of number of iterates required to achieve convergence
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