Quantum Natural Gradient Variational Circuits

Short answer: Quantum natural gradient is an optimisation method for variational quantum circuits that steps in the steepest-descent direction measured by the geometry of the quantum states, rather than by raw parameter changes. A 2020 paper introduced it with an efficient approximation, and a 2022 paper extended it to noisy, mixed states. These are algorithm papers, and neither claims a practical advantage.

Why optimisation matters

A 2020 Quantum paper (arXiv 1909.02108) says variational optimisation of parametrised circuits is an integral part of many hybrid quantum-classical algorithms, which it calls arguably the most promising applications of noisy intermediate-scale quantum computers. It names the variational quantum eigensolver, the quantum approximate optimisation algorithm and quantum neural networks as examples. A variety of optimisation methods has been proposed, and the method used affects how well these algorithms work.

The 2020 proposal

The paper presents a quantum generalisation of natural gradient descent as part of a general-purpose framework for variational circuits. The optimisation is interpreted as moving in the steepest-descent direction with respect to quantum information geometry, which corresponds to the real part of the quantum geometric tensor, also known as the Fubini-Study metric tensor. The authors present an efficient algorithm for computing a block-diagonal approximation to that metric tensor for parametrised circuits, and note that it may be of independent interest.

Extending to noisy states, 2022

A 2022 Physical Review A paper recalls that for noise-free unitary circuits, quantum natural gradient has been shown to be equivalent to imaginary time evolution, and effective because a metric tensor reconciles the classical parameter space with the device's Hilbert space. The authors generalise it to arbitrary quantum states, mixed and pure, using completely positive maps, so the circuits can include imperfect unitary gates and non-unitary operations such as measurements. They use the quantum Fisher information as the core metric on density operators. By modifying the error suppression by derangements and virtual distillation techniques, they obtain an accurate and experimentally efficient approximation to the quantum Fisher information through the Hilbert-Schmidt metric tensor, using prior results on the dominant eigenvector of noisy states.

Follow-on work

The reference list on the 2020 paper's journal page cites later work, such as papers on efficient circuits based on quantum natural gradient and on a weighted approximate version for the variational quantum eigensolver. This shows continued interest, though those papers are not covered here.

Limits

The two papers are methods papers. They define the optimiser and show how to approximate its metric, and neither paper's abstract reports a general speed-up over other optimisers on hardware. Metric computation adds measurement cost, which is why approximations matter.

FAQ

What is natural gradient descent?

Gradient descent that accounts for the geometry of the space being optimised, here the space of quantum states.

What is the Fubini-Study metric?

The real part of the quantum geometric tensor, used as the metric for the steepest-descent direction.

Why use a block-diagonal approximation?

The 2020 paper presents it as an efficient way to compute the metric tensor for parametrised circuits.

Does it work with noise?

The 2022 paper extends it to mixed states and non-unitary operations using the quantum Fisher information.

Is it faster than other optimisers?

These papers do not establish a general answer.