Gradient descent is well-known and widely used in many scientific applications. In many scenarios the aim is to find the minimum of a function; gradient descent does this by always moving in the direction of the steepest descent. This is analogous to how a person would find the way from the top of a mountain to a valley while navigating through heavy fog, which obscures the path ahead.
Cost functions, however, in practice depend on many variables which might inter-depend. Natural gradient descent, which is a technique known from machine learning, is a generalisation of the previous steepest descent concept which takes this parameter inter-dependence into account. This is achieved by correcting the gradient vector with the so-called Fisher information matrix — which reflects the natural metric in parameter space and characterises its information geometry.
The quantum version of natural gradient descent was recently introduced by Stokes et al. [1]. The authors showed that if a quantum state encodes a classical probability distribution, then the classical Fisher information matrix defines the core metric as the sensitivity with respect to ansatz-circuit parameters. Surprisingly, in this case quantum natural gradient descent is equivalent to imaginary time evolution [2] — and the former provides a geometrical explanation of the latter’s efficacy.
Unfortunately, current quantum computers cannot perform ideal unitary transformations due to experimental imperfections and, moreover, one might also want to incorporate fundamentally non-unitary operations in a quantum optimisation, such as measurements. In our preprint [3] we consider the most general scenario and propose the quantum natural gradient descent optimisation for arbitrary pure and mixed quantum states. We find that the quantum Fisher information defines the core metric — which naturally incorporates the previous pure state variants of [1-2] as special cases.
Refer also to the nice summary [4] published in Quantum Views.
References
[1] J. Stokes et al., Quantum natural gradient. Quantum 4 (2020): 269
[2] S. McArdle et al., Variational ansatz-based quantum simulation of imaginary time evolution. npj Quantum Information 5.1 (2019): 1-6
[3] B. Koczor & S. C. Benjamin, Quantum natural gradient generalised to non-unitary circuits. arXiv preprint: arXiv:1909.02108
[4] J. Napp, Variational quantum algorithms and geometry. Quantum Views 4 (2020), 37
