Random Natural Gradient

TL;DR

This work proposes two methods that reduce the resources/state preparations required for QNG, while keeping the advantages and performance of the QNG-based optimization, and proposes a novel approximation to the QNG which is inspired by stochastic-coordinate methods.

Abstract

Hybrid quantum-classical algorithms appear to be the most promising approach for near-term quantum applications. An important bottleneck is the classical optimization loop, where the multiple local minima and the emergence of barren plateaux make these approaches less appealing. To improve the optimization the Quantum Natural Gradient (QNG) method [Quantum 4, 269 (2020)] was introduced - a method that uses information about the local geometry of the quantum state-space. While the QNG-based optimization is promising, in each step it requires more quantum resources, since to compute the QNG one requires $O(m^2)$ quantum state preparations, where $m$ is the number of parameters in the parameterized circuit. In this work we propose two methods that reduce the resources/state preparations required for QNG, while keeping the advantages and performance of the QNG-based optimization. Specifically, we first introduce the Random Natural Gradient (RNG) that uses random measurements and the classical Fisher information matrix (as opposed to the quantum Fisher information used in QNG). The essential quantum resources reduce to linear $O(m)$ and thus offer a quadratic "speed-up", while in our numerical simulations it matches QNG in terms of accuracy. We give some theoretical arguments for RNG and then benchmark the method with the QNG on both classical and quantum problems. Secondly, inspired by stochastic-coordinate methods, we propose a novel approximation to the QNG which we call Stochastic-Coordinate Quantum Natural Gradient that optimizes only a small (randomly sampled) fraction of the total parameters at each iteration. This method also performs equally well in our benchmarks, while it uses fewer resources than the QNG.

Figures (10)

• Figure 1: Distance of random CFIMs from QFIM. As the number of layers increases, sampling a random measurement tends to have a small distance from the QFIM and, as such, carries more information. A total number of 10000 CFIMs were calculated for each choice of measurement basis.
• Figure 2: Ranks of QFIM $\mathcal{F}_Q$, CFIM with $Z$-basis measurements $\mathcal{F}_C^Z$, and CFIM with measurements on a random basis $\mathcal{F}_C^\mathcal{M}$ compared to the total number of parameters of the parameterized quantum circuit on the left of Figure \ref{['fig:quantum_circuits']}.
• Figure 3: Relative error for Gradient Descent, Random Natural Gradient and Quantum Natural Gradient for 30 12-qubit random weighted 3-regular graphs (in logarithmic scale).
• Figure 4: Performance of the Random Natural Gradient optimizer on a Heisenberg model of 10 qubits compared to the Quantum Natural Gradient and Gradient Descent on both the optimization iterations (left figure) and on quantum resources (right figure). The RNG and GD methods require less quantum resources to converge (compared to QNG), but, as seen in both figures, the GD method converges to a bad quality minimum.
• Figure 5: Comparison of SC-QNG (with sampling half of the total parameters at each iteration) with QNG both in terms of optimization iterations (left) and quantum calls (right).

Theorems & Definitions (10)

• Corollary 1
• Corollary 2
• Lemma 1
• proof
• Corollary 3
• Proposition 1
• proof
• Definition 1
• Lemma 2
• proof