Gradient descent reliably finds depth- and gate-optimal circuits for generic unitaries

TL;DR

This paper demonstrates that simple gradient descent can reliably find minimal depth and gate count circuits for generic unitaries, even with connectivity constraints, challenging prior beliefs about the necessity of combinatorial search.

Contribution

It shows that gradient descent effectively optimizes circuits for generic unitaries, avoiding the need for complex combinatorial methods, even under connectivity restrictions.

Findings

Gradient descent reliably finds optimal circuits for generic unitaries.

Optimal synthesis can be achieved without combinatorial search.

Connectivity constraints do not prevent gradient-based optimization.

Abstract

When the gate set has continuous parameters, synthesizing a unitary operator as a quantum circuit is always possible using exact methods, but finding minimal circuits efficiently remains a challenging problem. The landscape is very different for compiled unitaries, which arise from programming and typically have short circuits, as compared with generic unitaries, which use all parameters and typically require circuits of maximal size. We show that simple gradient descent reliably finds depth- and gate-optimal circuits for generic unitaries, including in the presence of restricted chip connectivity. This runs counter to earlier evidence that optimal synthesis required combinatorial search, and we show that this discrepancy can be explained by avoiding the random selection of certain parameter-deficient circuit skeletons.