Beyond Ans\"atze: Learning Quantum Circuits as Unitary Operators

TL;DR

This paper introduces an ansatz-free method for optimizing quantum circuits directly as unitary operators in the full space, offering a more general and computationally manageable approach compared to traditional ansatz-based methods.

Contribution

It proposes a gradient-based optimization in the Lie algebra of the unitary group, enabling direct, efficient, and general quantum circuit optimization without ansatz restrictions.

Findings

The method is faster and more general than ansatz-based approaches.

It provides an upper bound on the performance achievable by any ansatz.

The approach simplifies classical computation of quantum circuit optimization.

Abstract

This paper explores the advantages of optimizing quantum circuits on $N$ wires as operators in the unitary group $U(2^N)$. We run gradient-based optimization in the Lie algebra $\mathfrak u(2^N)$ and use the exponential map to parametrize unitary matrices. We argue that $U(2^N)$ is not only more general than the search space induced by an ansatz, but in ways easier to work with on classical computers. The resulting approach is quick, ansatz-free and provides an upper bound on performance over all ans\"atze on $N$ wires.