# Supervised learning of time-independent Hamiltonians for gate design

## TL;DR

This paper introduces a supervised learning framework to design time-independent Hamiltonians for quantum gates, enabling flexible and efficient gate synthesis without ancillae, demonstrated on Toffoli, Fredkin, and complex four-qubit gates.

## Contribution

It develops a novel inverse eigenvalue problem approach combined with supervised learning to find Hamiltonians for quantum gates, including those with experimentally friendly interactions.

## Key findings

- Successfully identified Hamiltonians for Toffoli and Fredkin gates without ancillae.
- Demonstrated the method's ability to design complex four-qubit gates with simple interactions.
- Showed the approach's flexibility in solving problems difficult for analytical methods.

## Abstract

We present a general framework to tackle the problem of finding time-independent dynamics generating target unitary evolutions. We show that this problem is equivalently stated as a set of conditions over the spectrum of the time-independent gate generator, thus transforming the task to an inverse eigenvalue problem. We illustrate our methodology by identifying suitable time-independent generators implementing Toffoli and Fredkin gates without the need for ancillae or effective evolutions. We show how the same conditions can be used to solve the problem numerically, via supervised learning techniques. In turn, this allows us to solve problems that are not amenable, in general, to direct analytical solution, providing at the same time a high degree of flexibility over the types of gate-design problems that can be approached. As a significant example, we find generators for the Toffoli gate using only diagonal pairwise interactions, which are easier to implement in some experimental architectures. To showcase the flexibility of the supervised learning approach, we give an example of a nontrivial four-qubit gate that is implementable using only diagonal, pairwise interactions.