# Convex optimization of programmable quantum computers

## TL;DR

This paper formulates the problem of optimizing program states in programmable quantum gate arrays as a convex optimization task, enabling efficient solutions using semidefinite programming and machine learning techniques.

## Contribution

It demonstrates that finding the optimal program state is a convex optimization problem, providing a practical method to improve programmable quantum processors.

## Key findings

- Optimal program states can be computed efficiently using convex optimization.
- The approach applies to various quantum processor designs, including teleportation-based and parametric circuits.
- This method advances the understanding of quantum processor programmability and simulation accuracy.

## Abstract

A fundamental model of quantum computation is the programmable quantum gate array. This is a quantum processor that is fed by a program state that induces a corresponding quantum operation on input states. While being programmable, any finite-dimensional design of this model is known to be nonuniversal, meaning that the processor cannot perfectly simulate an arbitrary quantum channel over the input. Characterizing how close the simulation is and finding the optimal program state have been open questions for the past 20 years. Here, we answer these questions by showing that the search for the optimal program state is a convex optimization problem that can be solved via semidefinite programming and gradient-based methods commonly employed for machine learning. We apply this general result to different types of processors, from a shallow design based on quantum teleportation, to deeper schemes relying on port-based teleportation and parametric quantum circuits.