# The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine

## TL;DR

This paper introduces the Ising Born Machine, a quantum generative model that cannot be efficiently simulated classically, and demonstrates novel training methods and potential quantum advantages in learning complex distributions.

## Contribution

The paper proposes the Ising Born Machine, new training methods using Stein Discrepancy and Sinkhorn Divergence, and demonstrates quantum advantage in learning complex distributions.

## Key findings

- The IBM cannot be efficiently simulated classically in the worst case.
- The proposed training methods outperform MMD in numerical experiments.
- Quantum kernels improve over classical kernels in the context of IBM.

## Abstract

The search for an application of near-term quantum devices is widespread. Quantum Machine Learning is touted as a potential utilisation of such devices, particularly those which are out of the reach of the simulation capabilities of classical computers. In this work, we propose a generative Quantum Machine Learning Model, called the Ising Born Machine (IBM), which we show cannot, in the worst case, and up to suitable notions of error, be simulated efficiently by a classical device. We also show this holds for all the circuit families encountered during training. In particular, we explore quantum circuit learning using non-universal circuits derived from Ising Model Hamiltonians, which are implementable on near term quantum devices. We propose two novel training methods for the IBM by utilising the Stein Discrepancy and the Sinkhorn Divergence cost functions. We show numerically, both using a simulator within Rigetti's Forest platform and on the Aspen-1 16Q chip, that the cost functions we suggest outperform the more commonly used Maximum Mean Discrepancy (MMD) for differentiable training. We also propose an improvement to the MMD by proposing a novel utilisation of quantum kernels which we demonstrate provides improvements over its classical counterpart. We discuss the potential of these methods to learn `hard' quantum distributions, a feat which would demonstrate the advantage of quantum over classical computers, and provide the first formal definitions for what we call `Quantum Learning Supremacy'. Finally, we propose a novel view on the area of quantum circuit compilation by using the IBM to `mimic' target quantum circuits using classical output data only.