The Born supremacy: quantum advantage and training of an Ising Born machine

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 out of reach of the simulation capabilities of classical computers. In this work, we study such an application in generative modelling, focussing on a class of quantum circuits known as Born machines. Specifically, we define a subset of this class based on Ising Hamiltonians and show that the circuits encountered during gradient-based training cannot be efficiently sampled from classically up to multiplicative error in the worst case. Our gradient-based training methods use cost functions known as the Sinkhorn divergence and the Stein discrepancy, which have not previously been used in the gradientbased training of quantum circuits, and we also introduce quantum kernels to generative modelling. We show that these methods outperform the previous standard method, which used maximum mean discrepancy (MMD) as a cost function, and achieve this with minimal overhead. Finally, we discuss the ability of the model to learn hard distributions and provide formal definitions for 'quantum learning supremacy'. We also exemplify the work of this paper by using generative modelling to perform quantum circuit compilation.

Application-Motivated, Holistic Benchmarking of a Full Quantum Computing Stack

“…Instantaneous Quantum Polytime (IQP) circuits [51] consist of commuting gates and, as well as being simpler to implement than universal quantum circuits, are believed, even in the presence of noise, to be impossible to simulate efficiently using classical computers [47,48,52]. This has allowed for the application of noisy quantum technology in areas such as machine learning [30,31,53] and interactive two-player games [51,54]. The shallow class of circuits introduced here, whose depth increases slowly with width, is a subclass of IQP circuits.…”

“…In summary, inspired by a variety of near-term applications of IQP circuits [30,31,51,53,54], we introduce shallow circuits in Algorithm 1. Performance when implementing these circuits is indicative of the performance when implementing those applications, but also, more generally, of applications requiring circuits which grow slowly in depth.…”

“…Shallow: Inspired by hardware-efficient ansatze [28,29] which may be useful for near-term quantum machine learning and chemistry applications [30][31][32]. The width of shallow circuits grow with a minimal increase in depth, allowing the impact of including many qubits to be explored.…”

Error-mitigated data-driven circuit learning on noisy quantum hardware

“…There are many different loss functions that can be utilized in training, and the optimization can be gradientbased or gradient-free. Recent studies have used the following loss functions: clipped log-likelihoods (Benedetti et al 2019), the Sinkhorn divergence (Coyle et al 2019), and the Jensen-Shannon divergence (Leyton-Ortega et al 2019). Following the work of Liu and Wang (2018b) and our earlier studies (Hamilton et al 2019), we use the maximum mean discrepancy loss function (L MMD ) with radial basis function kernels (σ = 0.1) and gradient-based optimization with Adam (Kingma and Ba 2014) using code adapted from Liu (2018a).…”

Integrated World Modeling Theory (IWMT) Revisited

“…To be clear, these are all (very) highly speculative analogies for information dynamics, and quantum physical phenomena are likely not directly relevant to the brain's computational abilities in any meaningful sense, given the hot and crowded nature of biological systems (Tegmark, 2000). However, it may also be worth considering that brains may be able to emulate quantum computational principles via classical means (Borders et al, 2019;Coyle et al, 2019;Guillet et al, 2019).…”