Protocols for Trainable and Differentiable Quantum Generative Modelling

Abstract

We propose an approach for learning probability distributions as differentiable quantum circuits (DQC) that enable efficient quantum generative modelling (QGM) and synthetic data generation. Contrary to existing QGM approaches, we perform training of a DQCbased model, where data is encoded in a latent space with a phase feature map, followed by a variational quantum circuit. We then map the trained model to the bit basis using a fixed unitary transformation, coinciding with a quantum Fourier transform circuit in the simplest case. This allows fast sampling from parametrized distributions using a single-shot readout. Importantly, latent space training provides models that are automatically differentiable, and we show how samples from solutions of stochastic differential equations (SDEs) can be accessed by solving stationary and time-dependent Fokker-Planck equations with a quantum protocol. Finally, our approach opens a route to multidimensional generative modelling with qubit registers explicitly correlated via a (fixed) entangling layer. In this case quantum computers can offer advantage as efficient samplers, which perform complex inverse transform sampling enabled by the fundamental laws of quantum mechanics. On a technical side the advances are multiple, as we introduce the phase feature map, analyze its properties, and develop frequency-taming techniques that include qubit-wise training and feature map sparsification.

Pricing multi-asset derivatives by variational quantum algorithms

“…We can obtain such α p and U p by solving the Fokker-Planck equation, which describes the time evolution of the probability density function, using VQS [22,41]. Alternatively, they can also be obtained by quantum generative models [33][34][35][36][37] since the probability density function of the underlying asset price at any t ∈ [0, T ] can be obtained analytically under the BS model (see Eq. (53) in Sec.…”

“…To find such α V and U V , we can use the quantum generative models [33][34][35][36][37]. Fourth, we solve the BSPDE from τ = 0 to τ ter using VQS and obtain an unnormalized state…”

“…Our algorithm has the following three parts; embedding the probability distribution of the underlying asset prices into the quantum state, solving the BSPDE with boundary conditions, and calculating the inner product. For the first part, we can use the quantum generative algorithms [33][34][35][36][37] or variational quantum simulation (VQS) for the Fokker-Planck equations [22,[38][39][40][41], which describe the time evolution of the probability density functions of the stochastic processes. For the second part, we discretize the BSPDE using the FDM and solve it using VQS.…”

Protocols for classically training quantum generative models on probability distributions

“…In the context of QGM, QCBM is an excellent example of implicit training where typically a MMD like loss-function is used. On the other hand, recent work showcases how explicit quantum models such as DQGM [38] and quantum quantile mechanics [11] benefit from a functional access to the model probability distributions, allowing input-differentiable quantum models [39,40] To enable the classical training, we propose a strategy described in a schematic shown in Fig. 2.…”

“…We explore different families of quantum circuits in detail and perform numerical studies to study their sampling complexity and expressivity. For expressivity studies in particular, we look at training a differentiable quantum generative model (DQGM) [38][39][40] architecture which allows training in the latent (or "frequency") space, and sampling in the bit-basis. This presents a good testing ground for applying the proposed method to explicit quantum generative models.…”