Provable superior accuracy in machine learned quantum models

Abstract

In modelling complex processes, the potential past data that influence future expectations are immense. Models that track all this data are not only computationally wasteful but also shed little light on what past data most influence the future. There is thus enormous interest in dimensional reduction -finding automated means to reduce the memory dimension of our models while minimizing its impact on its predictive accuracy. Here we construct dimensionally reduced quantum models by machine learning methods that can achieve greater accuracy than provably optimal classical counterparts. We demonstrate this advantage on present-day quantum computing hardware. Our algorithm works directly off classical time-series data and can thus be deployed in real-world settings. These techniques illustrate the immediate relevance of quantum technologies to time-series analysis and offer a rare instance where the resulting quantum advantage can be provably established.

Time-Warping Invariant Quantum Recurrent Neural Networks via Quantum-Classical Adaptive Gating

“…Finally, it is important to identify instances in which the proposed model class may offer improvements as compared to classical gated architectures, such as the gated LSTM model. An interesting direction could be the investigation of the memory cost of time warpinginvariant quantum recurrent models, since quantum models have been shown to describe temporal sequences with reduced memory [25,24]. Given the performance of the proposed model on prediction tasks, we expect it to be advantageous for tasks related to quantum phenomena, with extensions possibly operating directly on quantum data [39].…”

“…In this section, we present an alternative derivation of the TWI-QRNN model that follows directly the steps in [7]. To start, we introduce a continuous-time version of the update (7) via the approximation ρ(t + δt) ≈ ρ(t) + dρ(t) dt δt, (25) where we have used ρ(t) to denote a continuoustime version of the density matrix and δt > 0 is a time interval. The update (7) can be viewed as the standard Euler approximation of (25).…”

“…Accordingly, under suitable assumptions, the accuracy of the approximation is proportional to δt (see, e.g., [40,Chapter 4]). It is emphasized that (25) is not meant to be a description of the continuous-time dynamics of the system, which is dictated by the Liouville-von Neumann equation [41], but only a stepping stone in the derivation of time warping-invariant discrete-time models. From (7), setting δt = 1, we then have dρ(t) dt = V (x(t), θ)ρ(t)V (x(t), θ) † − ρ(t), (26) where we have kept the dependence of the unitary V (x(t), θ) on time t implicit.…”

Time-warping invariant quantum recurrent neural networks via quantum-classical adaptive gating

“…This is particularly useful for running the models on quantum hardware, rather than simulators, and it is also left for future work. An interesting direction could be the investigation of the memory cost of time warping-invariant quantum recurrent models, since quantum models have been shown to describe temporal sequences with reduced memory [24,25]. Another interesting application is the use of the proposed model for quantum optimal control problems, which have already been treated using classical LSTMs [40,41].…”

“…The model has been shown to be capable of learning sequential data [22], and its capacity was analyzed in [23] in comparison to counterpart classical models with the same number of memory units. In particular, it was shown in [24,25] that quantum models can describe temporal sequences with reduced memory, compared to their classical counterparts. A model with classical memory, was introduced in [26] by integrating quantum models within a classical LSTM architecture.…”