A Universal Training Algorithm for Quantum Deep Learning (1806.09729v1)

Abstract: We introduce the Backwards Quantum Propagation of Phase errors (Baqprop) principle, a central theme upon which we construct multiple universal optimization heuristics for training both parametrized quantum circuits and classical deep neural networks on a quantum computer. Baqprop encodes error information in relative phases of a quantum wavefunction defined over the space of network parameters; it can be thought of as the unification of the phase kickback principle of quantum computation and of the backpropagation algorithm from classical deep learning. We propose two core heuristics which leverage Baqprop for quantum-enhanced optimization of network parameters: Quantum Dynamical Descent (QDD) and Momentum Measurement Gradient Descent (MoMGrad). QDD uses simulated quantum coherent dynamics for parameter optimization, allowing for quantum tunneling through the hypothesis space landscape. MoMGrad leverages Baqprop to estimate gradients and thereby perform gradient descent on the parameter landscape; it can be thought of as the quantum-classical analogue of QDD. In addition to these core optimization strategies, we propose various methods for parallelization, regularization, and meta-learning as augmentations to MoMGrad and QDD. We introduce several quantum-coherent adaptations of canonical classical feedforward neural networks, and study how Baqprop can be used to optimize such networks. We develop multiple applications of parametric circuit learning for quantum data, and show how to perform Baqprop in each case. One such application allows for the training of hybrid quantum-classical neural-circuit networks, via the seamless integration of Baqprop with classical backpropagation. Finally, for a representative subset of these proposed applications, we demonstrate the training of these networks via numerical simulations of implementations of QDD and MoMGrad.

• The paper introduces Baqprop, a framework that fuses quantum phase kickback with classical backpropagation to enable coherent gradient propagation.
• It details Quantum Dynamical Descent (QDD) and Momentum Measurement Gradient Descent (MoMGrad) as novel optimization methods leveraging quantum dynamics and semiclassical measurements.
• Simulations demonstrate the potential for quantum advantage in hybrid training scenarios, offering actionable insights for deploying deep learning on NISQ devices.

An Analysis of "A Universal Training Algorithm for Quantum Deep Learning"

The paper presented introduces a novel framework for training both quantum and classical deep learning models on quantum computers, by leveraging a theoretical foundation they term Backwards Quantum Propagation of Phase errors (Baqprop). Baqprop is inspired by traditional backpropagation used in classical machine learning and the phase kickback phenomenon in quantum mechanics. This research puts forth two principal optimization techniques: Quantum Dynamical Descent (QDD) and Momentum Measurement Gradient Descent (MoMGrad), alongside proposed methods for regularization, parallelization, and hyper-parameter optimization within quantum computing contexts.

Baqprop: A Quantum Backpropagation Framework

Baqprop is central to the work, unifying quantum computation's phase kickback with error backpropagation from deep learning. In essence, Baqprop enables the encoding of error information within quantum wavefunctions through phase dynamics. This procedure facilitates the propagation of loss gradients back through the network, akin to traditional backpropagation in neural networks, but realized through unitary operations in quantum lattice or Hilbert spaces. This quantum-coherent backpropagation operates on superpositions, offering both quantum speedups and differentially richer information paths.

Core Heuristics

1. Quantum Dynamical Descent (QDD)

QDD uses simulated dynamical descent through coherent quantum oscillations under an effective potential landscape dictated by the loss function. Key to this operation is its basis in Schrödinger dynamics to move the parameter states through quantum tunneling, potentially offering advantages in navigating complex error surfaces that are classically hard, due to local minima and saddle points entrapment. The correspondence between QDD and Quantum Approximate Optimization Algorithm (QAOA) suggests inherent quantum advantages, particularly regarding entanglement handling in parameter space, potentially leading to a form of quantum advantage in certain optimization scenarios.

2. Momentum Measurement Gradient Descent (MoMGrad)

MoMGrad focuses on gradient estimation through momentum shifts in parameter wavefunctions, adopting a semiclassical approach where a hybrid classical-quantum system measures quantum-induced phase shifts to update parameters. This hybridization retains the quantum phase dynamics while employing classical computations to consolidate learning, potentially offering a more computationally feasible pathway for near-term quantum devices where coherence times are limited.

Numerical Simulations and Quantum Learning Models

The authors assert the efficacy of their approach through various quantum simulation models, proposing adapted quantum neural networks and parametric models that retain classical feedforward architectures but with parameter optimization carried out in quantum. Several application scenarios are addressed:

• Quantum state learning where the system learns to replicate quantum data distributions.
• Quantum unitary learning where the task involves approximating unitary quantum processes.
• Application to hybrid quantum-classical networks for data learning, whereby classical neural network layers seamlessly integrate with quantum parametric circuits via Baqprop.

Theoretical and Practical Implications

The theoretical implications are significant, positioning Baqprop as a potent crossroad between quantum computational theory and classical deep learning, using quantum resources more efficiently. Practically, the development of robust quantum optimization strategies like QDD and MoMGrad demonstrates pathways to leveraging quantum computers for practical machine learning tasks, particularly within noisy intermediate-scale quantum (NISQ) environments.

Future Directions

Future exploratory pathways include optimizing quantum resources for broader machine learning tasks, refining error correction mechanisms inherent in Baqprop’s quantum phase dynamics, and extensive profiling of quantum and classical comparative performance across diverse learning frameworks and data paradigms. Additionally, real-world implementations on NISQ devices could validate the hypothetical models, especially around resource overheads versus classical alternatives.

The presented universal training algorithm substantiates an intricate intersection between quantum computing advancements and machine learning paradigms, heralding a robust foundation for future quantum-enhanced computational progress.