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Journal of Physics A Mathematical and Theoretical

Training quantum neural networks using the quantum information bottleneck method

Ahmet Burak Çatlı, Nathan Wiebe

Published Aug 9, 2024• 3 citations
Publisher

Abstract

Abstract We provide in this paper a concrete method for training a quantum neural network to maximize the relevant information about a property that is transmitted through the network. This is significant because it gives an operationally well founded quantity to optimize when training autoencoders for problems where the inputs and outputs are fully quantum. We provide a rigorous algorithm for computing the value of the quantum information bottleneck quantity within error ε that requires O ( log 2 ⁡ ( 1 / ϵ ) + 1 / δ 2 ) queries to a purification of the input density operator if its spectrum is supported on { 0 } ⋃ [ δ , 1 − δ ] for δ > 0 and the kernels of the relevant density matrices are disjoint. We further provide algorithms for estimating the derivatives of the QIB function, showing that quantum neural networks can be trained efficiently using the QIB quantity given that the number of gradient steps required is polynomial.

DOI
10.1088/1751-8121/ad6daf

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Abstract We provide in this paper a concrete method for training a quantum neural network to maximize the relevant information about a property that is transmitted through the network. This is significant because it gives an operationally well founded quantity to optimize when training autoencoders for problems where the inputs and outputs are fully quantum. We provide a rigorous algorithm for computing the value of the quantum information bottleneck quantity within error ε that requires O ( log 2 ⁡ ( 1 / ϵ ) + 1 / δ 2 ) queries to a purification of the input density operator if its spectrum is supported on { 0 } ⋃ [ δ , 1 − δ ] for δ > 0 and the kernels of the relevant density matrices are disjoint. We further provide algorithms for estimating the derivatives of the QIB function, showing that quantum neural networks can be trained efficiently using the QIB quantity given that the number of gradient steps required is polynomial.
We provide in this paper a concrete method for training a quantum neural network to maximize the relevant information about a property that is transmitted through the network. This is significant because it gives an operationally well founded quantity to optimize when training autoencoders for problems where the inputs and outputs are fully quantum. We provide a rigorous algorithm for computing the value of the quantum information bottleneck quantity within error εεε that requires O(log⁡2(1/ε)+1/δ2)O(\log^2(1/ε) + 1/δ^2)O(log2(1/ε)+1/δ2) queries to a purification of the input density operator if its spectrum is supported on {0} ⋃ [δ,1−δ]\{0\}~\bigcup ~[δ,1-δ]{0} ⋃ [δ,1−δ] for δ>0δ>0δ>0 and the kernels of the relevant density matrices are disjoint. We further provide algorithms for estimating the derivatives of the QIB function, showing that quantum neural networks can be trained efficiently using the QIB quantity given that the number of gradient steps required is polynomial.
Quantum Neural Networks (QNNs) have been recently proposed as generalizations of classical neural networks to achieve the quantum speed-up. Despite the potential to outperform classical models, serious bottlenecks exist for training QNNs; namely, QNNs with random structures have poor trainability due to the vanishing gradient with rate exponential to the input qubit number. The vanishing gradient could seriously influence the applications of large-size QNNs. In this work, we provide a viable solution with theoretical guarantees. Specifically, we prove that QNNs with tree tensor and step controlled architectures have gradients that vanish at most polynomially with the qubit number. We numerically demonstrate QNNs with tree tensor and step controlled structures for the application of binary classification. Simulations show faster convergent rates and better accuracy compared to QNNs with random structures.
Quantum Machine Learning is an emerging sub-field in machine learning where one of the goals is to perform pattern recognition tasks by encoding data into quantum states. This extension from classical to quantum domain has been made possible due to the development of hybrid quantum-classical algorithms that allow a parameterized quantum circuit to be optimized using gradient based algorithms that run on a classical computer. The similarities in training of these hybrid algorithms and classical neural networks has further led to the development of Quantum Neural Networks (QNNs). However, in the current training regime for QNNs, the gradients w.r. t objective function have to be computed on the quantum device. This computation is highly non-scalable and is affected by hardware and sampling noise present in the current generation of quantum hardware. In this paper, we propose a training algorithm that does not rely on gradient information. Specifically, we introduce a novel meta-optimization algorithm that trains a meta-optimizer network to output parameters for the quantum circuit such that the objective function is minimized. We empirically and theoretically show that we achieve a better quality minima in fewer circuit evaluations than existing gradient based algorithms on different datasets.
Artificial neural networks, usually just called neural networks, computing systems indefinitely inspired by the biological neural networks and they are extensive in both research as well as industry. It is critical to design quantum Neural Networks for complete quantum learning tasks. In this project, we suggest a computational neural network model based on principles of quantum mechanics which form a quantum feed-forward neural network proficient in universal quantum computation. This structure takes input from one layer of qubits and drives that input onto another layer of qubits. This layer of qubits evaluates this information and drives on the output to the next layer. Eventually, the path leads to the final layer of qubits. The layers do not have to be of the same breadth, meaning they need not have the same number of qubits as the layer before and/or after it. This assembly is trained on which path to take identical to classical ANN. The intended project can be compiled by the subsequent points provided here: 1. The expert training of the quantum neural network utilizing the fidelity as a cost function, providing both conventional and efficient quantum implementations. 2. Use of methods that enable quick optimization with reduced memory requirements. 3. Benchmarking our proposal for the quantum task of learning an unknown unitary and find extraordinary generality and a remarkable sturdiness to noisy training data.
Quantum Machine Learning is an emerging sub-field in machine learning where one of the goals is to perform pattern recognition tasks by encoding data into quantum states. This extension from classical to quantum domain has been made possible due to the development of hybrid quantum-classical algorithms that allow a parameterized quantum circuit to be optimized using gradient based algorithms that run on a classical computer. The similarities in training of these hybrid algorithms and classical neural networks has further led to the development of Quantum Neural Networks (QNNs). However, in the current training regime for QNNs, the gradients w.r.t objective function have to be computed on the quantum device. This computation is highly non-scalable and is affected by hardware and sampling noise present in the current generation of quantum hardware. In this paper, we propose a training algorithm that does not rely on gradient information. Specifically, we introduce a novel meta-optimization algorithm that trains a \emph{meta-optimizer} network to output parameters for the quantum circuit such that the objective function is minimized. We empirically and theoretically show that we achieve a better quality minima in fewer circuit evaluations than existing gradient based algorithms on different datasets.
Quantum Neural Networks (QNNs) are a promising variational learning paradigm with applications to near-term quantum processors, however they still face some significant challenges. One such challenge is finding good parameter initialization heuristics that ensure rapid and consistent convergence to local minima of the parameterized quantum circuit landscape. In this work, we train classical neural networks to assist in the quantum learning process, also know as meta-learning, to rapidly find approximate optima in the parameter landscape for several classes of quantum variational algorithms. Specifically, we train classical recurrent neural networks to find approximately optimal parameters within a small number of queries of the cost function for the Quantum Approximate Optimization Algorithm (QAOA) for MaxCut, QAOA for Sherrington-Kirkpatrick Ising model, and for a Variational Quantum Eigensolver for the Hubbard model. By initializing other optimizers at parameter values suggested by the classical neural network, we demonstrate a significant improvement in the total number of optimization iterations required to reach a given accuracy. We further demonstrate that the optimization strategies learned by the neural network generalize well across a range of problem instance sizes. This opens up the possibility of training on small, classically simulatable problem instances, in order to initialize larger, classically intractably simulatable problem instances on quantum devices, thereby significantly reducing the number of required quantum-classical optimization iterations.