M. S. Kim

Alistair W. R. Smith, A. J. Paige, M. S. Kim

We present a new optimization method for small-to-intermediate scale variational algorithms on noisy near-term quantum processors which uses a Gaussian process surrogate model equipped with a classically-evaluated quantum kernel. Variational algorithms are typically optimized using gradient-based approaches however these are difficult to implement on current noisy devices, requiring large numbers of objective function evaluations. Our scheme shifts this computational burden onto the classical optimizer component of these hybrid algorithms, greatly reducing the number of queries to the quantum processor. We focus on the variational quantum eigensolver (VQE) algorithm and demonstrate numerically that such surrogate models are particularly well suited to the algorithm's objective function. Next, we apply these models to both noiseless and noisy VQE simulations and show that they exhibit better performance than widely-used classical kernels in terms of final accuracy and convergence speed. Compared to the typically-used stochastic gradient-descent approach for VQAs, our quantum kernel-based approach is found to consistently achieve significantly higher accuracy while requiring less than an order of magnitude fewer quantum circuit evaluations. We analyse the performance of the quantum kernel-based models in terms of the kernels' induced feature spaces and explicitly construct their feature maps. Finally, we describe a scheme for approximating the best-performing quantum kernel using a classically-efficient tensor network representation of its input state and so provide a pathway for scaling these methods to larger systems.

Tobias Haug, M. S. Kim

Generalization is the ability of machine learning models to make accurate predictions on new data by learning from training data. However, understanding generalization of quantum machine learning models has been a major challenge. Here, we introduce the data quantum Fisher information metric (DQFIM). It describes the capacity of variational quantum algorithms depending on variational ansatz, training data and their symmetries. We apply the DQFIM to quantify circuit parameters and training data needed to successfully train and generalize. Using the dynamical Lie algebra, we explain how to generalize using a low number of training states. Counter-intuitively, breaking symmetries of the training data can help to improve generalization. Finally, we find that out-of-distribution generalization, where training and testing data are drawn from different data distributions, can be better than using the same distribution. Our work provides a useful framework to explore the power of quantum machine learning models.

Dominic Lowe, M. S. Kim, Roberto Bondesan

Gaussian Process Regression is a well-known machine learning technique for which several quantum algorithms have been proposed. We show here that in a wide range of scenarios these algorithms show no exponential speedup. We achieve this by rigorously proving that the condition number of a kernel matrix scales at least linearly with the matrix size under general assumptions on the data and kernel. We additionally prove that the sparsity and Frobenius norm of a kernel matrix scale linearly under similar assumptions. The implications for the quantum algorithms runtime are independent of the complexity of loading classical data on a quantum computer and also apply to dequantised algorithms. We supplement our theoretical analysis with numerical verification for popular kernels in machine learning.

Tobias Haug, Chris N. Self, M. S. Kim

Quantum computers promise to enhance machine learning for practical applications. Quantum machine learning for real-world data has to handle extensive amounts of high-dimensional data. However, conventional methods for measuring quantum kernels are impractical for large datasets as they scale with the square of the dataset size. Here, we measure quantum kernels using randomized measurements. The quantum computation time scales linearly with dataset size and quadratic for classical post-processing. While our method scales in general exponentially in qubit number, we gain a substantial speed-up when running on intermediate-sized quantum computers. Further, we efficiently encode high-dimensional data into quantum computers with the number of features scaling linearly with the circuit depth. The encoding is characterized by the quantum Fisher information metric and is related to the radial basis function kernel. Our approach is robust to noise via a cost-free error mitigation scheme. We demonstrate the advantages of our methods for noisy quantum computers by classifying images with the IBM quantum computer. To achieve further speedups we distribute the quantum computational tasks between different quantum computers. Our method enables benchmarking of quantum machine learning algorithms with large datasets on currently available quantum computers.

Tobias Haug, M. S. Kim

Variational quantum algorithms (VQAs) promise efficient use of near-term quantum computers. However, training VQAs often requires an extensive amount of time and suffers from the barren plateau problem where the magnitude of the gradients vanishes with increasing number of qubits. Here, we show how to optimally train VQAs for learning quantum states. Parameterized quantum circuits can form Gaussian kernels, which we use to derive adaptive learning rates for gradient ascent. We introduce the generalized quantum natural gradient that features stability and optimized movement in parameter space. Both methods together outperform other optimization routines in training VQAs. Our methods also excel at numerically optimizing driving protocols for quantum control problems. The gradients of the VQA do not vanish when the fidelity between the initial state and the state to be learned is bounded from below. We identify a VQA for quantum simulation with such a constraint that thus can be trained free of barren plateaus. Finally, we propose the application of Gaussian kernels for quantum machine learning.

Tobias Haug, Kishor Bharti, M. S. Kim

To harness the potential of noisy intermediate-scale quantum devices, it is paramount to find the best type of circuits to run hybrid quantum-classical algorithms. Key candidates are parametrized quantum circuits that can be effectively implemented on current devices. Here, we evaluate the capacity and trainability of these circuits using the geometric structure of the parameter space via the effective quantum dimension, which reveals the expressive power of circuits in general as well as of particular initialization strategies. We assess the expressive power of various popular circuit types and find striking differences depending on the type of entangling gates used. Particular circuits are characterized by scaling laws in their expressiveness. We identify a transition in the quantum geometry of the parameter space, which leads to a decay of the quantum natural gradient for deep circuits. For shallow circuits, the quantum natural gradient can be orders of magnitude larger in value compared to the regular gradient; however, both of them can suffer from vanishing gradients. By tuning a fixed set of circuit parameters to randomized ones, we find a region where the circuit is expressive, but does not suffer from barren plateaus, hinting at a good way to initialize circuits. We show an algorithm that prunes redundant parameters of a circuit without affecting its effective dimension. Our results enhance the understanding of parametrized quantum circuits and can be immediately applied to improve variational quantum algorithms.

Alistair W. R. Smith, Johnnie Gray, M. S. Kim

We use a meta-learning neural-network approach to analyse data from a measured quantum state. Once our neural network has been trained it can be used to efficiently sample measurements of the state in measurement bases not contained in the training data. These samples can be used calculate expectation values and other useful quantities. We refer to this process as "state sample tomography". We encode the state's measurement outcome distributions using an efficiently parameterized generative neural network. This allows each stage in the tomography process to be performed efficiently even for large systems. Our scheme is demonstrated on recent IBM Quantum devices, producing a model for a 6-qubit state's measurement outcomes with a predictive accuracy (classical fidelity) > 95% for all test cases using only 100 random measurement settings as opposed to the 729 settings required for standard full tomography using local measurements. This reduction in the required number of measurements scales favourably, with training data in 200 measurement settings yielding a predictive accuracy > 92% for a 10 qubit state where 59,049 settings are typically required for full local measurement-based quantum state tomography. A reduction in number of measurements by a factor, in this case, of almost 600 could allow for estimations of expectation values and state fidelities in practicable times on current quantum devices.