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arXiv:1805.11089

Bayesian Quantum Circuit

Yuxuan Du, Tongliang Liu, Dacheng Tao

Published May 27, 2018• 4 citations
arXivHTMLPDF

Abstract

Parameterized quantum circuits (PQCs), as one of the most promising schemes to realize quantum machine learning algorithms on near-term quantum computers, have been designed to solve machine earning tasks with quantum advantages. In this paper, we explain why PQCs with different structures can achieve generative tasks in the discrete case from the perspective of probability estimation. Although different kinds of PQCs are proposed for generative tasks, the current methods often encounter the following three hurdles: (1) the mode contraction problem; (2) unexpected data are often generated with a high proportion; (3) target data cannot be sampled directly. For the purpose of tackling the above hurdles, we devise Bayesian quantum circuit (BQC) through introducing ancillary qubits to represent prior distributions. BQC advances both generative and semi-supervised quantum circuit learning tasks, where its effectiveness is validated by numerical simulations using the Rigetti Forest platform.

DOI
10.48550/arxiv.1805.11089

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Simulation is an indispensable tool in both engineering and the sciences. In simulation-based modeling, a parametric simulator is adopted as a mechanistic model of a physical system. The problem of designing algorithms that optimize the simulator parameters is the focus of the emerging field of simulation-based inference (SBI), which is often formulated in a Bayesian setting with the goal of quantifying epistemic uncertainty. This work studies Bayesian SBI that leverages a parameterized quantum circuit (PQC) as the underlying simulator. The proposed solution follows the well-established principle that quantum computers are best suited for the simulation of certain physical phenomena. It contributes to the field of quantum machine learning by moving beyond the likelihood-based methods investigated in prior work and accounting for the likelihood-free nature of PQC training. Experimental results indicate that well-motivated quantum circuits that account for the structure of the underlying physical system are capable of simulating data from two distinct tasks.
Variational quantum algorithms (VQAs) have emerged as a promising quantum alternative for solving optimization and machine learning problems using parameterized quantum circuits (PQCs). The design of these circuits influences the ability of the algorithm to efficiently explore the solution space and converge to more optimal solutions. Quantum architecture search (QAS) is therefore a highly relevant topic, in which the selection of an appropriate circuit topology, gate set, and parameterization scheme is determinant to achieve good performance. In addition, it is not only problem-dependent, but the quantum hardware used also has a significant impact on the results. Therefore, we present BPQCO, a Bayesian optimization-based strategy to search for optimal PQCs adapted to the problem to be solved and to the characteristics and limitations of the chosen quantum hardware. To this end, we experimentally demonstrate the influence of the circuit design on the performance obtained for two classification problems (a synthetic dataset and the well-known Iris dataset), focusing on the design of the circuit ansatz. In addition, we study the degradation of the obtained circuits in the presence of noise when simulating real quantum computers. To mitigate the effect of noise, two alternative optimization strategies based on the characteristics of the quantum system are proposed. The results obtained confirm the relevance of the presented approach and allow its adoption in further work based on the use of PQCs.
Parameterized quantum circuits (PQC) have emerged as a quantum analogue of deep neural networks and can be trained for discriminative or generative tasks and can be trained with gradient-based optimization on near-term quantum devices [1], [2], [3]. In the current era of quantum computing, known as the noisy intermediate scale quantum (NISQ) era [4], these devices contain a moderate number of qubits (< 100), and algorithmic performance is strongly impacted by hardware noise. Additionally, the training of PQCs are hybrid algorithms, in which the computational workflow is split between quantum and classical computing platforms.
Generative modeling is a flavor of machine learning with applications ranging from computer vision to chemical design. It is expected to be one of the techniques most suited to take advantage of the additional resources provided by near-term quantum computers. Here, we implement a data-driven quantum circuit training algorithm on the canonical Bars-and-Stripes dataset using a quantum-classical hybrid machine. The training proceeds by running parameterized circuits on a trapped ion quantum computer and feeding the results to a classical optimizer. We apply two separate strategies, Particle Swarm and Bayesian optimization to this task. We show that the convergence of the quantum circuit to the target distribution depends critically on both the quantum hardware and classical optimization strategy. Our study represents the first successful training of a high-dimensional universal quantum circuit and highlights the promise and challenges associated with hybrid learning schemes.
Parameterized quantum circuits (PQCs) have emerged as a promising approach for quantum neural networks. However, understanding their expressive power in accomplishing machine learning tasks remains a crucial question. This paper investigates the expressivity of PQCs for approximating general multivariate function classes. Unlike previous Universal Approximation Theorems for PQCs, which are either nonconstructive or rely on parameterized classical data processing, we explicitly construct data re-uploading PQCs for approximating multivariate polynomials and smooth functions. We establish the first non-asymptotic approximation error bounds for these functions in terms of the number of qubits, quantum circuit depth, and number of trainable parameters. Notably, we demonstrate that for approximating functions that satisfy specific smoothness criteria, the quantum circuit size and number of trainable parameters of our proposed PQCs can be smaller than those of deep ReLU neural networks. We further validate the approximation capability of PQCs through numerical experiments. Our results provide a theoretical foundation for designing practical PQCs and quantum neural networks for machine learning tasks that can be implemented on near-term quantum devices, paving the way for the advancement of quantum machine learning.