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IEEE Transactions on Knowledge and Data Engineering

Quantum Circuit Learning with Parameterized Boson Sampling

Jinjing Shi, Yongze Tang, Yuhu Lu, Yanyan Feng, Ronghua Shi, Shichao Zhang

Published Jan 1, 2021• 26 citations
Publisher

Abstract

A quantum circuit learning approach is studied to carry out the fast-fitting of Gaussian functions. First, a parameterized structure is designed for quantum circuits based on the boson sampling model. And then, the training procedure of exploiting gradient-based optimizations is presented to iteratively update the gradient of the loss function concerning circuit parameters. For efficiency, two kinds of circuit loss, the kernel maximum mean discrepancy and the mean absolute error, are used in the training procedure, which are both competent to achieve quantum circuit learning well. It is significant that the two circuit losses assist in reducing the variance to$2.54 \times 10^{-6}$and$6.91 \times 10^{-6}$, respectively. Finally, a kind of quantum circuit fixed structure is developed with the boson sampling model that can decrease the model complexity as the circuit depth d grows. Sets of experiments have been conducted to evaluate the proposed quantum circuit learning scheme, and demonstrate that our parameterized approach is efficient and promising, and it is worth looking forward to solving practical application problems with quantum computers since valid quantum circuits for Gaussian function fast-fitting can be designed indeed.

DOI
10.1109/tkde.2021.3095103

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A quantum circuit learning approach is studied to carry out the fast-fitting of Gaussian functions. First, a parameterized structure is designed for quantum circuits based on the boson sampling model. And then, the training procedure of exploiting gradient-based optimizations is presented to iteratively update the gradient of the loss function concerning circuit parameters. For efficiency, two kinds of circuit loss, the kernel maximum mean discrepancy and the mean absolute error, are used in the training procedure, which are both competent to achieve quantum circuit learning well. It is significant that the two circuit losses assist in reducing the variance to2.54×10−62.54 \times 10^{-6}2.54×10−6and6.91×10−66.91 \times 10^{-6}6.91×10−6, respectively. Finally, a kind of quantum circuit fixed structure is developed with the boson sampling model that can decrease the model complexity as the circuit depth d grows. Sets of experiments have been conducted to evaluate the proposed quantum circuit learning scheme, and demonstrate that our parameterized approach is efficient and promising, and it is worth looking forward to solving practical application problems with quantum computers since valid quantum circuits for Gaussian function fast-fitting can be designed indeed.
Quantum circuit Born machines are generative models which represent the probability distribution of classical dataset as quantum pure states. Computational complexity considerations of the quantum sampling problem suggest that the quantum circuits exhibit stronger expressibility compared to classical neural networks. One can efficiently draw samples from the quantum circuits via projective measurements on qubits. However, similar to the leading implicit generative models in deep learning, such as the generative adversarial networks, the quantum circuits cannot provide the likelihood of the generated samples, which poses a challenge to the training. We devise an efficient gradient-based learning algorithm for the quantum circuit Born machine by minimizing the kerneled maximum mean discrepancy loss. We simulated generative modeling of the BARS-AND-STRIPES dataset and Gaussian mixture distributions using deep quantum circuits. Our experiments show the importance of circuit depth and the gradient-based optimization algorithm. The proposed learning algorithm is runnable on near-term quantum device and can exhibit quantum advantages for probabilistic generative modeling.
Quantum generative modeling has emerged as a promising application of quantum computers, aiming to model complex probability distributions beyond the reach of classical methods. In practice, however, training such models often requires costly gradient estimation performed directly on the quantum hardware. Crucially, for certain structured quantum circuits, expectation values of local observables can be efficiently evaluated on a classical computer, enabling classical training without calls to the quantum hardware in the optimization loop. In these models, sampling from the resulting circuits can still be classically hard, so inference must be performed on a quantum device, yielding a potential computational advantage. In this work, we introduce a photonic quantum generative model built on parametrized Gaussian Boson Sampling circuits. The training is based on the efficient classical evaluation of expectation values enabled by the Gaussian structure of the state, allowing scalable optimization of the model parameters through the maximum mean discrepancy loss function. We demonstrate the effectiveness of the approach through numerical experiments on photonic systems with up to 805 modes and over a million trainable parameters, highlighting its scalability and suitability for near-term photonic quantum devices.
Hamiltonian learning, as an important quantum machine learning technique, provides a significant approach for determining an accurate quantum system. This paper establishes parameterized Hamiltonian learning (PHL) and explores its application and implementation on quantum computers. A parameterized quantum circuit for Hamiltonian learning is first created by decomposing unitary operators to excite the system evolution. Then, a PHL algorithm is developed to prepare a specific Hamiltonian system by iteratively updating the gradient of the loss function about circuit parameters. Finally, the experiments are conducted on Origin Pilot, and it demonstrates that the PHL algorithm can deal with the image segmentation problem and provide a segmentation solution accurately. Compared with the classical Grabcut algorithm, the PHL algorithm eliminates the requirement of early manual intervention. It provides a new possibility for solving practical application problems with quantum devices, which also assists in solving increasingly complicated problems and supports a much wider range of application possibilities in the future.
Parametrised quantum circuits learn by adjusting gate parameters, while their design shapes the functions they can represent and how readily they learn them. We introduce the circuit harmonic matrix, a fixed matrix organising Fourier expansions over inputs and parameters. It makes the effects of encoding, gates, initial state and observable explicit in coefficient variance, covariance and the quantum neural tangent kernel, linking variation across parameter space to local sensitivity. The construction also constrains representable functions and lower-bounds fitting error. For fixed Clifford gates and independently parametrised Pauli or controlled rotations, we develop a two-copy propagation method that computes covariance and the averaged tangent kernel analytically for uniform parameters, without constructing the full matrix or sampling parameters. Across 400 configurations spanning eight circuit families, we compare these quantities with actual learning. Larger coefficient variance is strongly associated with faster learning and lower error for the corresponding target frequency, revealing spectral bias towards lower frequencies. Increasing depth and qubit count generally suppresses total variance, yet deeper circuits usually learn faster and achieve lower error, while adding qubits slows early learning and has a frequency-dependent effect on final error. Matched targets show no robust overall advantage from covariance alignment. For one circuit setting, initial tangent kernels closely forecast the modest loss reductions reached in small-step gradient descent. These results connect circuit choices to representational constraints and learning performance, providing a basis for selecting designs suited to particular learning tasks.
Abstract Parameterized quantum circuits (PQCs) are crucial for quantum machine learning and circuit synthesis, enabling the practical implementation of complex quantum tasks. However, PQC learning has been largely confined to classical optimization methods, which suffer from issues like gradient vanishing.In this work, we introduce a nested optimization model, a hybrid approach that leverages quantum gradients to improve PQC learning for arbitrary polynomial-type cost functions. The proposed approach decomposes the learning problem into multiple subproblems, each aimed at learning the state identified by the quantum gradient using current circuit synthesis methods.By leveraging quantum gradients, the optimization procedure is free of a specific type of gradient vanishing, such as unfavorable local stationary points and barren plateaus.Specifically, with the guidance of quantum gradient, unfavorable local stationary points can be overcome by increasing the depth of the ansatz, while barren plateaus can be mitigated by constraining the optimization region.Numerically, we demonstrate the feasibility of the approach on two tasks: the Max-Cut problem and polynomial optimization. Additionally, we analyze the overheads introduced by this approach, which are primarily polynomial in the number of qubits in the system and can be controlled by adjusting the learning rate.The method can generate circuits while avoiding gradient vanishing, thus effectively optimizing the cost function.From the perspective of quantum algorithms, our model improves quantum optimization for polynomial-type cost functions, addressing the challenge of exponential sample complexity growth.
The topic of generative learning has gained traction within the field of quantum machine learning, in particular with the advent of train-on-classical, deploy-on-quantum methods. This approach exploits the properties of intermediate-complexity circuits whose training can be simulated classically efficiently, but that generally require quantum hardware for the corresponding sampling problem. Quantum linear optics possess similar properties, which allow us to propose an efficient training procedure for photon-native quantum generative models based on the maximum mean discrepancy, where the deployment of the model corresponds to the task of boson sampling. We provide numerical results, propose datasets, and we also explore how initialization strategies, kernel and ansatz choice affect the training.