Parameterized Quantum Circuits Research Articles

Variational denoising for variational quantum eigensolver

The variational quantum eigensolver (VQE) is a hybrid algorithm that has the potential to provide a quantum advantage in practical chemistry problems that are currently intractable on classical computers. VQE trains parameterized quantum circuits using a classical optimizer to approximate the eigenvalues and eigenstates of a given Hamiltonian. However, VQE faces challenges in task-specific design and machine-specific architecture, particularly when running on noisy quantum devices. This can have a negative impact on its trainability, accuracy, and efficiency, resulting in noisy quantum data. We propose variational denoising, an unsupervised learning method that employs a parameterized quantum neural network to improve the solution of VQE by learning from noisy VQE outputs. Our approach can significantly decrease energy estimation errors and increase fidelities with ground states compared to noisy input data for the H2, LiH, and BeH2 molecular Hamiltonians and the transverse field Ising model. Surprisingly, it only requires noisy data for training. Variational denoising can be integrated into quantum hardware, increasing its versatility as an end-to-end quantum processing for quantum data. Published by the American Physical Society 2024

Optimizing one-axis twists for variational Bayesian quantum metrology

Quantum metrology and sensing seek advantages in estimating an unknown parameter of some quantum state or channel, using entanglement such as spin squeezing produced by one-axis twists or other quantum resources. In particular, qubit phase estimation, or rotation sensing, appears as a ubiquitous problem with applications to electric field sensing, magnetometry, atomic clocks, and gyroscopes. By adopting the Bayesian formalism to the phase estimation problem to account for limited initial knowledge about the value of the phase, we formulate variational metrology and treat the state preparation (or encoding) and measurement (or decoding) procedures as parameterized quantum circuits. It is important to understand how effective various parametrized protocols are as well as how robust they are to the effects of complex noise such as spatially correlated noise. First, we propose a family of parametrized encoding and decoding protocols called arbitrary-axis twist ansatzes, and show that it can lead to a substantial reduction in the number of one-axis twists needed to achieve a target estimation error. Furthermore, we demonstrate that the estimation error associated with these strategies decreases with system size in a faster manner than classical (or no-twist) protocols, even in the less-explored regimes where the prior information is limited. Last, using a polynomial-sized tensor network algorithm, we numerically analyze practical variational metrology beyond the symmetric subspace of a collective spin and find that quantum advantage persists for the arbitrary-axis twist ansatzes with a few one-axis twists and smaller total twisting angles for practically relevant noise levels. Published by the American Physical Society 2024

Sequential optimal selections of single-qubit gates in parameterized quantum circuits

In variational quantum algorithms, it is important to balance conflicting requirements of expressibility and trainability of a parameterized quantum circuit (PQC). However, appropriate PQC designs are not necessarily trivial. Here, we propose an algorithm for optimizing the PQC structure, where single-qubit gates are sequentially replaced by the optimal ones via diagonalization of a matrix whose elements are evaluated on slightly modified circuits. This replacement leads to a better approximation of target states with limited circuit depth. Furthermore, we clarify the existence of a barren plateau in the sequential optimization in terms of the spectrum concentration of the matrix, which defines the cost landscape with respect to changes in the target gate. Then, we rigorously show the concentration is no faster than polynomials in the number of qubits when an n-qubit PQC depth is O(log⁡n) using local observables. Finally, numerical experiments are provided to show the convergence of our method which is faster than classical optimizers on both simulators and a real device. Our results provide evidences for sequential optimizers as better alternatives to optimize PQCs on near-term quantum devices.

Information flow in parameterized quantum circuits

In this work, we introduce a new way to quantify information flow in quantum systems, especially for parameterized quantum circuits (PQCs). We use a graph representation of the circuits and propose a new distance metric using the mutual information between gate nodes. We then present an optimization procedure for variational algorithms using paths based on the distance measure. We explore the features of the algorithm by means of the variational quantum eigensolver, in which we compute the ground state energies of the Heisenberg model. In addition, we employ the method to solve a binary classification problem using variational quantum classification. From numerical simulations, we show that our method can be successfully used for optimizing the PQCs primarily used in near-term algorithms. We further note that information-flow based paths can be used to improve convergence of existing stochastic gradient based methods.

On the universality of Sn -equivariant k-body gates

The importance of symmetries has recently been recognized in quantum machine learning from the simple motto: if a task exhibits a symmetry (given by a group G ), the learning model should respect said symmetry. This can be instantiated via G -equivariant quantum neural networks (QNNs), i.e. parametrized quantum circuits whose gates are generated by operators commuting with a given representation of G . In practice, however, there might be additional restrictions to the types of gates one can use, such as being able to act on at most k qubits. In this work we study how the interplay between symmetry and k-bodyness in the QNN generators affect its expressiveness for the special case of G=Sn , the symmetric group. Our results show that if the QNN is generated by one- and two-body Sn -equivariant gates, the QNN is semi-universal but not universal. That is, the QNN can generate any arbitrary special unitary matrix in the invariant subspaces, but has no control over the relative phases between them. Then, we show that in order to reach universality one needs to include n-body generators (if n is even) or (n−1) -body generators (if n is odd). As such, our results brings us a step closer to better understanding the capabilities and limitations of equivariant QNNs.

Prediction of anaerobic digestion performance by quantum convolutional reconstruction gated recurrent neural network* *This work was supported by the Open Fund of Advanced Cryptography and System Security Key Laboratory of Sichuan Province (Grant No. SKLACSS202208), National Natural Science Foundation of China (No.61772295),and the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant no.KJZD-M202000501).

Methane as a renewable energy source has become a hot topic in recent years. Methane is a bioenergy source produced during the anaerobic digestion of organic waste, and the anaerobic digestion process must be monitored and controlled to produce the required amount of methane in a stable manner. Mathematical modeling is used to simulate digester operation to predict the biogas production from anaerobic digestion, to avoid reactor loading or performance degradation, and to ensure efficient operation of the system. In this paper, a Quantum Convolutional Reconstruction Gated Recurrent Neural Network is proposed. The original data features are extracted by convolutional neural network to reduce the dimensionality and retain the information, the parameterized quantum circuit is integrated in the gating recurrent unit, and the quantum reset gate and quantum update gate are constructed. The information extracted by the Convolution Neural networks is input into the quantum gated recurrent neural network, and the quantum storage unit integrates the information into the hidden layer state, thus processing the hidden layer state information more efficiently. The experimental results show that the prediction accuracy of the A Quantum Convolution Reconstructed Gated Recurrent Neural Network is improved from 81.95 to 88.21%, and the MAE value is reduced from 54.53% to 37.38%.

Quantum optimization algorithms: Energetic implications

SummarySince the dawn of quantum computing (QC), theoretical developments like Shor's algorithm proved the conceptual superiority of QC over traditional computing. However, such quantum supremacy claims are difficult to achieve in practice because of the technical challenges of realizing noiseless qubits. In the near future, QC applications will need to rely on noisy quantum devices that offload part of their work to classical devices. One way to achieve this is by using parameterized quantum circuits in optimization or even in machine learning tasks. The energy requirements of quantum algorithms have not yet been studied extensively. In this article, we explore several optimization algorithms using both theoretical insights and numerical experiments to understand their impact on energy consumption. Specifically, we highlight why and how algorithms like quantum natural gradient descent, simultaneous perturbation stochastic approximations or circuit learning methods, are at least to more energy efficient than their classical counterparts; why feedback‐based quantum optimization is energy‐inefficient; and how techniques like Rosalin can improve the energy efficiency of other algorithms by a factor of 20. Finally, we use the NchooseK high‐level programming model to run optimization problems on both gate‐based quantum computers and quantum annealers. Empirical data indicate that these optimization problems run faster, have better success rates, and consume less energy on quantum annealers than on their gate‐based counterparts.

Simulation of a spin-boson model by iterative optimization of a parametrized quantum circuit

Time evolution of the populations of spin states coupled with bosons, which can be a model of photosynthetic excitation energy transfer of dye molecules surrounded by proteins, is simulated using the projected-variational quantum dynamics algorithm. By a transformation of the Hamiltonian describing the spin-boson model into the one-dimensional nearest-neighbor form, it is shown that the spin-boson model can be simulated using the sequential ansatz even if a quantum computer has limited connectivity. The optimization of the parametrized quantum circuits is performed by the gradient descent method on a classical computer using the automatic differentiation, and the population of the spins is simulated on a noisy intermediate-scale quantum computer. The error originating from the quantum computing is mitigated by the Clifford data regression, in which the noise channel is estimated using the data obtained from all the time steps.

Generating Universal Adversarial Perturbations for Quantum Classifiers

Quantum Machine Learning (QML) has emerged as a promising field of research, aiming to leverage the capabilities of quantum computing to enhance existing machine learning methodologies. Recent studies have revealed that, like their classical counterparts, QML models based on Parametrized Quantum Circuits (PQCs) are also vulnerable to adversarial attacks. Moreover, the existence of Universal Adversarial Perturbations (UAPs) in the quantum domain has been demonstrated theoretically in the context of quantum classifiers. In this work, we introduce QuGAP: a novel framework for generating UAPs for quantum classifiers. We conceptualize the notion of additive UAPs for PQC-based classifiers and theoretically demonstrate their existence. We then utilize generative models (QuGAP-A) to craft additive UAPs and experimentally show that quantum classifiers are susceptible to such attacks. Moreover, we formulate a new method for generating unitary UAPs (QuGAP-U) using quantum generative models and a novel loss function based on fidelity constraints. We evaluate the performance of the proposed framework and show that our method achieves state-of-the-art misclassification rates, while maintaining high fidelity between legitimate and adversarial samples.

Training-Free Quantum Architecture Search

Variational quantum algorithm (VQA) derives advantages from its error resilience and high flexibility in quantum resource requirements, rendering it broadly applicable in the noisy intermediate-scale quantum era. As the performance of VQA highly relies on the structure of the parameterized quantum circuit, it is worthwhile to propose quantum architecture search (QAS) algorithms to automatically search for high-performance circuits. Nevertheless, existing QAS methods are time-consuming, requiring circuit training to assess circuit performance. This study pioneers training-free QAS by utilizing two training-free proxies to rank quantum circuits, in place of the expensive circuit training employed in conventional QAS. Taking into account the precision and computational overhead of the path-based and expressibility-based proxies, we devise a two-stage progressive training-free QAS (TF-QAS). Initially, directed acyclic graphs (DAGs) are employed for circuit representation, and a zero-cost proxy based on the number of paths in the DAG is designed to filter out a substantial portion of unpromising circuits. Subsequently, an expressibility-based proxy, finely reflecting circuit performance, is employed to identify high-performance circuits from the remaining candidates. These proxies evaluate circuit performance without circuit training, resulting in a remarkable reduction in computational cost compared to current training-based QAS methods. Simulations on three VQE tasks demonstrate that TF-QAS achieves a substantial enhancement of sampling efficiency ranging from 5 to 57 times compared to state-of-the-art QAS, while also being 6 to 17 times faster.

Quafu-RL: The cloud quantum computers based quantum reinforcement learning

With the rapid advancement of quantum computing, hybrid quantum–classical machine learning has shown numerous potential applications at the current stage, with expectations of being achievable in the noisy intermediate-scale quantum (NISQ) era. Quantum reinforcement learning, as an indispensable study, has recently demonstrated its ability to solve standard benchmark environments with formally provable theoretical advantages over classical counterparts. However, despite the progress of quantum processors and the emergence of quantum computing clouds, implementing quantum reinforcement learning algorithms utilizing parameterized quantum circuits (PQCs) on NISQ devices remains infrequent. In this work, we take the first step towards executing benchmark quantum reinforcement problems on real devices equipped with at most 136 qubits on the BAQIS Quafu quantum computing cloud. The experimental results demonstrate that the policy agents can successfully accomplish objectives under modified conditions in both the training and inference phases. Moreover, we design hardware-efficient PQC architectures in the quantum model using a multi-objective evolutionary algorithm and develop a learning algorithm that is adaptable to quantum devices. We hope that the Quafu-RL can be a guiding example to show how to realize machine learning tasks by taking advantage of quantum computers on the quantum cloud platform.

Estimating quantum mutual information through a quantum neural network

We propose a method of quantum machine learning called quantum mutual information neural estimation (QMINE) for estimating von Neumann entropy and quantum mutual information, which are fundamental properties in quantum information theory. The QMINE proposed here basically utilizes a technique of quantum neural networks (QNNs), to minimize a loss function that determines the von Neumann entropy, and thus quantum mutual information, which is believed more powerful to process quantum datasets than conventional neural networks due to quantum superposition and entanglement. To create a precise loss function, we propose a quantum Donsker-Varadhan representation (QDVR), which is a quantum analog of the classical Donsker-Varadhan representation. By exploiting a parameter shift rule on parameterized quantum circuits, we can efficiently implement and optimize the QNN and estimate the quantum entropies using the QMINE technique. Furthermore, numerical observations support our predictions of QDVR and demonstrate the good performance of QMINE.

Hamiltonian variational ansatz without barren plateaus

Variational quantum algorithms, which combine highly expressive parameterized quantum circuits (PQCs) and optimization techniques in machine learning, are one of the most promising applications of a near-term quantum computer. Despite their huge potential, the utility of variational quantum algorithms beyond tens of qubits is still questioned. One of the central problems is the trainability of PQCs. The cost function landscape of a randomly initialized PQC is often too flat, asking for an exponential amount of quantum resources to find a solution. This problem, dubbed barren plateaus, has gained lots of attention recently, but a general solution is still not available. In this paper, we solve this problem for the Hamiltonian variational ansatz (HVA), which is widely studied for solving quantum many-body problems. After showing that a circuit described by a time-evolution operator generated by a local Hamiltonian does not have exponentially small gradients, we derive parameter conditions for which the HVA is well approximated by such an operator. Based on this result, we propose an initialization scheme for the variational quantum algorithms and a parameter-constrained ansatz free from barren plateaus.

Quantum Gaussian process regression for Bayesian optimization

Gaussian process regression is a well-established Bayesian machine learning method. We propose a new approach to Gaussian process regression using quantum kernels based on parameterized quantum circuits. By employing a hardware-efficient feature map and careful regularization of the Gram matrix, we demonstrate that the variance information of the resulting quantum Gaussian process can be preserved. We also show that quantum Gaussian processes can be used as a surrogate model for Bayesian optimization, a task that critically relies on the variance of the surrogate model. To demonstrate the performance of this quantum Bayesian optimization algorithm, we apply it to the hyperparameter optimization of a machine learning model which performs regression on a real-world dataset. We benchmark the quantum Bayesian optimization against its classical counterpart and show that quantum version can match its performance.

Quantum variational distance-based centroid classifier

Quantum machine learning is a rapidly growing field with the potential to surpass classical methods and offer novel solutions to learning problems. Supervised classification, a prominent pattern recognition task, has attracted significant attention for improving existing methods using quantum techniques. Although previous quantum algorithms for distance-based classification have shown promise, they are not compatible with current Noisy Intermediate-Scale Quantum (NISQ) devices and cannot handle multiclass problems. To address these limitations, we present the quantum Variational Distance-based Centroid Classifier (qVDCC), a hybrid approach that combines the Quantum One-Class Classifier (QOCC) and Parameterized Quantum Circuits (PQCs). This approach greatly improves the feasibility of multiclass classification on NISQ devices, as it eliminates the need for a label qubit and uses a single-qubit measurement, regardless of data size. Our simulations on seven benchmark datasets using both noisy and noise-free simulations show that qVDCC is competitive with the k-nearest neighbors algorithm, while providing exponential data compression and efficient quantum data processing.

Quantum image compression with autoencoders based on parameterized quantum circuits

Quantum image compression with autoencoders based on parameterized quantum circuits

ResQNets: a residual approach for mitigating barren plateaus in quantum neural networks

The barren plateau problem in quantum neural networks (QNNs) is a significant challenge that hinders the practical success of QNNs. In this paper, we introduce residual quantum neural networks (ResQNets) as a solution to address this problem. ResQNets are inspired by classical residual neural networks and involve splitting the conventional QNN architecture into multiple quantum nodes, each containing its own parameterized quantum circuit, and introducing residual connections between these nodes. Our study demonstrates the efficacy of ResQNets by comparing their performance with that of conventional QNNs and plain quantum neural networks through multiple training experiments and analyzing the cost function landscapes. Our results show that the incorporation of residual connections results in improved training performance. Therefore, we conclude that ResQNets offer a promising solution to overcome the barren plateau problem in QNNs and provide a potential direction for future research in the field of quantum machine learning.

Randomness-Enhanced Expressivity of Quantum Neural Networks.

As a hybrid of artificial intelligence and quantum computing, quantum neural networks (QNNs) have gained significant attention as a promising application on near-term, noisy intermediate-scale quantum devices. Conventional QNNs are described by parametrized quantum circuits, which perform unitary operations and measurements on quantum states. In this Letter, we propose a novel approach to enhance the expressivity of QNNs by incorporating randomness into quantum circuits. Specifically, we introduce a random layer, which contains single-qubit gates sampled from a trainable ensemble pooling. The prediction of QNN is then represented by an ensemble average over a classical function of measurement outcomes. We prove that our approach can accurately approximate arbitrary target operators using Uhlmann's theorem for majorization, which enables observable learning. Our proposal is demonstrated with extensive numerical experiments, including observable learning, Rényi entropy measurement, and image recognition. We find the expressivity of QNNs is enhanced by introducing randomness for multiple learning tasks, which could have broad application in quantum machine learning.

Explainable representation learning of small quantum states

Unsupervised machine learning models build an internal representation of their training data without the need for explicit human guidance or feature engineering. This learned representation provides insights into which features of the data are relevant for the task at hand. In the context of quantum physics, training models to describe quantum states without human intervention offers a promising approach to gaining insight into how machines represent complex quantum states. The ability to interpret the learned representation may offer a new perspective on non-trivial features of quantum systems and their efficient representation. We train a generative model on two-qubit density matrices generated by a parameterized quantum circuit. In a series of computational experiments, we investigate the learned representation of the model and its internal understanding of the data. We observe that the model learns an interpretable representation which relates the quantum states to their underlying entanglement characteristics. In particular, our results demonstrate that the latent representation of the model is directly correlated with the entanglement measure concurrence. The insights from this study represent proof of concept toward interpretable machine learning of quantum states. Our approach offers insight into how machines learn to represent small-scale quantum systems autonomously.

Improving Noisy Hybrid Quantum Graph Neural Networks for Particle Decay Tree Reconstruction

Improving Noisy Hybrid Quantum Graph Neural Networks for Particle Decay Tree Reconstruction