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The variational quantum eigensolver (VQE) stands as a prominent quantum-classical hybrid algorithm for near-term quantum computers to obtain the ground states of molecular Hamiltonians in quantum chemistry. However, due to the non-commutativity of the Pauli operators in the Hamiltonian, the number of measurements required on quantum computers increases significantly as the system size grows, which may hinder practical applications of VQE. In this work, we present a protocol termed joint Bell measurement VQE (JBM-VQE) to reduce the number of measurements and speed up the VQE algorithm. Our method employs joint Bell measurements, enabling the simultaneous measurement of the absolute values of all expectation values of Pauli operators present in the Hamiltonian. In the course of the optimization, JBM-VQE estimates the absolute values of the expectation values of the Pauli operators for each iteration by the joint Bell measurement, while the signs of them are measured less frequently by the conventional method to measure the expectation values. Our approach is based on the empirical observation that the signs do not often change during optimization. We illustrate the speed-up of JBM-VQE compared to conventional VQE by numerical simulations for finding the ground states of molecular Hamiltonians of small molecules, and the speed-up of JBM-VQE at the early stage of the optimization becomes increasingly pronounced in larger systems. Our approach based on the joint Bell measurement is not limited to VQE and can be utilized in various quantum algorithms whose cost functions are expectation values of many Pauli operators., Comment: 14 pages, 8 figures

Quantum state estimation (or state tomography) is an indispensable task in quantum information processing. Because full state tomography that determines all elements of the density matrix is computationally demanding, one usually takes the strategy of assuming a certain model of quantum states and identifying the model parameters. However, it is difficult to make a valid assumption given little prior knowledge on a quantum state of interest, and thus we need a reasonable model selection method for quantum state estimation. Actually, in the classical statistical estimation theory, several types of information criteria have been established and widely used in practice for appropriately choosing a classical statistical model. In this study, we propose quantum information criteria for evaluating the quality of the estimated quantum state in terms of the quantum relative entropy, which is a natural quantum analogue of the classical information criterion defined in terms of Kullback-Leibler divergence. In particular, we derive two quantum information criteria depending on the type of estimator for the quantum relative entropy; one uses the log-likelihood and the other uses the classical shadow. The general role of information criteria is to predict the performance of an estimated model for unseen data, although it is a function of only sampled data; this generalization capability of the proposed quantum information criteria is evaluated in numerical simulations.

Variational quantum algorithms (VQAs) are promising methods that leverage noisy quantum computers and classical computing techniques for practical applications. In VQAs, the classical optimizers such as gradient-based optimizers are utilized to adjust the parameters of the quantum circuit so that the objective function is minimized. However, they often suffer from the so-called vanishing gradient or barren plateau issue. On the other hand, the normalized gradient descent (NGD) method, which employs the normalized gradient vector to update the parameters, has been successfully utilized in several optimization problems. Here, we study the performance of the NGD methods in the optimization of VQAs for the first time. Our goal is two-fold. The first is to examine the effectiveness of NGD and its variants for overcoming the vanishing gradient problems. The second is to propose a new NGD that can attain the faster convergence than the ordinary NGD. We performed numerical simulations of these gradient-based optimizers in the context of quantum chemistry where VQAs are used to find the ground state of a given Hamiltonian. The results show the effective convergence property of the NGD methods in VQAs, compared to the relevant optimizers without normalization. Moreover, we make use of some normalized gradient vectors at the past iteration steps to propose the novel historical NGD that has a theoretical guarantee to accelerate the convergence speed, which is observed in the numerical experiments as well.

Quantum classifiers provide sophisticated embeddings of input data in Hilbert space promising quantum advantage. The advantage stems from quantum feature maps encoding the inputs into quantum states with variational quantum circuits. A recent work shows how to map discrete features with fewer quantum bits using Quantum Random Access Coding (QRAC), an important primitive to encode binary strings into quantum states. We propose a new method to embed discrete features with trainable quantum circuits by combining QRAC and a recently proposed strategy for training quantum feature map called quantum metric learning. We show that the proposed trainable embedding requires not only as few qubits as QRAC but also overcomes the limitations of QRAC to classify inputs whose classes are based on hard Boolean functions. We numerically demonstrate its use in variational quantum classifiers to achieve better performances in classifying real-world datasets, and thus its possibility to leverage near-term quantum computers for quantum machine learning., Comment: IEEE QCE21 Poster

Recent days have witnessed significant interests in applying quantum-enhanced techniques for solving a variety of machine learning tasks. Variational methods that use quantum resources of imperfect quantum devices with the help of classical computing techniques are popular for supervised learning. Variational quantum classification (VQC) is one of such methods with possible quantum advantage in using quantum-enhanced features that are hard to compute by classical methods. Its performance depends on the mapping of classical features into a quantum-enhanced feature space. Although there have been many quantum-mapping functions proposed so far, there is little discussion on efficient mapping of discrete features, such as age group, zip code, and others, which are often significant for classifying datasets of interest. We first introduce the use of quantum random-access coding (QRAC) to map such discrete features efficiently into limited number of qubits for VQC. In numerical simulations, we present a range of encoding strategies and demonstrate their limitations and capabilities. We experimentally show that QRAC can help speeding up the training of VQC by reducing its parameters via saving on the number of qubits for the mapping. We confirm the effectiveness of the QRAC in VQC by experimenting on classification of real-world datasets with both simulators and real quantum devices.

A method for analyzing the feature map for the kernel-based quantum classifier is developed; that is, we give a general formula for computing a lower bound of the exact training accuracy, which helps us to see whether the selected feature map is suitable for linearly separating the dataset. We show a proof of concept demonstration of this method for a class of 2-qubit classifier, with several 2-dimensional dataset. Also, a synthesis method, that combines different kernels to construct a better-performing feature map in a lager feature space, is presented., Comment: 9 pages, 10 figures, 4 tables

The effects of nitridation on the density of traps at SiO$_2$/SiC interfaces near the conduction band edge were qualitatively examined by a simple, newly developed characterization method that utilizes Hall effect measurements and split capacitance-voltage measurements. The results showed a significant reduction in the density of interface traps near the conduction band edge by nitridation, as well as the high density of interface traps that was not eliminated by nitridation.