Search

Efficient understanding of a quantum system fundamentally relies on the selection of observables. Pauli observables and mutually unbiased bases (MUBs) are widely used in practice and are often regarded as theoretically optimal for quantum state tomography (QST). However, Pauli observables require a large number of measurements for complete tomography and do not permit direct measurement of density matrix elements with a constant number of observables. For MUBs, the existence of complete sets of \(d+1\) bases in all dimensions remains unresolved, highlighting the need for alternative observables. In this work, we introduce a novel set of \(2d\) observables specifically designed to enable the complete characterization of any \(d\)-dimensional quantum state. To demonstrate the advantages of these observables, we explore two key applications. First, we show that direct measurement of density matrix elements is feasible without auxiliary systems, with any element extractable using only three selected observables. Second, we demonstrate that QST for unknown rank-\(r\) density matrices, excluding only a negligible subset, can be achieved with \(O(r \log d)\) observables. This significantly reduces the number of unitary operations compared to compressed sensing with Pauli observables, which typically require \(O(r d \log^2 d)\) operations. Each circuit is iteratively generated and can be efficiently decomposed into at most \(O(n^4)\) elementary gates for an \(n\)-qubit system. The proposed observables represent a substantial advancement in the characterization of quantum systems, enhancing both the efficiency and practicality of quantum state learning and offering a promising alternative to traditional methods.

Darwinian evolution of the biological brain is documented through multiple lines of evidence, although the modes of evolutionary changes remain unclear. Drawing inspiration from the evolved neural systems (e.g., visual cortex), deep learning models have demonstrated superior performance in visual tasks, among others. While the success of training deep neural networks has been relying on back-propagation (BP) and its variants to learn representations from data, BP does not incorporate the evolutionary processes that govern biological neural systems. This work proposes a neural network optimization framework based on evolutionary theory. Specifically, BP-trained deep neural networks for visual recognition tasks obtained from the ending epochs are considered the primordial ancestors (initial population). Subsequently, the population evolved with differential evolution. Extensive experiments are carried out to examine the relationships between Darwinian evolution and neural network optimization, including the correspondence between datasets, environment, models, and living species. The empirical results show that the proposed framework has positive impacts on the network, with reduced over-fitting and an order of magnitude lower time complexity compared to BP. Moreover, the experiments show that the proposed framework performs well on deep neural networks and big datasets., Comment: This work has been submitted to the IEEE for possible publication

"AI for Science" aims to solve fundamental scientific problems using AI techniques. As most physical phenomena can be described as Partial Differential Equations (PDEs) , approximating their solutions using neural networks has evolved as a central component of scientific-ML. Physics-Informed Neural Networks (PINNs) is the general method that has evolved for this task but its training is well-known to be very unstable. In this work we explore the possibility of changing the model being trained from being just a neural network to being a non-linear transformation of it - one that algebraically includes the boundary/initial conditions. This reduces the number of terms in the loss function than the standard PINN losses. We demonstrate that our modification leads to significant performance gains across a range of benchmark tasks, in various dimensions and without having to tweak the training algorithm. Our conclusions are based on conducting hundreds of experiments, in the fully unsupervised setting, over multiple linear and non-linear PDEs set to exactly solvable scenarios, which lends to a concrete measurement of our performance gains in terms of order(s) of magnitude lower fractional errors being achieved, than by standard PINNs. The code accompanying this manuscript is publicly available at, https://github.com/MorganREN/Improving-PINNs-By-Algebraic-Inclusion-of-Boundary-and-Initial-Conditions, Comment: 48 Pages, 25 Figures

Recent advancements in quantum hardware and classical computing simulations have significantly enhanced the accessibility of quantum system data, leading to an increased demand for precise descriptions and predictions of these systems. Accurate prediction of quantum Hamiltonian dynamics and identification of Hamiltonian parameters are crucial for advancements in quantum simulations, error correction, and control protocols. This study introduces a machine learning model with dual capabilities: it can deduce time-dependent Hamiltonian parameters from observed changes in local observables within quantum many-body systems, and it can predict the evolution of these observables based on Hamiltonian parameters. Our model's validity was confirmed through theoretical simulations across various scenarios and further validated by two experiments. Initially, the model was applied to a Nuclear Magnetic Resonance quantum computer, where it accurately predicted the dynamics of local observables. The model was then tested on a superconducting quantum computer with initially unknown Hamiltonian parameters, successfully inferring them. Our approach aims to enhance various quantum computing tasks, including parameter estimation, noise characterization, feedback processes, and quantum control optimization., Comment: 19 pages, 14 figures

This paper introduces a representative-based approach for distributed learning that transforms multiple raw data points into a virtual representation. Unlike traditional distributed learning methods such as Federated Learning, which do not offer human interpretability, our method makes complex machine learning processes accessible and comprehensible. It achieves this by condensing extensive datasets into digestible formats, thus fostering intuitive human-machine interactions. Additionally, this approach maintains privacy and communication efficiency, and it matches the training performance of models using raw data. Simulation results show that our approach is competitive with or outperforms traditional Federated Learning in accuracy and convergence, especially in scenarios with complex models and a higher number of clients. This framework marks a step forward in integrating human intuition with machine intelligence, which potentially enhances human-machine learning interfaces and collaborative efforts.

In this paper, we address the problem of designing an experimental plan with both discrete and continuous factors under fairly general parametric statistical models. We propose a new algorithm, named ForLion, to search for locally optimal approximate designs under the D-criterion. The algorithm performs an exhaustive search in a design space with mixed factors while keeping high efficiency and reducing the number of distinct experimental settings. Its optimality is guaranteed by the general equivalence theorem. We present the relevant theoretical results for multinomial logit models (MLM) and generalized linear models (GLM), and demonstrate the superiority of our algorithm over state-of-the-art design algorithms using real-life experiments under MLM and GLM. Our simulation studies show that the ForLion algorithm could reduce the number of experimental settings by 25% or improve the relative efficiency of the designs by 17.5% on average. Our algorithm can help the experimenters reduce the time cost, the usage of experimental devices, and thus the total cost of their experiments while preserving high efficiencies of the designs., Comment: 36 pages, 7 tables, 5 figures

In recent years, analysis methods for quantum states based on randomized measurements have been investigated extensively. Still, in the experimental implementations these methods were typically used for characterizing strongly entangled states and not to analyze the different families of multiparticle or weakly entangled states. In this work, we experimentally prepare various entangled states with path-polarization hyper-entangled photon pairs, and study their entanglement properties using the full toolbox of randomized measurements. First, we successfully characterize the correlations of a series of GHZ-W mixed states using the second moments of the random outcomes, and demonstrate the advantages of this method by comparing it with the well-known three-tangle and squared concurrence. Second, we generate bound entangled chessboard states of two three-dimensional systems and verify their weak entanglement with a criterion derived from moments of randomized measurements., Comment: 10 pages, 6 figures, Comments are welcome

A practical fault-tolerant quantum computer is worth looking forward to as it provides applications that outperform their known classical counterparts. However, millions of interacting qubits with stringent criteria are required, which is intractable with current quantum technologies. As it would take decades to make it happen, exploiting the power of noisy intermediate-scale quantum(NISQ) devices, which already exist, is becoming one of current goals. Matrix exponentiation, especially hermitian matrix exponentiation, is an essential element for quantum information processing. In this article, a heuristic method is reported as simulating a hermitian matrix exponentiation using parametrized quantum circuit(PQC). To generate PQCs for simulations, two strategies, each with its own advantages, are proposed, and can be deployed on near future quantum devices. Compared with the method such as product formula and density matrix exponentiation, the PQCs provided in our method require only low depth circuit and easily accessible gates, which benefit experimental realizations. Furthermore, in this paper, an ancilla-assisted parameterized quantum circuit is proposed to characterize and compress a unitary process, which is likely to be applicable to realizing applications on NISQ hardwares, such as phase estimation, principal component analyses, and matrix inversion. To support the feasibility of our method, numerical experiments were investigated via simulating evolutions by Bell state, GHZ state and Hamiltonian of Crotonic acid, which show an experimental friendly result when compared with their conventional methods. As pursuing a fault-tolerant quantum computer is still challenging and takes decades, our work, which gives a NISQ device friendly way, contributes to the field of NISQ algorithms and provides a possibility, exploiting the power with current quantum technology., Comment: all comments are welcomed

Extracting information from quantum devices has long been a crucial problem in the field of quantum mechanics. By performing elaborate measurements, quantum state tomography, an important and fundamental tool in quantum science and technology, can be used to determine unknown quantum states completely. In this study, we explore methods to determine multi-qubit pure quantum states uniquely and directly. Two adaptive protocols are proposed, with their respective quantum circuits. Herein, two or three observables are sufficient, while the number of measurement outcomes is either the same as or fewer than those in existing methods. Additionally, experiments on the IBM 5-qubit quantum computer, as well as numerical investigations, demonstrate the feasibility of the proposed protocols., Comment: 8 pages+9 figures. All comments are welcomed

Gradient-based algorithms, popular strategies to optimization problems, are essential for many modern machine-learning techniques. Theoretically, extreme points of certain cost functions can be found iteratively along the directions of the gradient. The time required to calculating the gradient of $d$-dimensional problems is at a level of $\mathcal{O}(poly(d))$, which could be boosted by quantum techniques, benefiting the high-dimensional data processing, especially the modern machine-learning engineering with the number of optimized parameters being in billions. Here, we propose a quantum gradient algorithm for optimizing general polynomials with the dressed amplitude encoding, aiming at solving fast-convergence polynomials problems within both time and memory consumption in $\mathcal{O}(poly (\log{d}))$. Furthermore, numerical simulations are carried out to inspect the performance of this protocol by considering the noises or perturbations from initialization, operation and truncation. For the potential values in high-dimension optimizations, this quantum gradient algorithm is supposed to facilitate the polynomial-optimizations, being a subroutine for future practical quantum computer., Comment: (5+10)pages + (3+1) figures. Comments are always welcome

Among existing approaches to holonomic quantum computing, the adiabatic holonomic quantum gates (HQGs) suffer errors due to decoherence, while the non-adiabatic HQGs either require additional Hilbert spaces or are difficult to scale. Here, we report a systematic, scalable approach based on dynamical invariants to realize HQGs without using additional Hilbert spaces. While presenting the theoretical framework of our approach, we design and experimentally evaluate single-qubit and two-qubits HQGs for the nuclear magnetic resonance system. The single-qubit gates acquire average fidelity 0.9972 by randomized benchmarking, and the controlled-NOT gate acquires fidelity 0.9782 by quantum process tomography. Our approach is also platform-independent, and thus may open a way to large-scale holonomic quantum computation., Comment: 9 pages, 3 figures

We propose a fast and efficient strategy, called the representative approach, for big data analysis with generalized linear models, especially for distributed data with localization requirements or limited network bandwidth. With a given partition of massive dataset, this approach constructs a representative data point for each data block and fits the target model using the representative dataset. In terms of time complexity, it is as fast as the subsampling approaches in the literature. As for efficiency, its accuracy in estimating parameters given a homogeneous partition is comparable with the divide-and-conquer method. Supported by comprehensive simulation studies and theoretical justifications, we conclude that mean representatives (MR) work fine for linear models or generalized linear models with a flat inverse link function and moderate coefficients of continuous predictors. For general cases, we recommend the proposed score-matching representatives (SMR), which may improve the accuracy of estimators significantly by matching the score function values. As an illustrative application to the Airline on-time performance data, we show that the MR and SMR estimates are as good as the full data estimate when available., Comment: 44 pages, 11 figures

We develop a new nonparametric approach for estimating the risk-neutral density of asset prices and reformulate its estimation into a double-constrained optimization problem. We evaluate our approach using the S\&P 500 market option prices from 1996 to 2015. A comprehensive cross-validation study shows that our approach outperforms the existing nonparametric quartic B-spline and cubic spline methods, as well as the parametric method based on the Normal Inverse Gaussian distribution. As an application, we use the proposed density estimator to price long-term variance swaps, and the model-implied prices match reasonably well with those of the variance future downloaded from the CBOE website., Comment: 19 pages, 2 figures

Gradient descent method, as one of the major methods in numerical optimization, is the key ingredient in many machine learning algorithms. As one of the most fundamental way to solve the optimization problems, it promises the function value to move along the direction of steepest descent. For the vast resource consumption when dealing with high-dimensional problems, a quantum version of this iterative optimization algorithm has been proposed recently[arXiv:1612.01789]. Here, we develop this protocol and implement it on a quantum simulator with limited resource. Moreover, a prototypical experiment was shown with a 4-qubit Nuclear Magnetic Resonance quantum processor, demonstrating a optimization process of polynomial function iteratively. In each iteration, we achieved an average fidelity of 94\% compared with theoretical calculation via full-state tomography. In particular, the iterative point gradually converged to the local minimum. We apply our method to multidimensional scaling problem, further showing the potentially capability to yields an exponentially improvement compared with classical counterparts. With the onrushing tendency of quantum information, our work could provide a subroutine for the application of future practical quantum computers., Comment: 6+4 pages, 8 figures

We experimentally simulate the spin networks -- a fundamental description of quantum spacetime at the Planck level. We achieve this by simulating quantum tetrahedra and their interactions. The tensor product of these quantum tetrahedra comprises spin networks. In this initial attempt to study quantum spacetime by quantum information processing, on a four-qubit nuclear magnetic resonance quantum simulator, we simulate the basic module -- comprising five quantum tetrahedra -- of the interactions of quantum spacetime. By measuring the geometric properties on the corresponding quantum tetrahedra and simulate their interactions, our experiment serves as the basic module that represents the Feynman diagram vertex in the spin-network formulation of quantum spacetime.

As of today, no one can tell when a universal quantum computer with thousands of logical quantum bits (qubits) will be built. At present, most quantum computer prototypes involve less than ten individually controllable qubits, and only exist in laboratories for the sake of either the great costs of devices or professional maintenance requirements. Moreover, scientists believe that quantum computers will never replace our daily, every-minute use of classical computers, but would rather serve as a substantial addition to the classical ones when tackling some particular problems. Due to the above two reasons, cloud-based quantum computing is anticipated to be the most useful and reachable form for public users to experience with the power of quantum. As initial attempts, IBM Q has launched influential cloud services on a superconducting quantum processor in 2016, but no other platforms has followed up yet. Here, we report our new cloud quantum computing service -- NMRCloudQ (http://nmrcloudq.com/zh-hans/), where nuclear magnetic resonance, one of the pioneer platforms with mature techniques in experimental quantum computing, plays as the role of implementing computing tasks. Our service provides a comprehensive software environment preconfigured with a list of quantum information processing packages, and aims to be freely accessible to either amateurs that look forward to keeping pace with this quantum era or professionals that are interested in carrying out real quantum computing experiments in person. In our current version, four qubits are already usable with in average 1.26% single-qubit gate error rate and 1.77% two-qubit controlled-NOT gate error rate via randomized benchmaking tests. Improved control precisions as well as a new seven-qubit processor are also in preparation and will be available later., Comment: NMRCloudQ at http://nmrcloudq.com:8000/zh-hans/. Interface is Chinese; English available soon

The quantum clock synchronization (QCS) is to measure the time difference among the spatially separated clocks with the principle of quantum mechanics. The first QCS algorithm proposed by Chuang and Jozsa is merely based on two parties, which is further extended and generalized to the multiparty situation by Krco and Paul. They present a multiparty QCS protocol based upon W states that utilizes shared prior entanglement and broadcast of classical information to synchronize spatially separated clocks. Shortly afterwards, Ben-Av and Exman came up with an optimized multiparty QCS using Z state. In this work, we firstly report an implementation of Krco and Ben-AV multiparty QCS algorithm using a four-qubit Nuclear Magnetic Resonance (NMR). The experimental results show a great agreement with the theory and also prove Ben-AV multiparty QCS algorithm more accurate than Krco., Comment: 6 pages and 6 figures. All comments are welcome!

The problem of determining whether a given quantum state is entangled lies at the heart of quantum information processing, which is known to be an NP-hard problem in general. Despite the proposed many methods such as the positive partial transpose (PPT) criterion and the k-symmetric extendibility criterion to tackle this problem in practice, none of them enables a general, effective solution to the problem even for small dimensions. Explicitly, separable states form a high-dimensional convex set, which exhibits a vastly complicated structure. In this work, we build a new separability-entanglement classifier underpinned by machine learning techniques. Our method outperforms the existing methods in generic cases in terms of both speed and accuracy, opening up the avenues to explore quantum entanglement via the machine learning approach., Comment: 8 pages, 4 figures; comments welcome

Quantum state tomography is an indispensable but costly part of many quantum experiments. Typically, it requires measurements to be carried in a number of different settings on a fixed experimental setup. The collected data is often informationally overcomplete, with the amount of information redundancy depending on the particular set of measurement settings chosen. This raises a question about how should one optimally take data so that the number of measurement settings necessary can be reduced. Here, we cast this problem in terms of integer programming. For a given experimental setup, standard integer programming algorithms allow us to find the minimum set of readout operations that can realize a target tomographic task. We apply the method to certain basic and practical state tomographic problems in nuclear magnetic resonance experimental systems. The results show that, considerably less readout operations can be found using our technique than it was by using the previous greedy search strategy. Therefore, our method could be helpful for simplifying measurement schemes so as to minimize the experimental effort.

Quantum simulation promises to have wide applications in many fields where problems are hard to model with classical computers. Various quantum devices of different platforms have been built to tackle the problems in, say, quantum chemistry, condensed matter physics, and high-energy physics. Here, we report an experiment towards the simulation of quantum gravity by simulating the holographic entanglement entropy. On a six-qubit nuclear magnetic resonance quantum simulator, we demonstrate a key result of Anti-de Sitter/conformal field theory(\adscft) correspondence---the Ryu-Takayanagi formula is demonstrated by measuring the relevant entanglement entropies on the perfect tensor state. The fidelity of our experimentally prepared the six-qubit state is 85.0\% via full state tomography and reaches 93.7\% if the signal-decay due to decoherence is taken into account. Our experiment serves as the basic module of simulating more complex tensor network states that exploring \adscft correspondence. As the initial experimental attempt to study \adscft via quantum information processing, our work opens up new avenues exploring quantum gravity phenomena on quantum simulators., Comment: To appear in NPJ quantum information. All comments are welcome!

Accurate and efficient control of quantum systems is one of the central challenges for quantum information processing. Current state-of-the-art experiments rarely go beyond 10 qubits and in most cases demonstrate only limited control. Here we demonstrate control of a 12-qubit system, and show that the system can be employed as a quantum processor to optimize its own control sequence by using measurement-based feedback control (MQFC). The final product is a control sequence for a complex 12-qubit task: preparation of a 12-coherent state. The control sequence is about 10% more accurate than the one generated by the standard (classical) technique, showing that MQFC can correct for unknown imperfections. Apart from demonstrating a high level of control over a relatively large system, our results show that even at the 12-qubit level, a quantum processor can be a useful lab instrument. As an extension of our work, we propose a method for combining the MQFC technique with a twirling protocol, to optimize the control sequence that produces a desired Clifford gate., Comment: The published version in npj Quantum Information

Topological orders can be used as media for topological quantum computing --- a promising quantum computation model due to its invulnerability against local errors. Conversely, a quantum simulator, often regarded as a quantum computing device for special purposes, also offers a way of characterizing topological orders. Here, we show how to identify distinct topological orders via measuring their modular $S$ and $T$ matrices. In particular, we employ a nuclear magnetic resonance quantum simulator to study the properties of three topologically ordered matter phases described by the string-net model with two string types, including the $\Z_2$ toric code, doubled semion, and doubled Fibonacci. The third one, non-Abelian Fibonacci order is notably expected to be the simplest candidate for universal topological quantum computing. Our experiment serves as the basic module, built on which one can simulate braiding of non-Abelian anyons and ultimately topological quantum computation via the braiding, and thus provides a new approach of investigating topological orders using quantum computers., Comment: 5 pages, 4 figures; to appear in Phys. Rev. Lett

Superposition, arguably the most fundamental property of quantum mechanics, lies at the heart of quantum information science. However, how to create the superposition of any two unknown pure states remains as a daunting challenge. Recently, it is proved that such a quantum protocol does not exist if the two input states are completely unknown, whereas a probabilistic protocol is still available with some prior knowledge about the input states [M. Oszmaniec \emph{et al.}, Phys. Rev. Lett. 116, 110403 (2016)]. The knowledge is that both of the two input states have nonzero overlaps with some given referential state. In this work, we experimentally realize the probabilistic protocol of superposing two pure states in a three-qubit nuclear magnetic resonance system. We demonstrate the feasibility of the protocol by preparing a families of input states, and the average fidelity between the prepared state and expected superposition state is over 99%. Moreover, we experimentally illustrate the limitation of the protocol that it is likely to fail or yields very low fidelity, if the nonzero overlaps are approaching zero. Our experimental implementation can be extended to more complex situations and other quantum systems., Comment: 6 pages, 7 figures

We investigate quantum state tomography (QST) for pure states and quantum process tomography (QPT) for unitary channels via $adaptive$ measurements. For a quantum system with a $d$-dimensional Hilbert space, we first propose an adaptive protocol where only $2d-1$ measurement outcomes are used to accomplish the QST for $all$ pure states. This idea is then extended to study QPT for unitary channels, where an adaptive unitary process tomography (AUPT) protocol of $d^2+d-1$ measurement outcomes is constructed for any unitary channel. We experimentally implement the AUPT protocol in a 2-qubit nuclear magnetic resonance system. We examine the performance of the AUPT protocol when applied to Hadamard gate, $T$ gate ($\pi/8$ phase gate), and controlled-NOT gate, respectively, as these gates form the universal gate set for quantum information processing purpose. As a comparison, standard QPT is also implemented for each gate. Our experimental results show that the AUPT protocol that reconstructing unitary channels via adaptive measurements significantly reduce the number of experiments required by standard QPT without considerable loss of fidelity., Comment: 8 pages, 2 figures and 2 tables. All comments are welcome!

Each free homotopy class of directed closed curves on a surface with boundary can be described by a cyclic reduced word in the generators of the fundamental group and their inverses. The word length is the number of letters of the cyclic word. If the surface has a hyperbolic metric with geodesic boundary, the geometric length of the class is the length of the unique geodesic. By computer experiments, we investigate the distribution of the geometric length among all classes with a given word length in the pair of pants surface. Our experiments strongly suggest that the distribution is normal.