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Comparison of quantum (left) vs classical (right) learning. U A and U B are unitaries that act on entangled Bell pairs distributed between Alice and Bob. C A and C B are operations on random classical bit strings belonging to Alice and Bob.
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![FIG. 9. Feed-forward neural network with dimension [k, m].](publication/365240511/figure/fig5/AS:11431281096447371@1668152451957/Feed-forward-neural-network-with-dimension-k-m_Q320.jpg)
Modeling joint probability distributions is an important task in a wide variety of fields. One popular technique for this employs a family of multivariate distributions with uniform marginals called copulas. While the theory of modeling joint distributions via copulas is well understood, it gets practically challenging to accurately model real data...
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... us analyze the implication of this for the quantum learning algorithms described in the paper versus classical learning schemes. The communication flow in the quantum learning scheme is shown in the left part of Fig. 8. Once the Bell pairs are created, the classical computer executing the optimization scheme can separately send x and y to Alice and Bob respectively. The measurements a and b are then collectively processed by the classical optimizer to produce the input to the quantum computer for the next iteration. Now consider how one would ...
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... can be written as a combination of operations that only acts on the bits of Alice or Bob separately, and operations that act on both sets of bits. The latter can be further modeled as operations on the bits belonging to Alice, followed by communication to Bob and then operations on just on the bits belonging to Bob, and vice versa (right part of Fig. 8). Then, without loss of generality, we can restrict Alice's operation C A to be of the form C A (x, d AB ), i.e., parametrized by x from the optimizer, and the sequence of bits d AB communicated from Bob to Alice. Similarly, for Bob we have C B (y, d BA ). Note that d AB can contain information about y and d BA about x, so we have not ...
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... operation can be written as a combination of operations that only acts on the bits of A or B separately, and operations that act on both sets of bits. The latter can be further modeled as operations on the bits belonging to A, followed by communication to B and then operations on just on the bits belonging to B, and vice versa (right part of Fig. 8). Then, without loss of generality, we can restrict C A to be of the form C A (x, d AB ), i.e., parametrized by x from the optimizer, and the sequence of bits d AB communicated from B to A. Similarly, for B we have C B (y, d BA ). Note that d AB can contain information about y and d BA about x, so we have not restricted the form of the ...
Citations
... For this, we assume, that the joint distribution of a target MN is learnable by both an arbitrary classical model and a QCMRF model to a given precision ε and analyze the complexity of sampling the trained models. Previous works [13,53] presented similar arguments relying on the results of Refs. [7,54]. ...
Leveraging the intrinsic probabilistic nature of quantum systems, generative quantum machine learning (QML) offers the potential to outperform classical learning models. Current generative QML algorithms mostly rely on general-purpose models that, while being very expressive, face several training challenges. A potential way to address these setbacks involves constructing problem-informed models capable of more efficient training on structured problems. In particular, probabilistic graphical models provide a flexible framework for representing structure in generative learning problems and can thus be exploited to incorporate inductive bias in QML algorithms. In this work, we propose a problem-informed quantum circuit Born machine Ansatz for learning the joint probability distribution of random variables, with independence relations efficiently represented by a Markov network (MN). We further demonstrate the applicability of the MN framework in constructing generative learning benchmarks and compare our model's performance to previous designs, showing it outperforms problem-agnostic circuits. Based on a preliminary analysis of trainability, we narrow down the class of MNs to those exhibiting favorable trainability properties. Finally, we discuss the potential of our model to offer quantum advantage in the context of generative learning.
... Another application of current trapped-ion quantum computers is quantum machine learning (QML). Especially in the field of generative models, quantum versions of the neural networks can lead to an exponential advantage, such as the learning of joint probability distributions using copulas, which can be directly mapped to multipartite maximally entangled states [149]. In [149], a quantum generative adversarial network and a quantum circuit Born machine are being used for said task. ...
... Especially in the field of generative models, quantum versions of the neural networks can lead to an exponential advantage, such as the learning of joint probability distributions using copulas, which can be directly mapped to multipartite maximally entangled states [149]. In [149], a quantum generative adversarial network and a quantum circuit Born machine are being used for said task. In [150], these methods are then applied to model risk aggregation of three-and four-dimensional datasets. ...
This is the second paper in a series of papers providing an overview of different quantum computing hardware platforms from an industrial end-user perspective. It follows our first paper on neutral-atom quantum computing. In the present paper, we provide a survey on the current state-of-the-art in trapped-ion quantum computing, taking up again the perspective of an industrial end-user. To this end, our paper covers, on the one hand, a comprehensive introduction to the physical foundations and mechanisms that play an important role in operating a trapped-ion quantum computer. On the other hand, we provide an overview of the key performance metrics that best describe and characterise such a device's current computing capability. These metrics encompass performance indicators such as qubit numbers, gate times and errors, native gate sets, qubit stability and scalability as well as considerations regarding the general qubit types and trap architectures. In order to ensure that these metrics reflect the current state of trapped-ion quantum computing as accurate as possible, they have been obtained by both an extensive review of recent literature and, more importantly, from discussions with various quantum hardware vendors in the field. We combine these factors and provide - again from an industrial end-user perspective - an overview of what is currently possible with trapped-ion quantum computers, which algorithms and problems are especially suitable for this platform, what are the relevant end-to-end wall clock times for calculations, and what might be possible with future fault-tolerant trapped-ion quantum computers.
... In this regard, trapped-ion quantum processors seem to offer some advantages, thanks to long coherence time, all-to-all connectivity, and high-fidelity gate operations (Bruzewicz et al. 2019). Previous studies have demonstrated the implementation of different NISQ algorithms on trapped-ion quantum computers; in particular, researchers have recently used the IonQ Harmony quantum processor and reported interesting results in quantum machine learning (Johri et al. 2021;Ishiyama et al. 2022;Rudolph et al. 2022), finance (Zhu et al. 2022), quantum chemistry (Nam et al. 2020;Zhao et al. 2023), and the generation of pseudo-random quantum state (Cenedese et al. 2023). A recent study has shown the feasibility of implementing QSVM with a simple quantum circuit on a trapped-ion quantum computer (Moradi et al. 2022); nonetheless, further investigation is necessary to understand the full potential of the quantum kernel method on this platform using a different quantum kernel and various datasets. ...
Quantum machine learning is a rapidly growing field at the intersection of quantum computing and machine learning. In this work, we examine our quantum machine learning models, which are based on quantum support vector classification (QSVC) and quantum support vector regression (QSVR). We investigate these models using a quantum circuit simulator, both with and without noise, as well as the IonQ Harmony quantum processor. For the QSVC tasks, we use a dataset containing fraudulent credit card transactions and image datasets (the MNIST and the Fashion-MNIST datasets); for the QSVR tasks, we use a financial dataset and a materials dataset. For the classification tasks, the performance of our QSVC models using 4 qubits of the trapped-ion quantum computer was comparable to that obtained from noiseless quantum circuit simulations. The result is consistent with the analysis of our device noise simulations with varying qubit gate error rates. For the regression tasks, applying a low-rank approximation to the noisy quantum kernel, in combination with hyperparameter tuning in ε-SVR, improved the performance of the QSVR models on the near-term quantum device. The alignment, as measured by the Frobenius inner product between the noiseless and noisy quantum kernels, can serve as an indicator of the relative prediction performance on noisy quantum devices in comparison with their ideal counterparts. Our results suggest that the quantum kernel, as described by our shallow quantum circuit, can be effectively used for both QSVC and QSVR tasks, indicating its resistance to noise and its adaptability to various datasets.
... Until recently, most work in quantum algorithms was theoretical, because these algorithms could not be run on quantum hardware. This has changed rapidly, and in the 2020's, new results from quantum computers in artificial intelligence are being published almost every month, with applications including probabilistic reasoning [55], financial modeling [56], and image classification [57,58]. Similar questions arise across these areas, partly because the availability of some of the key quantum properties like indeterminacy, interference, and entanglement pose the question "How can these mathematical properties help to model a given situation?" ...
This paper describes experiments showing that some tasks in natural language processing (NLP) can already be performed using quantum computers, though so far only with small datasets. We demonstrate various approaches to topic classification. The first uses an explicit word-based approach, in which word-topic weights are implemented as fractional rotations of individual qubits, and a phrase is classified based on the accumulation of these weights onto a scoring qubit, using entangling quantum gates. This is compared with more scalable quantum encodings of word embedding vectors, which are used to compute kernel values in a quantum support vector machine: this approach achieved an average of 62% accuracy on classification tasks involving over 10000 words, which is the largest such quantum computing experiment to date. We describe a quantum probability approach to bigram modeling that can be applied to understand sequences of words and formal concepts, investigate a generative approximation to these distributions using a quantum circuit Born machine, and introduce an approach to ambiguity resolution in verb-noun composition using single-qubit rotations for simple nouns and 2-qubit entangling gates for simple verbs. The smaller systems presented have been run successfully on physical quantum computers, and the larger ones have been simulated. We show that statistically meaningful results can be obtained, but the quality of individual results varies much more using real datasets than using artificial language examples from previous quantum NLP research. Related NLP research is compared, partly with respect to contemporary challenges including informal language, fluency, and truthfulness.
... In contrast, we present a data driven quantum method in this paper which does not make assumptions about the parametric forms of the dependence structure, and thus has a higher degree of modeling flexibility. Following the machine learning approaches to copulas, it has been demonstrated that a generative learning algorithm on trapped ion quantum computers for up to 8 qubits outperformed equivalent classical generative learning models with the same number of parameters in terms of the Kolmogorov-Smirnov (KS) test 16 . In that work, a Quantum Generative Adversarial Network (QGAN) and a Quantum Circuit Born Machine (QCBM) were trained to generate samples from joint distributions of historical returns of two individual stocks from the technology sector. ...
... The previous work 16 proposed a parametric quantum circuit ansatz that prepares a quantum state corresponding to a discretized copula distribution. This ansatz was used to model the correlation between two variables by optimizing the circuit parameters based on the dataset consisting of the returns of two individual stocks. ...
... Figure 4a,b show two examples of training with hybrid quantum-classical optimizations involving 3 (DJI, VIX and N225) and 4 (DJI, VIX, N225 and RUT) variables, each with 2 qubits per variable. In both cases, the training on both the simulator and hardware converges, indicating that the training is practically scalable to higher than 2 dimensions studied in 16 . In Fig. 4b, due to noise in the hardware, the experiment is unable to converge to as low of a minimum as the simulator. ...
Copulas are mathematical tools for modeling joint probability distributions. In the past 60 years they have become an essential analysis tool on classical computers in various fields. The recent finding that copulas can be expressed as maximally entangled quantum states has revealed a promising approach to practical quantum advantages: performing tasks faster, requiring less memory, or, as we show, yielding better predictions. Studying the scalability of this quantum approach as both the precision and the number of modeled variables increase is crucial for its adoption in real-world applications. In this paper, we successfully apply a Quantum Circuit Born Machine (QCBM) based approach to modeling 3- and 4-variable copulas on trapped ion quantum computers. We study the training of QCBMs with different levels of precision and circuit design on a simulator and a state-of-the-art trapped ion quantum computer. We observe decreased training efficacy due to the increased complexity in parameter optimization as the models scale up. To address this challenge, we introduce an annealing-inspired strategy that dramatically improves the training results. In our end-to-end tests, various configurations of the quantum models make a comparable or better prediction in risk aggregation tasks than the standard classical models.
... QGANs combine the power of quantum mechanics with the ingenuity of generative adversarial networks [5] and are characterized by their potential for generating quantum data in a wide range of fields, from real-world classical distribution loading [6-8] and image generation [9] to drug discovery [10]. In terms of experiments, several QGANs capable of generating high-quality quantum states have been demonstrated in superconducting circuits [9,[11][12][13][14][15] and trapped-ion processors [16]. These previous results are impressive; however, whether QGANs can work normally on NISQ-era devices, which are ubiquitously plagued by noise and defects [17][18][19][20][21], is in urgent need of further investigation. ...
Quantum generative adversarial networks (QGANs), an intersection of quantum computing and machine learning, have attracted widespread attention due to their potential advantages over classical analogs. However, in the current era of noisy intermediate-scale quantum (NISQ) computing, it is essential to investigate whether QGANs can perform learning tasks on near-term quantum devices usually affected by noise and even defects. In this Letter, using a programmable silicon quantum photonic chip, we experimentally demonstrate the QGAN model in photonics for the first time to our knowledge and investigate the effects of noise and defects on its performance. Our results show that QGANs can generate high-quality quantum data with a fidelity higher than 90%, even under conditions where up to half of the generator’s phase shifters are damaged, or all of the generator and discriminator’s phase shifters are subjected to phase noise up to 0.04π. Our work sheds light on the feasibility of implementing QGANs on the NISQ-era quantum hardware.
... In this regard, trapped-ion quantum processors seem to offer some advantages, thanks to long coherence time, all-to-all connectivity, and highfidelity gate operations (Bruzewicz et al. 2019). Previous studies have demonstrated the implementation of different NISQ algorithms on trapped-ion quantum computers; in particular, researchers have recently used the IonQ Harmony quantum processor and reported interesting results in quantum machine learning (Johri et al. 2021;Ishiyama et al. 2022;Rudolph et al. 2022), finance (Zhu et al. 2022), quantum chemistry (Nam et al. 2020;Zhao et al. 2023), and the generation of pseudo-random quantum state (Cenedese et al. 2023). A recent study has shown the feasibility of implementing QSVM with a simple quantum circuit on a trapped-ion quantum computer (Moradi et al. 2022); nonetheless, further investigation is necessary to understand the full potential of the quantum kernel method on this platform using a different quantum kernel and various datasets. ...
Quantum machine learning is a rapidly growing field at the intersection of quantum computing and machine learning. In this work, we examine our quantum machine learning models, which are based on quantum support vector classification (QSVC) and quantum support vector regression (QSVR). We investigate these models using a quantum-circuit simulator, both with and without noise, as well as the IonQ Harmony quantum processor. For the QSVC tasks, we use a dataset containing fraudulent credit card transactions and image datasets (the MNIST and the Fashion-MNIST datasets); for the QSVR tasks, we use a financial dataset and a materials dataset. For the classification tasks, the performance of our QSVC models using 4 qubits of the trapped-ion quantum computer was comparable to that obtained from noiseless quantum-circuit simulations. The result is consistent with the analysis of our device-noise simulations with varying qubit-gate error rates. For the regression tasks, applying a low-rank approximation to the noisy quantum kernel, in combination with hyperparameter tuning in ε-SVR, improved the performance of the QSVR models on the near-term quantum device. Our results suggest that the quantum kernel, as described by our shallow quantum circuit, can be effectively used for both QSVC and QSVR tasks, indicating its resistance to noise and its adaptability to various datasets.
... qGANs are relatively very new and active research area, especially in financial domain. Research in qGANs are being done in several areas in finance, such as, to generate probability distributions for univariate distributions [22,23] and for multivariate variate distributions [24,25 QIF]. A classical GAN process. ...
Quantum machine learning (QML) is a cross-disciplinary subject made up of two of the most exciting research areas: quantum computing and classical machine learning (ML), with ML and artificial intelligence (AI) being projected as the first fields that will be impacted by the rise of quantum machines. Quantum computers are being used today in drug discovery, material & molecular modelling and finance. In this work, we discuss some upcoming active new research areas in application of quantum machine learning (QML) in finance. We discuss certain QML models that has become areas of active interest in the financial world for various applications. We use real world financial dataset and compare models such as qGAN (quantum generative adversarial networks) and QCBM (quantum circuit Born machine) among others, using simulated environments. For the qGAN, we define quantum circuits for discriminators and generators and show promises of future quantum advantage via QML in finance.
... Scaling the quantum circuits by varying the number of qubits and circuit depth enables adjusting the workload. The copula architecture is optimized to learn the probability distribution whose cumulative marginals are uniformly distributed like the image of the PIT [50]. Figure 6 depicts the quantum circuits. ...
Benchmarking of quantum machine learning (QML) algorithms is challenging due to the complexity and variability of QML systems, e.g., regarding model ansatzes, data sets, training techniques, and hyper-parameters selection. The QUantum computing Application benchmaRK (QUARK) framework simplifies and standardizes benchmarking studies for quantum computing applications. Here, we propose several extensions of QUARK to include the ability to evaluate the training and deployment of quantum generative models. We describe the updated software architecture and illustrate its flexibility through several example applications: (1) We trained different quantum generative models using several circuit ansatzes, data sets, and data transformations. (2) We evaluated our models on GPU and real quantum hardware. (3) We assessed the generalization capabilities of our generative models using a broad set of metrics that capture, e.g., the novelty and validity of the generated data.
... In general, there exist other elliptical and Archimedean copulas that can model the tail-risk correlations more precisely and come in various functional forms (e.g. Ali-Mikhail-Haq, Clayton, Gumbel, Independence, Joe [69][70][71][72][73][74]). However, they all suffer from the curse of dimensionality and thus, computationally, there is not a clear winner. ...
Noisy intermediate-scale quantum (NISQ) devices are valuable platforms for testing the tenets of quantum computing, but these devices are susceptible to errors arising from de-coherence, leakage, cross-talk and other sources of noise. This raises concerns for ensuring the stability of program results when using NISQ devices as strategies for mitigating errors generally require well-characterized and reliable error models. Here, we quantify the reliability of NISQ devices by assessing the necessary conditions for generating stable results within a given tolerance. We use similarity metrics derived from device characterization data to analyze the stability of performance across several key features: gate fidelities, de-coherence time, SPAM error, and cross-talk error. We bound the behavior of these metrics derived from their joint probability distribution, and we validate these bounds using numerical simulations of the Bernstein-Vazirani circuit tested on a superconducting transmon device. Our results enable the rigorous testing of reliability in NISQ devices and support the long-term goals of stable quantum computing.
