Generative Modelling has become a promising use case for near term quantum computers. In particular, due to the fundamentally probabilistic nature of quantum mechanics, quantum computers naturally model and learn probability distributions, perhaps more efficiently than can be achieved classically. The Born machine is an example of such a model, easily implemented on near term quantum computers. However, in its original form, the Born machine only naturally represents discrete distributions. Since probability distributions of a continuous nature are commonplace in the world, it is essential to have a model which can efficiently represent them. Some proposals have been made in the literature to supplement the discrete Born machine with extra features to more easily learn continuous distributions, however, all invariably increase the resources required to some extent. In this work, we present the continuous variable Born machine, built on the alternative architecture of continuous variable quantum computing, which is much more suitable for modelling such distributions in a resource-minimal way. We provide numerical results indicating the models ability to learn both quantum and classical continuous distributions, including in the presence of noise.
Rapid improvement in quantum hardware has opened the door to complex problems, but the precise characterization of quantum systems itself remains a challenge. To address this obstacle, novel tomography schemes have been developed that employ generative machine learning models, enabling quantum state reconstruction from limited classical data. In particular, quantum-inspired Born machines provide a natural way to encode measured data into a model of a quantum state. Born machines have shown great success in learning from classical data; however, the full potential of a Born machine in learning …
Quantum phase transitions are ubiquitous in quantum many body systems. The quantum fluctuations that occur at very low temperatures are known to be responsible for driving the system across different phases as a function of an external control parameter. The XY Hamiltonian with a transverse field is a basic model that manifests two distinct quantum phase transitions, including spontaneous $Z_2$ symmetry breaking from an ordered to a disordered state. While programmable quantum devices have shown great success in investigating the various exotic quantum phases of matter, in parallel, the quest …
Near-term noisy intermediate-scale quantum circuits can efficiently implement implicit probabilistic models in discrete spaces, supporting distributions that are practically infeasible to sample from using classical means. One of the possible applications of such models, also known as Born machines, is probabilistic inference, which is at the core of Bayesian methods. This paper studies the use of Born machines for the problem of training binary Bayesian neural networks. In the proposed approach, a Born machine is used to model the variational distribution of the binary weights of the neural n…
We revise contextual probability theory and the associated inverse Born problem, solved by the Quantum-Like Representation Algorithm (QLRA) in case of two discrete random variables. After pointing out the special feature and limitations of QLRA for two binary random variables, we generalize the QLRA procedure to solve the inverse Born problem to the case of three binary random variables. We analyze the quantum spin model in the light of our results and showed that can be easily obtained from a contextual probability model. We furthermore study the inverse Born problem in the case of two contin…
The argument of environment-assisted invariance (known as envariance) implying Born's rule is widely used in models for quantum measurement to reason that they must yield the correct statistics, specifically for linear models. However, it has recently been shown that linear collapse models can never give rise to Born's rule. Here, we address this apparent contradiction and point out an inconsistency in the assumptions underlying the arguments based on envariance. We use a construction in which the role of the measurement machine is made explicit and show that the presence of envariance does no…
European Organization for Nuclear Research (CERN)（欧洲核子研究组织）；University of Geneva（日内瓦大学）；Technion（以色列理工学院）
Generative modeling is a promising task for near-term quantum devices, which can use the stochastic nature of quantum measurements as a random source. So called Born machines are purely quantum models and promise to generate probability distributions in a quantum way, inaccessible to classical computers. This paper presents an application of Born machines to Monte Carlo simulations and extends their reach to multivariate and conditional distributions. Models are run on (noisy) simulators and IBM Quantum superconducting quantum hardware. More specifically, Born machines are used to generate muo…
We combine the formalisms of Floquet theory and full counting statistics with a Markovian embedding strategy to access the dynamics and thermodynamics of a periodically driven thermal machine beyond the conventional Born-Markov approximation. The working medium is a two-level system and we drive the tunneling as well as the coupling to one bath with the same period. We identify four different operating regimes of our machine which include a heat engine and a refrigerator. As the coupling strength with one bath is increased, the refrigerator regime disappears, the heat engine regime narrows and…
We compare and contrast the statistical physics and quantum physics inspired approaches for unsupervised generative modeling of classical data. The two approaches represent probabilities of observed data using energy-based models and quantum states respectively.Classical and quantum information patterns of the target datasets therefore provide principled guidelines for structural design and learning in these two approaches. Taking the restricted Boltzmann machines (RBM) as an example, we analyze the information theoretical bounds of the two approaches. We verify our reasonings by comparing the…
University of Edinburgh（爱丁堡大学）；CNRS（法国国家科学研究中心）；École Normale Supérieure（巴黎高等师范学院）；Sorbonne Université（索邦大学）
The search for an application of near-term quantum devices is widespread. Quantum Machine Learning is touted as a potential utilisation of such devices, particularly those which are out of the reach of the simulation capabilities of classical computers. In this work, we propose a generative Quantum Machine Learning Model, called the Ising Born Machine (IBM), which we show cannot, in the worst case, and up to suitable notions of error, be simulated efficiently by a classical device. We also show this holds for all the circuit families encountered during training. In particular, we explore quant…
Sequential learning on a Tensor Network Born machine with Trainable Token Embedding
Department of Physics, University of California, San Diego, CA 92093, USA（加州大学圣地亚哥分校物理系）；School of Physics, Zhejiang University, Hangzhou 310027, China（浙江大学物理学院）；Department of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA（佐治亚理工学院物理系）
Generative modeling with Gaussian Boson Sampling: classically trainable Bosonic Born Machines
HUN-REN Wigner Research Centre for Physics（匈牙利科学院维格纳研究物理中心）；Eötvös Loránd University（罗兰大学）；Center for Quantum Enabled-Computing, Center for Theoretical Physics of the Polish Academy of Sciences（波兰科学院理论物理中心量子计算中心）；Department of Physics, Korea Advanced Institute of Science and Technology（韩国科学技术院物理系）；University of Helsinki（赫尔辛基大学）；Algorithmiq Ltd（算法有限公司）
Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所）；CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences（中国科学院大学拓扑量子计算卓越创新中心）
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 gen…
Finding a concrete use case for quantum computers in the near term is still an open question, with machine learning typically touted as one of the first fields which will be impacted by quantum technologies. In this work, we investigate and compare the capabilities of quantum versus classical models for the task of generative modelling in machine learning. We use a real world financial dataset consisting of correlated currency pairs and compare two models in their ability to learn the resulting distribution - a restricted Boltzmann machine, and a quantum circuit Born machine. We provide extens…
Everett's Relative State Interpretation has gained increasing interest due to the progress of understanding the role of decoherence. In order to fulfill its promise as a realistic description of the physical world, two postulates are formulated. In short they are 1) for a system with continuous coordinates $\vec{x}$, discrete variable $j$, and state $ψ_j(\vec{x})$, the density $ρ_j(\vec{x})=|ψ_j(\vec{x})|^2$ gives the distribution of the location of the system with the respect to the variables $\vec{x}$ and $j$; 2) an equation of motion for the state $i\hbar \partial_t ψ= Hψ$. The first postul…
The quantum circuit Born machine (QCBM) is a quantum physics inspired implicit generative model naturally suitable for learning binary images, with a potential advantage of modeling discrete distributions that are hard to simulate classically. As data samples are generated quantum-mechanically, QCBMs encompass a unique optimization landscape. However, pioneering works on QCBMs do not consider the practical scenario where only small batch sizes are allowed during training. QCBMs trained with a statistical two-sample test objective in the image space require large amounts of projective measureme…
QuKAN: A Quantum Circuit Born Machine approach to Quantum Kolmogorov Arnold Networks
Department of Computer Science and Research Initiative QC-AI, RPTU Kaiserslautern-Landau（计算机科学系和QC-AI研究计划，鲁尔-帕绍大学（RPTU）凯斯贝赫-劳恩道分校）；Embedded Intelligence, German Research Center for Artificial Intelligence (DFKI)（嵌入式智能，德国人工智能研究中心（DFKI））；Department of Physics, RPTU Kaiserslautern-Landau（物理系，鲁尔-帕绍大学（RPTU）凯斯贝赫-劳恩道分校）
本研究利用量子电路 Born 机（QCBM）在混合与全量子形式下实现了 KAN 架构，通过预训练残差函数发挥参数化量子电路的表示能力，验证了 QuKAN 架构的可行性、可解释性与性能。
Cambridge Quantum Computing Limited（剑桥量子计算有限公司）；Imperial College London（伦敦帝国理工学院）；University of Edinburgh（爱丁堡大学）
Generative modelling is an important unsupervised task in machine learning. In this work, we study a hybrid quantum-classical approach to this task, based on the use of a quantum circuit Born machine. In particular, we consider training a quantum circuit Born machine using $f$-divergences. We first discuss the adversarial framework for generative modelling, which enables the estimation of any $f$-divergence in the near term. Based on this capability, we introduce two heuristics which demonstrably improve the training of the Born machine. The first is based on $f$-divergence switching during tr…
Quantum gravitational effects suggest a minimal length, or spacetime interval, of order the Planck length. This in turn suggests that Hilbert space itself may be discrete rather than continuous. One implication is that quantum states with norm below some very small threshold do not exist. The exclusion of what Everett referred to as maverick branches is necessary for the emergence of the Born Rule in no collapse quantum mechanics. We discuss this in the context of quantum gravity, showing that discrete models (such as simplicial or lattice quantum gravity) indeed suggest a discrete Hilbert spa…
Setting the minimal-time bound for a quantum system to evolve between two distinguishable states, the quantum speed limit (QSL) characterizes the latent capability in speeding up of the system. It has found applications in determining the quantum superiority in many quantum technologies. However, previous results showed that such a speedup capability is generally destroyed by the environment induced decoherence in the Born-Markovian approximate dynamics. We here propose a scheme to recover the speedup capability in a dissipative continuous-variable system within the exact non-Markovian framewo…
Due to the linearity of quantum mechanics, it remains a challenge to design quantum generative machine learning models that embed non-linear activations into the evolution of the statevector. However, some of the most successful classical generative models, such as those based on neural networks, involve highly non-linear dynamics for quality training. In this paper, we explore the effect of these dynamics in quantum generative modeling by introducing a model that adds non-linear activations via a neural network structure onto the standard Born Machine framework - the Quantum Neuron Born Machi…
Massachusetts Institute of Technology（麻省理工学院）；IBM Quantum, MIT-IBM Watson AI Lab（IBM量子，MIT-IBM沃森人工智能实验室）；Harvard University（哈佛大学）；IBM Quantum, IBM T.J. Watson Research Center（IBM量子，IBM T.J.沃森研究中心）
In this treatise I introduce the time dependent Generalized Born's Rule for the probabilities of quantum events, including conditional and consecutive probabilities, as the unique fundamental time evolution equation of quantum theory. Then these probabilities, computed from states and events, are to be compared with relative frequencies of observations. Schrodinger's equation still is valid in one model of the axioms of quantum theory, which I call the Schrodinger model. However, the role of Schrodinger's equation is auxiliary, since it serves to help compute the continuous temporal evolution …
Unsupervised training of generative models is a machine learning task that has many applications in scientific computing. In this work we evaluate the efficacy of using quantum circuit-based generative models to generate synthetic data of high energy physics processes. We use non-adversarial, gradient-based training of quantum circuit Born machines to generate joint distributions over 2 and 3 variables.
In this work we introduce a novel approach to the pulsar classification problem in time-domain radio astronomy using a Born machine, often referred to as a quantum neural network. Using a single-qubit architecture, we show that the pulsar classification problem maps well to the Bloch sphere and that comparable accuracies to more classical machine learning approaches are achievable. We introduce a novel single-qubit encoding for the pulsar data used in this work and show that this performs comparably to a multi-qubit QAOA encoding.
Annealing machines specialized for combinatorial optimization problems have been developed, and some companies offer services to use those machines. Such specialized machines can only handle binary variables, and their input format is the quadratic unconstrained binary optimization (QUBO) formulation. Therefore, discretization is necessary to solve problems with continuous variables. However, there is a severe constraint on the number of binary variables with such machines. Although the simple binary expansion in the previous research requires many binary variables, we need to reduce the numbe…
Projective measurement is used as a fundamental axiom in quantum mechanics, even though it is discontinuous and cannot predict which measured operator eigenstate will be observed in which experimental run. The probabilistic Born rule gives it an ensemble interpretation, predicting proportions of various outcomes over many experimental runs. Understanding gradual weak measurements requires replacing this scenario with a dynamical evolution equation for the collapse of the quantum state in individual experimental runs. We revisit the quantum trajectory framework that models quantum measurement a…
