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Discrete stochastic processes (DSP) are instrumental for modelling the dynamics of probabilistic systems and have a wide spectrum of applications in science and engineering. DSPs are usually analyzed via Monte Carlo methods since the number of realizations increases exponentially with the number of time steps, and importance sampling is often required to reduce the variance. We propose a quantum algorithm for calculating the characteristic function of a DSP, which completely defines its probability distribution, using the number of quantum circuit elements that grows only linearly with the number of time steps. The quantum algorithm takes all stochastic trajectories into account and hence eliminates the need of importance sampling. The algorithm can be further furnished with the quantum amplitude estimation algorithm to provide quadratic speed-up in sampling. Both of these strategies improve variance beyond classical capabilities. The quantum method can be combined with Fourier approximation to estimate an expectation value of any integrable function of the random variable. Applications in finance and correlated random walks are presented to exemplify the usefulness of our results. Proof-of-principle experiments are performed using the IBM quantum cloud platform.
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We explore the connections between the theories of stochastic analysis and discrete quantum mechanical systems. Naturally these connections include the Feynman-Kac formula, and the Cameron-Martin-Girsanov theorem. More precisely, the notion of the quantum canonical transformation is employed for computing the time propagator, in the case of generic dynamical diffusion coefficients. Explicit computation of the path integral leads to a universal expression for the associated measure regardless of the form of the diffusion coefficient and the drift. This computation also reveals that the drift plays the role of a super potential in the usual super-symmetric quantum mechanics sense. Some simple illustrative examples such as the Ornstein-Uhlenbeck process and the multidimensional Black-Scholes model are also discussed. Basic examples of quantum integrable systems such as the quantum discrete non-linear hierarchy (DNLS) and the XXZ spin chain are presented providing specific connections between quantum (integrable) systems and stochastic differential equations (SDEs). The continuum limits of the SDEs for the first two members of the NLS hierarchy turn out to be the stochastic transport and the stochastic heat equations respectively. The quantum Darboux matrix for the discrete NLS is also computed as a defect matrix and the relevant SDEs are derived.
In this paper, we consider a spectral analysis of discrete time quantum walks on the path. For isospectral coin cases, we show that the time averaged distribution and stationary distributions of the quantum walks are described by the pair of eigenvalues of the coins as well as the eigenvalues and eigenvectors of the corresponding random walks which are usually referred as the birth and death chains. As an example of the results, we derive the time averaged distribution of so-called Szegedys walk which is related to the Ehrenfest model. It is represented by Krawtchouk polynomials which is the eigenvectors of the model and includes the arcsine law.
This brief article gives an overview of quantum mechanics as a {em quantum probability theory}. It begins with a review of the basic operator-algebraic elements that connect probability theory with quantum probability theory. Then quantum stochastic processes is formulated as a generalization of stochastic processes within the framework of quantum probability theory. Quantum Markov models from quantum optics are used to explicitly illustrate the underlying abstract concepts and their connections to the quantum regression theorem from quantum optics.
Error mitigation has been one of the recently sought after methods to reduce the effects of noise when computation is performed on a noisy near-term quantum computer. Interest in simulating stochastic processes with quantum models gained popularity after being proven to require less memory than their classical counterparts. With previous work on quantum models focusing primarily on further compressing memory, this work branches out into the experimental scene; we aim to bridge the gap between theoretical quantum models and practical use with the inclusion of error mitigation methods. It is observed that error mitigation is successful in improving the resultant expectation values. While our results indicate that error mitigation work, we show that its methodology is ultimately constrained by hardware limitations in these quantum computers.
Adaptive filtering is a powerful class of control theoretic concepts useful in extracting information from noisy data sets or performing forward prediction in time for a dynamic system. The broad utilization of the associated algorithms makes them attractive targets for similar problems in the quantum domain. To date, however, the construction of adaptive filters for quantum systems has typically been carried out in terms of stochastic differential equations for weak, continuous quantum measurements, as used in linear quantum systems such as optical cavities. Discretized measurement models are not as easily treated in this framework, but are frequently employed in quantum information systems leveraging projective measurements. This paper presents a detailed analysis of several technical innovations that enable classical filtering of discrete projective measurements, useful for adaptively learning system-dynamics, noise properties, or hardware performance variations in classically correlated measurement data from quantum devices. In previous work we studied a specific case of this framework, in which noise and calibration errors on qubit arrays could be efficiently characterized in space; here, we present a generalized analysis of filtering in quantum systems and demonstrate that the traditional convergence properties of nonlinear classical filtering hold using single-shot projective measurements. These results are important early demonstrations indicating that a range of concepts and techniques from classical nonlinear filtering theory may be applied to the characterization of quantum systems involving discretized projective measurements, paving the way for broader adoption of control theoretic techniques in quantum technology.
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