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The \(\langle \)Im|Possibility\(\rangle \) of Quantum Annealing for Maximum Likelihood Estimation

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Credible Asset Allocation, Optimal Transport Methods, and Related Topics (TES 2022)

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 429))

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Abstract

This paper provides a casual exploration of quantum ideas and quantum computing to solve maximum likelihood problems common in statistical and econometric estimations. A brief description of quantum ideas, and in particular, the quantum bit and its properties, is followed by how quantum annealing, specifically conducted by the D-Wave quantum computer, takes advantage of “quantum tunneling” to solve optimization problems that may typically be handled by simulated annealing. Although there is now established theoretical grounds in favor of quantum annealing over simulated annealing, there remain several limitations of its usefulness and useability in MLE, namely, improving its accuracy by formulating the problem more fully for quantum computers and/or reducing hardware noise which may mask the true solution. Currently, the process of embedding quantum maximum likelihood into the quantum machine reduces the scale of problems available and as of now non-artificial problems in which quantum annealing outperforms simulated annealing are hard to find.

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Notes

  1. 1.

    There is a lot of work from these pioneers, but perhaps see Stapp [35], Byrne [5], Rovelli [31], for a small glimpse into their ideas.

  2. 2.

    See for example Jacquier et al. [17], Byshkin et al. [6], Crosson and Harrow [7], and others. Even apparently simple MLE problem for a 3-parameter Weibull distribution can pose serious challenges (see for example Yang et al. [40]).

  3. 3.

    The D-Wave quantum computer is a rather controversial machine within the quantum computing community, with even some questioning how “quantum” the machine really is [34]. Johnson et al. [18] compared the results of quantum annealing using the D-Wave to classical techniques and showed that the D-Wave exhibited results that could only have occurred with quantum tunneling.

  4. 4.

    Nguyen and Dong [28] provides a more technical discussion on quantum properties and probability calculus.

  5. 5.

    Scientists only recently photographed entangled particle in July 2019.

  6. 6.

    The original paper discussing the Ising model for quantum annealing is Bian et al. [2]. For more on the Ising model see Irback et al. [16], Majewski et al. [22], Stauffer [36] and McGeoch [24].

  7. 7.

    This is derived from the original quantum Hamiltonian, an operator in the Hilbert space.

  8. 8.

    We can easily extend this to negative numbers by introducing a sign qubit [4].

  9. 9.

    The original simulated annealing paper is Metropolis et al. [25].

  10. 10.

    The tunneling process cannot be directly observed nor adequately explained by classical physics and is rather explained by the Heisenberg uncertainty principle and the wave-particle duality of matter in quantum physics [32, 33].

  11. 11.

    D-Wave in 2019 announced their next generation of their quantum computing topology, Pegasus, a significant improvement to the Chimera technology.

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Authors and Affiliations

  1. Faculty of Economics, Chulalongkorn University, Bangkok, Thailand

    Yong Yoon

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  1. Yong Yoon
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  1. Faculty of Economics, Chiang Mai University, Chiang Mai, Thailand

    Songsak Sriboonchitta

  2. Department of Computer Science, University of Texas, El Paso, TX, USA

    Vladik Kreinovich

  3. Faculty of Economics, Center of Excellence in Econometrics, Chiang Mai University, Chiang Mai, Thailand

    Woraphon Yamaka

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Yoon, Y. (2022). The \(\langle \)Im|Possibility\(\rangle \) of Quantum Annealing for Maximum Likelihood Estimation. In: Sriboonchitta, S., Kreinovich, V., Yamaka, W. (eds) Credible Asset Allocation, Optimal Transport Methods, and Related Topics. TES 2022. Studies in Systems, Decision and Control, vol 429. Springer, Cham. https://doi.org/10.1007/978-3-030-97273-8_31

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