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Quantum digital-to-analog conversion algorithm using decoherence

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Abstract

We consider the problem of mapping digital data encoded on a quantum register to analog amplitudes in parallel. It is shown to be unlikely that a fully unitary polynomial-time quantum algorithm exists for this problem; NP becomes a subset of BQP if it exists. In the practical point of view, we propose a nonunitary linear-time algorithm using quantum decoherence. It tacitly uses an exponentially large physical resource, which is typically a huge number of identical molecules. Quantumness of correlation appearing in the process of the algorithm is also discussed.

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Notes

  1. It is a tacit assumption that any DAC is designed to satisfy \(V_\mathrm{LSB}\gtrapprox \varepsilon _\mathrm{th}\) [30, 39]. Thus, setting \(n\) to a value slightly larger than \(m\) should be enough to satisfy the inequality \(2^{n-1}V_\mathrm{LSB} > 2^m (\varepsilon _\mathrm{th}+c)\). Even if the tacit assumption does not hold, a value of \(n\) enough to satisfy the inequality scales linearly in \(m\) as long as \(\varepsilon _\mathrm{th}\) is a constant.

  2. Note that fetching one resultant datum takes at least one step for a human in any parallel processing model. On the other hand, resultant data are usable for subsequent parallel processing. In fact, \(\rho _2\) can be directly used for further parallel processing within the ensemble quantum computing model.

  3. Photons are not physically connected like molecular spins; a projection using e.g. polarizing beam splitters for register \(\mathrm{R}\) does not separate signals in the ancilla register.

  4. Here, we are discussing an algorithmic structure. It is, of course, a formidable challenge to implement a large quantum circuit with linear optics [31].

  5. In general, the von Neumann entropy of a density matrix \(\rho \) is calculated as \(S(\rho )=-\mathrm{Tr}\rho \log _2\rho =-\sum _\lambda \lambda \log _2\lambda \) where \(\lambda \)’s are the eigenvalues of \(\rho \).

  6. Thus, the theory of quantum discord has a focus on the measurement stage. This is in contrast to the entanglement theory, which has a focus on the state generation stage [18, 38].

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Acknowledgments

This work was supported by the Grant-in-Aid for Scientific Research from JSPS (Grant No. 25871052).

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  1. Department of Computer Science and Engineering, Toyohashi University of Technology, 1-1 Hibarigaoka, Tenpaku-cho, Toyohashi, Aichi, 441-8580, Japan

    Akira SaiToh

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Correspondence to Akira SaiToh.

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SaiToh, A. Quantum digital-to-analog conversion algorithm using decoherence. Quantum Inf Process 14, 2729–2748 (2015). https://doi.org/10.1007/s11128-015-1033-x

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