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A Relational Time-Symmetric Framework for Analyzing the Quantum Computational Speedup

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Abstract

The usual representation of quantum algorithms is limited to the process of solving the problem. We extend it to the process of setting the problem. Bob, the problem setter, selects a problem-setting by the initial measurement. Alice, the problem solver, unitarily computes the corresponding solution and reads it by the final measurement. This simple extension creates a new perspective from which to see the quantum algorithm. First, it highlights the relevance of time-symmetric quantum mechanics to quantum computation: the problem-setting and problem solution, in their quantum version, constitute pre- and post-selection, hence the process as a whole is bound to be affected by both boundary conditions. Second, it forces us to enter into relational quantum mechanics. There must be a representation of the quantum algorithm with respect to Bob, and another one with respect to Alice, from whom the outcome of the initial measurement, specifying the setting and thus the solution of the problem, must be concealed. Time-symmetrizing the quantum algorithm to take into account both boundary conditions leaves the representation to Bob unaltered. It shows that the representation to Alice is a sum over histories in each of which she remains shielded from the information coming to her from the initial measurement, not from that coming to her backwards in time from the final measurement. In retrospect, all is as if she knew in advance, before performing her problem-solving action, half of the information that specifies the solution of the problem she will read in the future and could use this information to reach the solution with fewer computation steps (oracle queries). This elucidates the quantum computational speedup in all the quantum algorithms examined.

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Notes

  1. We use the term classical algorithm instead of just algorithm, i.e. Turing machine, to avoid confusion with the term quantum algorithm. However, with this, we do not intend to make any comparison between quantum and classical physics. Here it is quantum physics and classical logic that face each other.

  2. By the way, being from \(\left\{ 0 ,1\right\} ^{n} \rightarrow \left\{ 0 ,1\right\} ^{n -1}\), the functions in question have just two periods.

  3. It is anyhow almost optimal. in fact a \({\hbox {O}} \left( n\right) \) increase in the number of function evaluations does not affect the exponential character of the speedup.

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Acknowledgements

We wish to thank Yakir Aharonov and David Ritz Finkelstein for many helpful discussions.

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Authors and Affiliations

  1. Elsag Bailey ICT Division and Quantum Information Laboratory, Via Puccini 2, 16154, Genova, Italy

    G. Castagnoli

  2. Faculty of Engineering and the Institute of Nanotechnology and Advanced Materials, Bar Ilan University, 5290002, Ramat Gan, Israel

    E. Cohen

  3. Mathematical Institute, University of Oxford, Oxford, UK

    A. K. Ekert

  4. Centre for Quantum Technologies, National University of Singapore, Singapore, Singapore

    A. K. Ekert

  5. Institute for Quantum Studies, Chapman University, Orange, CA, 92866, USA

    A. C. Elitzur

  6. Iyar, The Israeli Institute for Advanced Research, POB 651, Zichron Ya’akov, 3095303, Israel

    A. C. Elitzur

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  1. G. Castagnoli
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  2. E. Cohen
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  3. A. K. Ekert
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  4. A. C. Elitzur
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Correspondence to G. Castagnoli.

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Castagnoli, G., Cohen, E., Ekert, A.K. et al. A Relational Time-Symmetric Framework for Analyzing the Quantum Computational Speedup. Found Phys 49, 1200–1230 (2019). https://doi.org/10.1007/s10701-019-00300-z

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