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Quantum communication protocols as a benchmark for programmable quantum computers

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Abstract

We point out that realization of quantum communication protocols in programmable quantum computers provides a deep benchmark for capabilities of real quantum hardware. Particularly, it is prospective to focus on measurements of entropy-based characteristics of the performance and to explore whether a “quantum regime” is preserved. We perform proof-of-principle implementations of superdense coding and quantum key distribution BB84 using 5- and 16-qubit superconducting quantum processors of IBM Quantum Experience. We focus on the ability of these quantum machines to provide an efficient transfer of information between distant parts of the processors by placing Alice and Bob at different qubits of the devices. We also examine the ability of quantum devices to serve as quantum memory and to store entangled states used in quantum communication. Another issue we address is an error mitigation. Although it is at odds with benchmarking, this problem is nevertheless of importance in a general context of quantum computation with noisy quantum devices. We perform such a mitigation and noticeably improve some results.

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Acknowledgements

We acknowledge use of the IBM Quantum Experience for this work. The viewpoints expressed are those of the authors and do not reflect the official policy or position of IBM or the IBM Quantum Experience team. E. O. K. was supported by RFBR (Project No. 18-37-00096). W. V. P. acknowledges a support from RFBR (Project No. 15-02-02128). Yu. E. L. acknowledges a support from RFBR (Project No. 17-02-01134) and the Program of Basic Research of HSE.

Author information

Authors and Affiliations

  1. Dukhov Research Institute of Automatics (VNIIA), Moscow, Russia, 127055

    A. A. Zhukov, A. A. Elistratov, W. V. Pogosov & Yu. E. Lozovik

  2. National Research Nuclear University (MEPhI), Moscow, Russia, 115409

    A. A. Zhukov

  3. Russian Quantum Center, Skolkovo, Moscow, Russia, 143025

    E. O. Kiktenko

  4. Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia, 119991

    E. O. Kiktenko

  5. Geoelectromagnetic Research Center of Schmidt Institute of Physics of the Earth, Russian Academy of Sciences, Troitsk, Moscow, Russia, 142190

    E. O. Kiktenko

  6. Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Moscow, Russia, 125412

    W. V. Pogosov

  7. Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, Russia, 141700

    W. V. Pogosov

  8. Institute of Spectroscopy, Russian Academy of Sciences, Moscow, Russia, 142190

    Yu. E. Lozovik

  9. Moscow Institute of Electronics and Mathematics, National Research University Higher School of Economics, Moscow, Russia, 101000

    Yu. E. Lozovik

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  1. A. A. Zhukov
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  2. E. O. Kiktenko
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  3. A. A. Elistratov
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  4. W. V. Pogosov
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Correspondence to W. V. Pogosov.

Additional information

E. O. K. was supported by RFBR (Project No. 18-37-00096). W. V. P. acknowledges a support from RFBR (Project No. 15-02-02128). Yu. E. L. acknowledges a support from RFBR (Project No. 17-02-01134) and the Program of Basic Research of HSE.

Appendices

Appendix A: Output distributions for superdense coding

Table 1 shows output distributions for the superdense coding protocol for the situation corresponding to Fig. 2a for different number of SWAPs, as obtained from 16-qubit IBMqx5 device. Here, \((a_1, a_2)\) is Alice’s input, while \((b_1,b_2)\) is Bob’s output. Results presented in the table provide output distributions in connection to the input data. In the ideal situation, the input and output must be the same, so that the corresponding matrix for each given \((a_1, a_2)\) should be identity (unit) matrix. We see from Table 1 that, in reality, even for the zero number of SWAPs this matrix is rather different from the identity matrix.

Table 1 The output distribution for superdense coding protocol for different number of SWAPs
Full size table

Table 2 presents similar data for different values of delay time, as obtained from 5-qubit IBMqx4 device. We again see noticeable deviations from the ideal distribution even for zero waiting time.

Table 2 The output distribution for superdense coding protocol for different values of time delay
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Tables 3 and 4 provide output distributions without error correction and with error correction, respectively, for different values of waiting time, as obtained from 16-qubit IBMQx5 device. We again see that the distributions are rather different from ideal ones even at \(t=0\), but the error correction, in general, indeed improves the results.

Table 3 The output distribution for superdense coding protocol for different values of time delay without any correction of coherent errors
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Table 4 The output distribution for superdense coding protocol for different values of time delay with the correction of the coherent error
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Measurements for the superdense coding protocol have been taken between April 25, 2018, and May 21, 2018.

Appendix B: Correction of the coherent error

Figure 12 shows the experimentally determined overlap (fidelity) between the prepared state and the Bell states \(|\varPsi _+ \rangle \) (blue circles) and \(|\varPsi _- \rangle \) (brown triangles) as a function of time, provided the initial target state was \(|\varPsi _+ \rangle \). Figure 12a corresponds to direct measurements, while Fig. 12b deals with the results after our error correction, which compensates the drift of the internal phase. Similar oscillations have been also revealed for Bell states \(|\varPhi _+ \rangle \) and \(|\varPhi _- \rangle \).

Fig. 12
figure 12

An overlap between the prepared state and the Bell states \(|\varPsi _+ \rangle \) (blue circles) and \(|\varPsi _- \rangle \) (brown triangles) as a function of time, provided the initial target state for \(|\varPsi \rangle \) was \(|\varPsi _+ \rangle \); a corresponds to direct measurements, b deals with the results after the correction of the coherent error (see in the text) (Color figure online)

Full size image

Appendix C: Error distributions for BB84 protocol

Table 5 gives error distribution for different time delays and each possible choice of the basis and bit of information, as obtained from 5-qubit IBMQx4 device. In the ideal case, the errors should be absent.

Table 5 The error distribution for BB84 protocol for different values of time delay
Full size table

Table 6 provides error distribution for different number of SWAPs and each possible choice of the basis and bit of information, as obtained from 5-qubit IBMQx4 device. Table 7 gives similar data, but using the encoding of the logical qubit into two physical qubits supplemented by post-selection procedure. The brackets contain fraction of algorithm’s runs used after the post-selection. The post-selection allowed us to improve the results, as seen from the comparison of data from Tables 6 and 7. We also note that the fraction of discarded data grows with the number of SWAPs, and this leads to the improvement in the performance.

Table 6 The error distribution for BB84 protocol for different number of SWAPs
Full size table
Table 7 The error distribution for BB84 protocol for different number of SWAPs
Full size table

Measurements for the BB84 protocol have been taken between April 4, 2018, and May 21, 2018.

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Zhukov, A.A., Kiktenko, E.O., Elistratov, A.A. et al. Quantum communication protocols as a benchmark for programmable quantum computers. Quantum Inf Process 18, 31 (2019). https://doi.org/10.1007/s11128-018-2144-y

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