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  • Open Access

Universal Deterministic Quantum Operations in Microwave Quantum Links

Guillermo F. Peñas, Ricardo Puebla, Tomás Ramos, Peter Rabl, and Juan José García-Ripoll
Phys. Rev. Applied 17, 054038 – Published 24 May 2022
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  • INTRODUCTION
  • SETUP
  • QUANTUM STATE TRANSFER
  • PHOTON PHASE VIA SCATTERING
  • APPLICATIONS
  • CONCLUSIONS AND OUTLOOK
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We propose a realistic setup, inspired by already existing experiments, within which we develop a general formalism for the implementation of distributed quantum gates. Mediated by a quantum link that establishes a bidirectional quantum channel between distant nodes, our proposal works both for inter- and intranode communication and handles scenarios ranging from the few to the many modes limit of the quantum link. We are able to design fast and reliable state transfer protocols in every regime of operation, which, together with a detailed description of the scattering process, allows us to engineer two sets of deterministic universal distributed quantum gates: gates whose implementation in quantum networks does not need entanglement distribution nor measurements. By employing a realistic description of the physical setup we identify the most relevant imperfections in the quantum links as well as optimal points of operation with resulting infidelities of 1−F≈10−2–10−3.

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    • Received 13 October 2021
    • Revised 9 February 2022
    • Accepted 15 April 2022

    DOI:https://doi.org/10.1103/PhysRevApplied.17.054038

    Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

    Published by the American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    PhotonicsQuantum communicationQuantum gatesQuantum information architectures & platformsQuantum networksQuantum protocolsSuperconducting quantum optics
    1. Techniques
    Microwave techniques
    Quantum Information, Science & Technology

    Authors & Affiliations

    Guillermo F. Peñas1,*, Ricardo Puebla1,2, Tomás Ramos1, Peter Rabl3, and Juan José García-Ripoll1,†

    • 1Instituto de Física Fundamental, IFF-CSIC, Calle Serrano 113b, Madrid 28006, Spain
    • 2Centre for Theoretical Atomic, Molecular and Optical Physics, Queen’s University Belfast, Belfast BT7 1NN, United Kingdom
    • 3Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, Vienna 1040, Austria

    • *guillermof.pens.fdez@iff.csic.es
    • †juanjose.garcia.ripoll@csic.es

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    Vol. 17, Iss. 5 — May 2022

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    • Figure 1
      Figure 1

      (a) Sketch of a quantum network, where multiple quantum nodes connected via quantum links (QLs) of different lengths allowing for an inter- and intranode bidirectional exchange of quantum information via photon pulses. (b) Building block of the physical setup, where two quantum nodes, consisting of a qubit coupled to a resonator, Q1 and R1 and Q2 and R2, respectively, are connected via a waveguide of length L that acts as a QL [cf. Eq. (1)].

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    • Figure 2
      Figure 2

      (a) Schematic illustration of the quantum state transfer protocol: by applying the controls g1,2(t) one can transfer the excitation from Q1 to Q2 via a photon propagating through the QL that distorts it, and, thus, |q2(T)|2≈1 at the end of the protocol. Panel (b) shows the quantum state transfer efficiency 1−|q2(T)|2 as a function of κ for three different QL lengths (L=30, 5, and 1 m), and distinct photon bandwidths, η=1 and η=4. The dash-dot lines correspond to the limit imposed by diffraction (short-photon limit); cf. Eq. (8). The dotted lines represent the length-independent limit (long-photon limit). The crossover between these regimes takes place when 2σt≈tp, i.e., κ/η≈2πvg/(3L). Panel (c) shows the impact in quantum state transfer efficiency when the resonant condition is not met, that is, ΩR1=ΩR2=ωc (resonant) versus ΩR1,R2=ωc+Δωc/2 (off resonant), i.e., R1 and R2 lie in between two TE10m modes. Panel (d) shows the impact that a truncation of the control g(t) has on the quantum state transfer efficiency for κ=2π×90,18,3 MHz and L=1, 5, 30 m, respectively [same style as in (b)]. This indicates that the diffraction limit can already be reached with g1(−T)≈2π×10−3 or 10−1 MHz, depending on the parameters.

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    • Figure 3
      Figure 3

      (a) Comparison of the quantum state transfer efficiency 1−|q2(T)|2 as a function of the protocol duration for the direct-swap, STIRAP-like, and pulse-shaping protocols following Eq. (7), for L=1, 5, 30 m and Δωc≈κ=2π×90, 18, 3 MHz, respectively. The horizontal dotted lines correspond to the efficiency limit posed by the wavepacket distortion [cf. Eq. (8)], to which the pulse shaping saturates. (b) State transfer efficiency for the same three protocols but including a finite coherence time of the qubits of T1=11.5μs, following Ref. [3]. The decoherence has been introduced according to 1−|q2(T)|2e−τ/T1.

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    • Figure 4
      Figure 4

      (a) Schematic illustration of the photon scattering process, where the photon interacts with R2, which results in a phase ϕscatt(ΩR2) imprinted on the photon. (b) Phase gained by each mode of frequency ωm of the QL after a scattering process with R2 with frequency ΩR2 and decay rate κ2. The phase for each of the discrete modes (points) of a QL with L=30, 15, and 5 m with κ2/2π=20, 25, and 80 MHz, respectively, closely follows the theoretical expression (9) for any L and κ2. Note that the spacing between the points is given by the free spectral range of the QL. Panel (c) shows the distortion of a realistic wavepacket due to the phase profile ϕscatt(ω) as well as due to the residual population that remains in R2 (dotted lines) for different lengths as in (b) but with κ=2π×100 MHz. Solid lines corresponds to the numerical simulation compared to the ideal photon after scattering |z|2=|⟨ξ~(T)|ψ(T)⟩|2, while the dash-dot lines have been obtained using the theoretical expression (9), |z|2=|⟨ξ~(T)|ξ(T)⟩|2. Note that, for η≲10, 1−|z|2∼η−4, as predicted by Eq. (10), while QLs with larger length L are more prone to distortion due to propagation (cf. Sec. 3). For η≫1, one enters the long-photon limit where the photon does not fit within the QL and residual populations remain (see the main text for further details).

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    • Figure 5
      Figure 5

      (a) Schematic illustration of a quantum gate transfer between Q1 and Q3, as given in Eq. (11), that involves two state transfer operations and a two-qubit gate U2,3 on qubits within the same node. (b) Duration of the gate transfer protocol t=2×2T, assuming that the local gate implemented at the second node is instantaneous, for three different waveguide lengths with η=1 (omitted in the legend) and one case with L=30 m and η=4. For small κ, the propagation time is negligible, and the limitation comes from the time-dependent control g(t) and all lines with η=1 overlap. (c) Fidelity of the process for the same set of parameters, where a trade-off between protocol duration and distortion of the wave packet is observed, especially in the L=30 m, η=1 case.

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    • Figure 6
      Figure 6

      (a) Schematic illustration for the realization of a controlled-phase gate, where the phase of the scattered photon depends on Q2’s state, |x∈{1,0}⟩. (b) Phase profile ϕx(ω) as a function of the detuning of the incident photon with respect to the effective R2 frequency. (c) Infidelity of the controlled-phase gate 1−Fg for two different L and κ. The solid lines correspond to the numerical simulations optimizing χ so one can access to the long-photon regime, while the theoretical predictions (dashed lines) are calculated from the joint diffraction and scattering expressions. Decoherence times and the effect of the Purcell filter are not considered. (d) Results for 1−Fg for the same set of parameters but including coherence times and a Purcell filter between the QL and R2. We set g2/Δ=0.1 and 0.125 for κ=2π×100 and 50 MHz with gp/Δ=0.03 and 0.04, respectively.

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    • Figure 7
      Figure 7

      Dependence of the distortion S [cf. Eq. (B1)] in the control g(t) [cf. Eq. (7)] produced by a low-pass filter with cutoff frequency ωc. Recall that κ is the resonator decay rate.

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    • Figure 8
      Figure 8

      (a) Robustness of each of the three protocols studied in this manuscript against slight variations of the protocol time for κ=2π×90MHz≈Δωc and a L=1 m waveguide. The lines plotted correspond to a single simulation of those shown in Fig. 3 around the 200 ns point. It can be seen that, whereas the pulse-shaping and the STIRAP protocols (solid and dash-dot lines, respectively) are robust, the direct-swap protocol (dotted line) undergoes large variations even for just a few nanoseconds. (b) Comparison of the quantum state transfer efficiency 1−|q2(T)|2 as a function of the protocol duration for an improved STIRAP-like protocol (cf. Appendix pp5) and pulse shaping following Eq. (7), for L=1, 5, and 30 m, as in Fig. 3. The horizontal dotted lines correspond to the efficiency limit posed by the wavepacket distortion [cf. Eq. (8)], to which the pulse shaping saturates.

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