Deterministic generation of remote entanglement with active quantum feedback

Leigh Martin, Felix Motzoi, Hanhan Li, Mohan Sarovar, and K. Birgitta Whaley

Phys. Rev. A 92, 062321 – Published 10 December 2015

Abstract

We consider the task of deterministically entangling two remote qubits using joint measurement and feedback, but no directly entangling Hamiltonian. In order to formulate the most effective experimentally feasible protocol, we introduce the notion of average-sense locally optimal feedback protocols, which do not require real-time quantum state estimation, a difficult component of real-time quantum feedback control. We use this notion of optimality to construct two protocols that can deterministically create maximal entanglement: a semiclassical feedback protocol for low-efficiency measurements and a quantum feedback protocol for high-efficiency measurements. The latter reduces to direct feedback in the continuous-time limit, whose dynamics can be modeled by a Wiseman-Milburn feedback master equation, which yields an analytic solution in the limit of unit measurement efficiency. Our formalism can smoothly interpolate between continuous-time and discrete-time descriptions of feedback dynamics and we exploit this feature to derive a superior hybrid protocol for arbitrary nonunit measurement efficiency that switches between quantum and semiclassical protocols. Finally, we show using simulations incorporating experimental imperfections that deterministic entanglement of remote superconducting qubits may be achieved with current technology using the continuous-time feedback protocol alone.

Received 10 July 2015

DOI:https://doi.org/10.1103/PhysRevA.92.062321

Authors & Affiliations

Leigh Martin^1,2,*, Felix Motzoi^1,2,3,†, Hanhan Li^1,2, Mohan Sarovar⁴, and K. Birgitta Whaley^1,3

¹Berkeley Center for Quantum Information and Computation, Berkeley, California 94720, USA
²Department of Physics, University of California, Berkeley, Berkeley, California 94720, USA
³Department of Chemistry, University of California, Berkeley, Berkeley, California 94720, USA
⁴Digital & Quantum Information Systems, Sandia National Laboratories, Livermore, California 94550, USA

^*Leigh@Berkeley.edu
^†Current address: Theoretical Physics, Saarland University, 66123 Saarbrücken, Germany.

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Vol. 92, Iss. 6 — December 2015

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Images

Figure 1
Histograms of the integrated measurement signal for the three eigenstates $ρ_{i}$ of the measurement, given by $Tr [Ω_{Δ V, η} ρ_{i} Ω_{Δ V, η}^{†}]$ . Plots are for $η = 1$ (thin lines) and $η = 0.2$ (thick lines) and use the parameters $Δ t = 3 μ s$ and $k = 1 (2 π) MHz$ .
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Figure 2
Discrete-time feedback simulations, showing fidelity with the $|t 0〉$ state as a function of time, starting from the maximally mixed state in the triplet subspace. For these simulations, $η = 0.1, k = 1 (2 π) MHz$ , and $Δ t = 20 μ s$ . The inset shows the optimal threshold voltage as a function of time. Note that for a smaller threshold, fidelity increases quickly at first but then saturates to a value significantly less than one, while for a larger constant threshold, fidelity increases slowly at first but then surpasses the former and approaches unity, though it does not asymptotically approach 1 [see Eq. (15)]. The locally optimal strategy, which increases the threshold as a function of time, matches or surpasses both fixed-threshold strategies at all times.
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Figure 3
(a) Discrete feedback simulations, showing fidelity with the $|t 0〉$ state as a function time under the semiclassical and quantum feedback [Eq. (9)] strategies, starting in the equal superposition state $|ψ_{0}〉$ and assuming a unit efficiency measurement and $k = 1 (2 π) MHz$ . The performance also changes according to the discrete time step $Δ t$ : we show two representative cases here. (b) Feedback $θ_{opt} (V)$ applied by the quantum protocol shown in (a) for the case $Δ t = 0.2 / k$ at four distinct time steps. At the latest time step, $θ_{opt}$ resembles the semiclassical protocol because the off-diagonal elements of the density matrix have decayed to zero.
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Figure 4
Continuous feedback simulations, showing fidelity with the $|t 0〉$ state $\bar{F}$ as a function of time under the continuous-time ASLO protocol given in Eq. (17). The initial state is the separable state $|ψ_{0}〉$ . The simulation parameters are $k = 1 (2 π) MHz$ and $η = 1$ , with time step $d t ≪ 1 / k$ . Fidelity as a function of time for several values of constant proportionality coefficient $P$ are also shown for comparison. The inset shows the steady-state fidelity ${\bar{F}}_{s s, f}$ achieved for constant $P$ , according to Eq. (23). Since we have taken the small $d t$ limit, the threshold strategy would not change from its fidelity at $t = 0$ . Note that if we were to plot the locally optimal strategy in this continuous-time situation, which would nomically require dynamical state estimation, it would coincide exactly with the curve for $P_{opt} (t)$ .
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Figure 5
Fidelity $\bar{F}$ starting from the $|ψ_{0}〉$ state as a function of time under the optimal variable-time-step protocol (blue circles), compared to fidelity obtained with constant-time-step protocols [discrete-time semiclassical protocol (red crosses) and continuous-time quantum protocol (yellow dotted line)]. The measurement is on continuously at a constant rate and feedback is applied instantaneously at each point in the graph. The parameters $k = 1 (2 π) MHz$ and $η = 0.4$ are chosen to illustrate the benefit of this optimized strategy for nonunit but not too small efficiency. The dense cluster of points before $t = 2 μ s$ shows that the gradient search resulted in almost all of the times for application of feedback to be located in the early stages of the evolution. Also shown is the fidelity obtained from a full non-Markovian strategy using dynamical state estimation and locally optimal feedback (see the text).
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Figure 6
Continuous-feedback simulations, showing fidelity with the $|t 0〉$ state as a function of time under the continuous-time ASLO protocol prescribed by Eq. (17) and incorporating experimentally realistic parameters and imperfections as described in the text. The fidelity reaches a much lower peak value than it would in the ideal case before decaying due to finite dephasing. Despite this, the fidelity substantially surpasses $50 %$ , which is the threshold for entanglement. To directly quantify entanglement, we also plot the concurrence [45] as a function of time. Dashed and dotted curves show fidelity using a 50% postselection criterion with and without feedback, respectively.
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