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arXiv:2308.03923 · Physical Review Applied

Stabilizing two-qubit entanglement with dynamically decoupled active feedback

Sacha Greenfield, Leigh S. Martin, Felix Motzoi, K. Birgitta Whaley, Justin Dressel, Eli M. Levenson-Falk

Published Feb 12, 2024• 8 citations
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Abstract

We propose and analyze a protocol for stabilizing a maximally entangled state of two noninteracting qubits using active state-dependent feedback from a continuous two-qubit half-parity measurement in coordination with a concurrent, noncommuting dynamical decoupling drive. We demonstrate that such a drive can be simultaneous with the measurement and feedback, while also playing a key part in the feedback protocol itself. We show that robust stabilization with near-unit fidelity can be achieved even in the presence of realistic nonidealities, such as time delay in the feedback loop, imperfect state-tracking, inefficient measurements, dephasing from $1/f$-distributed qubit-frequency noise, and relaxation. We mitigate feedback-delay error by introducing a forward-state-estimation strategy in the feedback controller that tracks the effects of control signals already in transit. More generally, the steady state is globally attractive without the need for ancillas, regardless of the error state, in contrast to most known feedback and error-correction schemes.

DOI
10.1103/physrevapplied.21.024022

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We propose and analyze a protocol for stabilizing a maximally entangled state of two noninteracting qubits using active state-dependent feedback from a continuous two-qubit half-parity measurement in coordination with a concurrent, noncommuting dynamical decoupling drive. We demonstrate that such a drive can be simultaneous with the measurement and feedback, while also playing a key part in the feedback protocol itself. We show that robust stabilization with near-unit fidelity can be achieved even in the presence of realistic nonidealities, such as time delay in the feedback loop, imperfect state-tracking, inefficient measurements, dephasing from 1/f1/f1/f-distributed qubit-frequency noise, and relaxation. We mitigate feedback-delay error by introducing a forward-state-estimation strategy in the feedback controller that tracks the effects of control signals already in transit. More generally, the steady state is globally attractive without the need for ancillas, regardless of the error state, in contrast to most known feedback and error-correction schemes.
We demonstrate and contrast two approaches to the stabilization of qubit entanglement by feedback.Our demonstration is built on a feedback platform consisting of two superconducting qubits coupled to a cavity, which are measured by a nearly quantum-limited measurement chain and controlled by high-speed classical logic circuits.This platform is used to stabilize entanglement by two nominally distinct schemes: a "passive" reservoir engineering method and an "active" correction based on conditional parity measurements.In view of the instrumental roles that these two feedback paradigms play in quantum error correction and quantum control, we directly compare them on the same experimental setup.Furthermore, we show that a second layer of feedback can be added to each of these schemes, which heralds the presence of a high-fidelity entangled state in real time.This "nested" feedback brings about a marked entanglement fidelity improvement without sacrificing success probability.
Generating on-demand maximally entangled states is one of the cornerstones for quantum information processing. Parity measurements can serve to create Bell states and have been implemented via an electronic Mach-Zehnder interferometer among others. However, the entanglement generation is necessarily harmed by measurement-induced dephasing processes in one of the two parity subspaces. In this work, we propose two different schemes of continuous feedback for a parity measurement. They enable us to avoid both the measurement-induced dephasing process and the experimentally unavoidable dephasing, e.g., due to fluctuations of the gate voltages controlling the initialization of the qubits. We show that we can generate maximally entangled steady states in both parity subspaces. Importantly, the measurement scheme we propose is valid for implementation of parity measurements with feedback loops in various solid-state environments.
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.
Abstract Fault-tolerant quantum computing relies on the ability to detect and correct errors, which in quantum error correction codes is typically achieved by projectively measuring multi-qubit parity operators and by conditioning operations on the observed error syndromes. Here, we experimentally demonstrate the use of an ancillary qubit to repeatedly measure the ZZ and XX parity operators of two data qubits and to thereby project their joint state into the respective parity subspaces. By applying feedback operations conditioned on the outcomes of individual parity measurements, we demonstrate the real-time stabilization of a Bell state with a fidelity of F ≈ 74% in up to 12 cycles of the feedback loop. We also perform the protocol using Pauli frame updating and, in contrast to the case of real-time stabilization, observe a steady decrease in fidelity from cycle to cycle. The ability to stabilize parity over multiple feedback rounds with no further reduction in fidelity provides strong evidence for the feasibility of executing stabilizer codes on timescales much longer than the intrinsic coherence times of the constituent qubits.
Abstract Recent work has shown that the use of quantum feedback can significantly enhance both the speed and success rate of measurement-based remote entanglement generation, but it is generally unknown what feedback protocols are optimal for these tasks. Here we consider two common measurements that are capable of projecting into pairwise entangled states, namely half- and full-parity measurements of two qubits, and determine in each case a globally optimal protocol for generation of entanglement. For the half-parity measurement, we rederive a previously described protocol using more general methods and prove that it is globally optimal for several figures of merit, including maximal concurrence or fidelity and minimal time to reach a specified concurrence or fidelity. For the full-parity measurement, we derive a protocol for rapid entanglement generation related to that of (Hill, Ralph, Phys. Rev. A 77 , 014305), and then map the dynamics of the concurrence of the state to the Bloch vector length of an effective qubit. This mapping allows us to prove several optimality results for feedback protocols with full-parity measurements. We further show that our full-parity protocol transfers entanglement optimally from one qubit to the other amongst all measurement-based schemes. The methods developed here will be useful for deriving feedback protocols and determining their optimality properties in many other quantum systems subject to measurement and unitary operations.
Recent work has shown that use of quantum feedback can significantly enhance both the speed and success rate of measurement-based remote entanglement generation, but it is generally unknown what feedback protocols are optimal for these tasks. Here we consider two common measurements that are capable of projecting into pairwise entangled states, namely half- and full-parity measurements of two qubits, and determine in each case a globally optimal protocol for generation of entanglement. For the half-parity measurement, we rederive a previously described protocol using more general methods and prove that it is globally optimal for several figures of merit, including maximal concurrence or fidelity, and minimal time to reach a specified concurrence or fidelity. For the full-parity measurement, we derive a protocol for rapid entanglement generation related to that of (C. Hill, J. Ralph, Phys. Rev. A 77, 014305 (2008)), and then map the dynamics of the concurrence of the state to the Bloch vector length of an effective qubit. This mapping allows us to prove several optimality results for feedback protocols with full-parity measurements. We further show that our full-parity protocol transfers entanglement optimally from one qubit to the other amongst all measurement-based schemes. The methods developed here will be useful for deriving feedback protocols and determining their optimality properties in many other quantum systems subject to measurement and unitary operations.