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arXiv:1411.7625 · Physical Review Letters

Nonperturbative Leakage Elimination Operators and Control of a Three-Level System

Jun Jing, Lian-Ao Wu, Mark Byrd, J. Q. You, Ting Yu, Zhao-Ming Wang

Published May 15, 2015• 53 citations
arXivHTMLPDF

Abstract

Dynamical decoupling operations have been shown to reduce errors in quantum information processing. Leakage from an encoded subspace to the rest of the system space is a particularly serious problem for which leakage elimination operators (LEOs) were introduced. Here we provide an analysis of nonideal pulses, rather than the well-understood idealization or bang-bang controls. Under realistic conditions, we show that these controls will provide the same protection from errors as idealized controls. Our work indicates that the effectiveness of LEOs depends on the integral of the pulse sequence in the time domain, which has been missing because of the idealization of pulse sequences. Our results are applied to a three-level system for the nitrogen-vacancy centers under an external magnetic field and are illustrated by the fidelity dynamics of LEO sequences, ranging from regular rectangular pulses, random pulses, and even disordered (noisy) pulses.

DOI
10.1103/physrevlett.114.190502

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Dynamical decoupling operations have been shown to reduce errors in quantum information processing. Leakage from an encoded subspace to the rest of the system space is a particularly serious problem for which leakage elimination operators (LEOs) were introduced. Here we provide an analysis of nonideal pulses, rather than the well-understood idealization or bang-bang controls. Under realistic conditions, we show that these controls will provide the same protection from errors as idealized controls. Our work indicates that the effectiveness of LEOs depends on the integral of the pulse sequence in the time domain, which has been missing because of the idealization of pulse sequences. Our results are applied to a three-level system for the nitrogen-vacancy centers under an external magnetic field and are illustrated by the fidelity dynamics of LEO sequences, ranging from regular rectangular pulses, random pulses, and even disordered (noisy) pulses.
Decoherence-induced leakage errors can potentially damage physical or logical qubits embedded in a subspace of the entire Hilbert space by coupling them to other system levels. Here we report the first experimental implementation of Leakage Elimination Operators (LEOs) that aims to reduce this undermining. LEOs are a type of dynamical decoupling control that have been previously introduced to counteract leakage from a chosen subspace into the rest of a Hilbert space, and have been widely explored theoretically. Different from other error correction strategies, LEOs are compatible with any gate sequence in a code space, and thus, compatible with universal quantum computation. Using IBM's cloud quantum computer (QC), we design three potentially applicable examples of subspaces in two- and three-qubit Hilbert spaces and derive the explicit forms of the corresponding LEOs for these subspaces. For the first time, we experimentally demonstrate that these LEOs significantly suppress leakage. The results also show that the LEO time-scale condition can be satisfied with noise in the IBM's cloud QC and pave a way for quantum setups to get rid of leakage trouble.
Adiabatic evolution is used in a variety of quantum information processing tasks. However, the elimination of errors is not as well developed as it is for circuit model processing. Here, we present a strategy to improve the performance of a quantum adiabatic process by adding leakage elimination operators (LEOs) to the evolution. These are a sequence of pulse controls acting in an adiabatic subspace to eliminate errors by suppressing unwanted transitions. Using the Feshbach PQPQPQ partitioning technique, we obtain an analytical solution for a set of pulse controls. The effectiveness of the LEO is independent of the specific form of the pulse but depends on the average frequency of the control function. By observing that the evolution of the target eigenstate is governed by a periodic function appearing in the integral of the control function, we show that control parameters can be chosen in such a way that the instantaneous eigenstates of the system are unchanged, yet a speedup can be achieved by suppressing transitions. Furthermore, we give the exact expression of the control function for a counter unitary transformation to be used in experiments which provides a clear physical meaning for the LEO, aiding in the implementation.
Continuous-variable (CV) systems are attracting increasing attention in the realization of universal quantum computation. Several recent experiments have shown the feasibility of using CV systems to, e.g., encode a qubit into a trapped-ion mechanical oscillator and perform logic gates [Nature 566, 513-517 (2019)]. The essential next step is to protect the encoded qubit from quantum decoherence, e.g., the motional decoherence due to the interaction between a mechanical oscillator and its environment. Here we propose a scheme to suppress quantum decoherence of a single-mode harmonic oscillator used to encode qubits by introducing a nonperturbative leakage elimination operator (LEO) specifically designed for this purpose. Remarkably, our nonperturbative LEO can be used to analytically derive exact equations of motion without approximations. It also allows us to prove that the effectiveness of these LEOs only depends on the integral of the pulse sequence in the time domain, while details of the pulse shape does not make a significant difference when the time period is chosen appropriately. This control method can be applied to a system at an arbitrary temperature and arbitrary system-bath coupling strength which makes it extremely useful for general open quantum systems.
We present a method aimed at protecting unitary dynamics in the presence of decoherence, by integrating leakage elimination operators (LEOs) into the system's evolution to create dynamical leakage-free paths. Deriving the dynamical equation for an open quantum system with general drives can be challenging. Our approach avoids the rotating wave approximation and instead uses the coarse-grained averaging technique to derive a quantum master equation for such systems. The combination of the coarse-graining approach and LEO operators appears suitable to study Markovian control methods. We show that employing LEO pulses in specific subspaces can reduce errors arising from undesired transitions due to decoherence. Notably, satisfactory final fidelity can still be achieved even when the reservoir is at a finite temperature. By looking into the dynamical equation governing the quantum state on the dynamical leakage-free path, we provide analytical insights into the effectiveness of the LEO method in suppressing decoherence effects.
Effective quantum control is essential for quantum computing and information processing, yet level leakage remains a bottleneck limiting control fidelity. The leakage elimination operator (LEO) approach is highly effective at suppressing leakage from a target subspace to other subspaces. Analytical control conditions under the high-frequency driving limit have been derived via the Feshbach PQ partitioning technique. However, the driving intensity is subject to fundamental physical bounds, and low-frequency driving is experimentally more feasible. In this work, we reformulate LEO within the Magnus framework and extend it beyond the high-frequency limit. We show that the analytical control conditions obtained from the Feshbach PQ partitioning technique are equivalent to the zeroth-order Magnus expansion. We then analyze the higher-order Magnus corrections. By minimizing the leakage, we obtain a modified pulse condition suitable for the low-frequency regime. As an illustrative example, we study near-perfect state transfer through a one-dimensional spin chain under LEO control. We find that the second-order Magnus correction improves the transmission fidelity at low-frequency driving, while the improvement becomes negligible at high frequencies. Our results provide an effective route for designing control pulses in the low-frequency regime, with potential applications in superconducting qubits, trapped ions, and solid-state spin systems.
Quantum error prevention strategies will be required to produce a scalable quantum computing device and are of central importance in this regard. Progress in this area has been quite rapid in the past few years. In order to provide an overview of the achievements in this area, we discuss the three major classes of error prevention strategies, the abilities of these methods and the shortcomings. We then discuss the combinations of these strategies which have recently been proposed in the literature. Finally, we present recent results in reducing errors on encoded subspaces using decoupling controls. We show how to generally remove mixing of an encoded subspace with external states (termed leakage errors) using decoupling controls. Such controls are known as ‘leakage elimination operations’ or ‘LEOs’.