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arXiv:1507.07303 · Physical Review A

Method for generating all uniform π -pulse sequences used in deterministic dynamical decoupling

Haoyu Qi, Jonathan P. Dowling

Published Sep 3, 2015• 2 citations
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Abstract

Dynamical decoupling has been actively investigated since Viola first suggested using a pulse sequence to protect a qubit from decoherence. Since then, many schemes of dynamical decoupling have been proposed to achieve high-order suppression, both analytically and numerically. However, hitherto, there has not been a systematic framework to understand all existing uniform $\ensuremath{\pi}$-pulse dynamical decoupling schemes. In this report, we use the projection pulse sequences as basic building blocks and concatenation as a way to combine them. We derived a concatenated-projection dynamical decoupling, a framework in which we can systematically construct pulse sequences to achieve arbitrary high suppression order. All previously known uniform dynamical decoupling sequences using $\ensuremath{\pi}$ pulse can be fit into this framework. Understanding uniform dynamical decoupling as successive projections on the Hamiltonian will also give insights on how to invent new ways to construct better pulse sequences.

DOI
10.1103/physreva.92.032303

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Dynamical decoupling has been actively investigated since Viola first suggested using a pulse sequence to protect a qubit from decoherence. Since then, many schemes of dynamical decoupling have been proposed to achieve high-order suppression, both analytically and numerically. However, hitherto, there has not been a systematic framework to understand all existing uniform \ensuremathπ\ensuremath{\pi}\ensuremathπ-pulse dynamical decoupling schemes. In this report, we use the projection pulse sequences as basic building blocks and concatenation as a way to combine them. We derived a concatenated-projection dynamical decoupling, a framework in which we can systematically construct pulse sequences to achieve arbitrary high suppression order. All previously known uniform dynamical decoupling sequences using \ensuremathπ\ensuremath{\pi}\ensuremathπ pulse can be fit into this framework. Understanding uniform dynamical decoupling as successive projections on the Hamiltonian will also give insights on how to invent new ways to construct better pulse sequences.
Aperiodic dynamical decoupling (DD) sequences of π pulses are of great interest to decoherence control and have been recently extended from single-qubit to two-qubit systems. If the environmental noise power spectrum is made available, then one may further optimize aperiodic DD sequences to reach higher efficiency of decoherence suppression than known universal schemes. This possibility is investigated in this work for the protection of two-qubit states, using an exactly solvable pure dephasing model including both local and nonlocal noise. The performance of optimized DD sequences in protecting two-qubit states is compared with that achieved by nested Uhrig's DD (nested-UDD) sequences, for several different types of noise spectrum. Except for the cases with noise spectrum decaying slowly in the high-frequency regime, optimized DD sequences with tens of control pulses can perform orders of magnitude better than that of nested-UDD. A two-qubit system with highly unbalanced local noise is also examined to shed more light on a recent experiment. Possible experiments that may be motivated by this work are discussed.
We introduce a high-order dynamical decoupling (DD) scheme for arbitrary bounded system-bath interactions in the weak-coupling regime. Given any decoupling group G\mathcal GG that averages the interaction to zero, our construction {guarantees the existence of pulse sequences with at most (∣G∣−1)K(|\mathcal G|-1)K(∣G∣−1)K pulses}, while canceling all error terms linear in the system-bath coupling strength up to order KKK in the total evolution time. As a corollary, for an nnn-qubit system with kkk-local system-bath interactions, we obtain an O(nk−1K)\mathcal{O}(n^{k-1}K)O(nk−1K)-pulse sequence, a significant improvement over existing schemes with O(exp⁡(n))\mathcal{O}(\exp(n))O(exp(n)) pulses (for k=O(1)k=\mathcal{O}(1)k=O(1)). The construction is obtained via a mapping to the continuous necklace-splitting problem, which asks how to cut a multi-colored interval into pieces that give each party the same share of every color. We provide explicit pulse sequences for suppressing general single-qubit decoherence, prove that the pulse count is asymptotically optimal, and verify the predicted error scaling in numerical simulations. For the same number of pulses, we observe that our sequences outperform the state-of-the-art Quadratic DD in the weak-coupling regime. We also construct explicit high-order sequences for suppressing representative 2-local noise models. Finally, the same construction extends to suppress slow, time-dependent classical noise and to filter-function design.
Dynamical decoupling can be used to preserve arbitrary quantum states despite undesired interactions with the environment, using control Hamiltonians affecting the system only. We present a system-independent analysis of dynamical decoupling based on leading order decoupling error estimates, valid for bounded-strength environments. Using as a key tool a renormalization transformation of the effective system-bath coupling Hamiltonian, we delineate the reliability domain of dynamical decoupling used for quantum state preservation, in a general setting for a single qubit. We specifically analyze and compare two deterministic dynamical decoupling schemes---periodic and concatenated---and distinguish between two limiting cases of fast versus slow environments. We prove that concatenated decoupling outperforms periodic decoupling over a wide range of parameters. These results are obtained for both ``ideal'' (zero-width) and realistic (finite-width) pulses This work extends and generalizes our earlier work [Phys. Rev. Lett. 95, 180501 (2005)].
Dynamical decoupling is a powerful method for protecting quantum information against unwanted interactions with the help of open-loop control pulses. Realistic control pulses are not ideal and may introduce additional systematic errors. We introduce a class of self-stabilizing pulse sequences capable of suppressing such systematic control errors efficiently in qubit systems. Embedding already known decoupling sequences into these self-stabilizing sequences offers powerful means to achieve robustness against unwanted external perturbations and systematic control errors. As these self-stabilizing sequences are based on single-qubit operations, they offer interesting perspectives for future applications in quantum information processing.
The loss of quantum information due to interactions with external degrees of freedom, which is known as decoherence, remains one of the main obstacles for large-scale implementations of quantum computing. Accordingly, different measures are being explored for reducing its effect. One of them is dynamical decoupling (DD) which offers a practical solution because it only requires the application of control pulses to the system qubits. Starting from basic DD sequences, more sophisticated schemes were developed that eliminate higher-order terms of the system-environment interaction and are also more robust against experimental imperfections. A particularly successful scheme, called concatenated DD (CDD), gives a recipe for generating higher-order sequences by inserting lower-order sequences into the delays of a generating sequence. Here, we show how this scheme can be improved further by converting some of the pulses to virtual (and thus ideal) pulses. The resulting scheme, called (XY4)n{(XY4)}^{n}(XY4)n, results in lower power deposition and is more robust against pulse imperfections than the original CDD scheme.
We present a randomized dynamical decoupling (DD) protocol that can substantially improve the performance of any given deterministic DD scheme for suppressing coherent noise by using no more than two additional pulses. Our construction is implemented by probabilistically applying sequences of pulses, which, when combined, effectively eliminate the error terms that scale linearly with the system-environment coupling strength. As a result, we show that a randomized protocol using a few pulses can outperform deterministic DD protocols that require considerably more pulses. Furthermore, we prove that the randomized protocol provides an improvement compared to deterministic DD sequences that aim to reduce the error in the system's Hilbert space, such as Uhrig DD, which had been previously regarded to be optimal. To rigorously evaluate the performance, we introduce new analytical methods suitable for analyzing higher-order DD protocols that might be of independent interest. We also present numerical simulations confirming the significant advantage of using randomized protocols compared to widely used deterministic protocols.