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arXiv:0810.5616 · Physical Review Letters

Concatenated Control Sequences Based on Optimized Dynamic Decoupling

Götz Silvester Uhrig

Published Mar 27, 2009• 98 citations
arXivHTMLPDF

Abstract

Two recent developments in quantum control, concatenation and optimization of pulse intervals, are combined to yield a strategy to suppress unwanted couplings in quantum systems to high order. Longitudinal relaxation and transverse dephasing can be suppressed so that systems with a small splitting between their energy levels can be kept isolated from their environment. The required number of pulses grows exponentially with the desired order but is only the square root of the number needed if only concatenation is used. An approximate scheme even brings the number down to polynomial growth. The approach is expected to be useful for quantum information and for high-precision nuclear magnetic resonance.

DOI
10.1103/physrevlett.102.120502

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Two recent developments in quantum control, concatenation and optimization of pulse intervals, are combined to yield a strategy to suppress unwanted couplings in quantum systems to high order. Longitudinal relaxation and transverse dephasing can be suppressed so that systems with a small splitting between their energy levels can be kept isolated from their environment. The required number of pulses grows exponentially with the desired order but is only the square root of the number needed if only concatenation is used. An approximate scheme even brings the number down to polynomial growth. The approach is expected to be useful for quantum information and for high-precision nuclear magnetic resonance.
Both longitudinal and transverse relaxations exist in the practical environment. Their simultaneous eliminations are extremely demanding in real applications. Previous experimental work has focused mainly on the suppression of transverse relaxation. In this paper we investigate the performance of three important dynamical decoupling schemes---quadratic dynamical decoupling, periodic dynamical decoupling, and concatenated dynamical decoupling---in an environment with hybrid errors. We propose a method to engineer arbitrary environment by modulating the control field. The technique developed here is universal and can be applied to other quantum information processing systems. Three-dimensional filter functions technique is utilized to analyze the fidelity decay of a one-qubit state protected by dynamical decoupling sequences. This enables us to quantitatively compare the performance of different dynamical decoupling sequences and demonstrate the superiority of quadratic dynamical decoupling in experiments for the first time. Our work reveals that quadratic dynamical decoupling is optimal conditioned on the appropriate noise properties. The difference of constructing dynamical decoupling sequences with various Pauli pulses is also investigated.
Over the past decade we have seen an explosion of demonstrations of quantum coherence in atomic, optical, and condensed matter systems. These developments have placed a new emphasis on the production of robust and optimal quantum control techniques in the presence of environmental noise. We discuss the use of dynamical decoupling as a form of open-loop quantum control capable of suppressing the effects of dephasing in quantum coherent systems. We introduce the concept of dynamical decoupling pulse-sequence construction as a filter-design problem, making connections with filter design from control theory and electrical engineering in the analysis of pulse-sequence performance for the preservation of the phase degree of freedom in a quantum superposition. A detailed mathematical description of how dephasing and the suppression of dephasing can be reduced to a linear control problem is provided, and used as motivation and context for studies of the filtration properties of various dynamical decoupling sequences. Our work then takes this practical perspective in addressing both "standard" sequences derived from nuclear magnetic resonance and novel optimized sequences developed in the context of quantum information. Additionally, we review new techniques for the numerical construction of optimized pulse sequences using the filter-design perspective. We show how the filter-design perspective permits concise comparisons of the relative capabilities of these sequences and reveals the physics underlying their functionality. The use of this new analytical framework allows us to derive new insights into the performance of these sequences and reveals important limiting issues, such as the effect of digital clocking on optimized sequence performance.
Over the past decade we have seen an explosion of demonstrations of quantum coherence in atomic, optical, and condensed matter systems.These developments have placed a new emphasis on the production of robust and optimal quantum control techniques in the presence of environmental noise.We discuss the use of dynamical decoupling as a form of open-loop quantum control capable of suppressing the effects of dephasing in quantum coherent systems.We introduce the concept of dynamical decoupling pulse-sequence construction as a filter-design problem, making connections with filter design from control theory and electrical engineering in the analysis of pulse-sequence performance for the preservation of the phase degree of freedom in a quantum superposition.A detailed mathematical description of how dephasing and the suppression of dephasing can be reduced to a linear control problem is provided, and used as motivation and context for studies of the filtration properties of various dynamical decoupling sequences.Our work then takes this practical perspective in addressing both "standard" sequences derived from nuclear magnetic resonance and novel optimized sequences developed in the context of quantum information.Additionally, we review new techniques for the numerical construction of optimized pulse sequences using the filter-design perspective.We show how the filter-design perspective permits concise comparisons of the relative capabilities of these sequences and reveals the physics underlying their functionality.The use of this new analytical framework allows us to derive new insights into the performance of these sequences and reveals important limiting issues, such as the effect of digital clocking on optimized sequence performance.
The need for strategies able to accurately manipulate quantum dynamics is ubiquitous in quantum control and quantum information processing. We investigate two scenarios where randomized dynamical decoupling techniques become more advantageous than standard deterministic methods in switching off unwanted dynamical evolution in a closed quantum system: when dealing with decoupling cycles which involve a large number of control actions and/or when seeking long-time quantum information storage. Highly effective hybrid decoupling schemes, which combine deterministic and stochastic features, are discussed, as well as the benefits of sequentially implementing a concatenated method, applied for short times, followed by a hybrid protocol, employed for longer times. A quantum register consisting of a chain of spin-1/2 particles interacting via the Heisenberg interaction is used as a model for the analysis throughout.
The need for strategies able to accurately manipulate quantum dynamics is ubiquitous in quantum control and quantum information processing. We investigate two scenarios where randomized dynamical decoupling techniques become more advantageous with respect to standard deterministic methods in switching off unwanted dynamical evolution in a closed quantum system: when dealing with decoupling cycles which involve a large number of control actions and/or when seeking long-time quantum information storage. Highly effective hybrid decoupling schemes, which combine deterministic and stochastic features are discussed, as well as the benefits of sequentially implementing a concatenated method, applied at short times, followed by a hybrid protocol, employed at longer times. A quantum register consisting of a chain of spin-1/2 particles interacting via the Heisenberg interaction is used as a model for the analysis throughout.
Introduction In this chapter we continue the introduction to dynamical decoupling techniques for open quantum systems that was started in Chapter 4. Here we focus on the construction of efficient schemes for dynamical decoupling and we highlight some combinatorial constructions. Efficiency of decoupling schemes is measured in terms of the number of control operations to be applied to the system. This number should be small, ideally a polynomial in the number of qubits in the system. If we assume that the quantum system is governed by a general Hamiltonian H S , having interactions involving any subset of the qubits, then any scheme that achieves decoupling of H S is necessarily inefficient in the above sense. The efficient schemes we consider here arise in physically realistic situations where there are much more stringent restrictions on the types of interactions. The most important example is the case of pair-interaction Hamiltonians. These Hamiltonians can be expressed as sums of interaction terms that involve at most two qubits. While there is a large body of work on dynamical decoupling schemes for pair-interaction Hamiltonians [S90, EBW94,WHH68, H76], which historically has been used mainly in the context of nuclear magnetic resonance (NMR) theory, the past few years have seen a development of new techniques for decoupling that are based on group theory. We therefore call such techniques, or rather the pulse sequences that an experimenter can apply to a given Hamiltonian in order to selectively switch off unwanted couplings, combinatorial decoupling schemes.