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arXiv:quant-ph/0507265 · Physical Review A

Geometric continuous dynamical decoupling with bounded controls

Pochung Chen

Published Feb 24, 2006• 20 citations
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Abstract

We develop a framework to implement the dynamical decoupling of open quantum systems using continuous pulse sequences. By exploiting the geometric perspective of the continuous dynamical decoupling, we show that the decoupling pulses can be intuitively designed in the basis of the structure function of the $\mathrm{SU}(n)$ corresponding to the control Hamiltonian. Several examples are given to illustrate the basic idea. We argue that in practice the efficiency of the decoupling is determined by the minimum attainable decoupling cycle time ${T}_{c}$ instead of the order of the discrete decoupling group.

DOI
10.1103/physreva.73.022343

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We develop a framework to implement the dynamical decoupling of open quantum systems using continuous pulse sequences. By exploiting the geometric perspective of the continuous dynamical decoupling, we show that the decoupling pulses can be intuitively designed in the basis of the structure function of the SU(n)\mathrm{SU}(n)SU(n) corresponding to the control Hamiltonian. Several examples are given to illustrate the basic idea. We argue that in practice the efficiency of the decoupling is determined by the minimum attainable decoupling cycle time Tc{T}_{c}Tc​ instead of the order of the discrete decoupling group.
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.
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.
We propose a general procedure for implementing dynamical decoupling of quantum systems without requiring arbitrarily strong, impulsive control actions. This is accomplished by designing continuous decoupling propagators according to Eulerian paths in the decoupling group for the system. Such Eulerian decoupling schemes offer two important advantages over their impulsive counterparts: they are able to enforce the same dynamical symmetrization but with more realistic control resources and, at the same time, they are intrinsically tolerant against a large class of systematic implementation errors.
We introduce a framework for designing Hamiltonian engineering pulse sequences that systematically accounts for the effects of higher-order contributions to the Floquet-Magnus expansion. Our techniques result in simple, intuitive decoupling rules, despite the higher-order contributions naively involving complicated, nonlocal-in-time commutators. We illustrate how these rules can be used to efficiently design improved Hamiltonian engineering pulse sequences for a wide variety of tasks such as dynamical decoupling, quantum sensing, and quantum simulation.
We address the problem of decoupling the interactions in a spin network governed by a pair-interaction Hamiltonian. Combinatorial schemes for decoupling and for manipulating the couplings of Hamiltonians have been developed, which use selective pulses. In this paper, we consider an additional requirement on these pulse sequences: as few different control operations as possible should be used. This requirement is motivated by the fact that to find an optimal implementation of each individual selective pulse will be expensive since it requires to solve a pulse shaping problem. Hence, it is desirable to use as few different selective pulses as possible. As a first result, we show that for d-dimensional systems, where d⩾2, the ability to implement only two control operations is sufficient to turn off the time evolution. Next, we address the case of a bipartite system with local control and show that four different control operations are sufficient. Finally, turning to networks consisting of several d-dimensional nodes, we show that decoupling can be achieved if one is able to control a number of different control operations, which is logarithmic in the number of nodes. We give an explicit family of efficient decoupling schemes with logarithmic number of different pulses based on the classic Hamming codes. We also provide a table of the best known decoupling schemes for small networks of qubits.