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.
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 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.
Motivation and overview The aim of this chapter is to illustrate the basic physical principles and mathematical framework of dynamical decoupling (DD) techniques for open quantum systems, as relevant to quantum information processing (QIP) applications. Historically, the physical origins of DD date back to the idea of coherent averaging of interactions, as pioneered in high-resolution solid-state nuclear magnetic resonance (NMR) by Haeberlen and Waugh using elegantly designed multiple-pulse sequences [HW68, WHH68, H76]. It was in the same landmark work [HW68] that average Hamiltonian theory was developed as a formalism on which the design and analysis of DD sequences has largely relied since then. In the original context of NMR spectroscopy, decoupling serves the purpose of enhancing resolution by simplifying complex spectra. This is achieved by realizing that an otherwise static spin Hamiltonian “can be made to appear time-dependent in a controlled way,” so that “as the characteristic repetition period of the pulses becomes [sufficiently] small, the spin system comes to behave over long times as though under the influence of a time-independent average Hamiltonian” [HW68]. Some basic insight may be gained by revisiting the paradigmatic example offered by the so-called Hahn echo [H50] and Carr–Purcell (CP) sequences [CP54] in the simplest setting of a two spin-1/2 system.
Decoupling the interactions in a spin network governed by a pair-interaction Hamiltonian is a well-studied problem. 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 {\em different} control operations as possible should be used. This requirement is motivated by the fact that optimizing each individual selective pulse will be expensive, i. e., it is desirable to use as few different selective pulses as possible. For an arbitrary d-dimensional system we show that the ability to implement only two control operations is sufficient to turn off the time evolution. In case of a bipartite system with local control we show that four different control operations are sufficient. Turning to networks consisting of several d-dimensional nodes which are governed by a pair-interaction Hamiltonian, 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.
Dynamical decoupling is a powerful technique to suppress errors in quantum systems originating from environmental couplings or from unwanted interparticle interactions. However, it can also be used to selectively decouple specific couplings in a quantum system. We present a simple and easy-to-use general method to construct such selective decoupling schemes on qubit and qudit networks by means of (generalized) Pauli operations. As these constructed schemes can suppress Hamiltonian interactions on general qudit networks selectively, they are well suited for purposes of approximate quantum simulation. Some examples are presented, demonstrating the use of our method and the resulting decoupling schemes.
In the NISQ era, where quantum information processing is hindered by the decoherence and dissipation of elementary quantum systems, developing new protocols to extend the lifetime of quantum states is of considerable practical and theoretical importance. A well-known technique, known as dynamical decoupling, uses a carefully designed sequence of pulses applied to a quantum system, such as a spin- j (which represents a qudit with d=2j+1 levels), to suppress the coupling Hamiltonian between the system and its environment, thereby mitigating dissipation. While dynamical decoupling of qubit systems has been widely studied, the decoupling of qudit systems has been far less explored and often involves complex sequences and operations. In this work, we design efficient decoupling sequences composed solely of global SU(2) rotations and based on tetrahedral, octahedral, and icosahedral point groups, which we call Platonic sequences. We extend the Majorana representation for Hamiltonians to develop a simple framework that establishes the decoupling properties of each Platonic sequence and show its effectiveness on many examples. These sequences are universal in their ability to cancel any type of interaction with the environment for single spin- j with spin quantum number j⩽5/2 , and they are capable of decoupling up to 5 -body interactions in an ensemble of interacting spin- 1/2 with only global pulses, provided that the interaction Hamiltonian has no isotropic component, with the exception of the global identity. We also discuss their inherent robustness to finite pulse duration and a wide range of pulse errors, as well as their potential application as building blocks for dynamically corrected gates.
One of the fundamental problems of chemistry is the difficulty in determining full quantum mechanical descriptions of molecules. Molecules contain many degrees of freedom and as a result the classical description scales factorially with the number of electrons. One way to overcome this mismatch between quantum and classical pictures is to use one quantum system, a quantum computer, to simulate another. Quantum computation is based upon a two-level system, a qubit, which is analogous to a bit in classical computation. Unlike a classical bit, a qubit can exist in a superposition of quantum states. The interaction between a quantum system and its environment results in the decay of the superposition state of the system into a probability density of classical states [1]. This is known as decoherence. A realistic quantum computer requires qubits with long coherence times; with respect to the achievable gating times, for a formal relation refer to section 1.2.2. The spin echo effect in nuclear magnetic resonance (NMR) offers a way to increase the coherence time of a qubit and has been studied as a way to mitigate decoherence in quantum computation[2][3]. By applying an appropriate Π pulse to a system under the influence of a large external field, as in NMR, the interaction term of the Hamiltonian between the qubit and environment can be eliminated[3][4]. Periodic implementation of such pulses is known as periodic dynamic decoupling (PDD). Similar to PDD, concatenated dynamic decoupling (CDD) is a periodic pulse sequence which removes the interaction term of the Hamiltonian, however in CDD pulse sequences are recursively embedded within the overall pulse train. Such pulse sequences have been shown theoretically to work for model quantum systems such as GaAs quantum dots, and theoretically such pulse sequences vastly improve the coherence time of a qubit [5]. There have been many calculations showing that by progressively increasing the number of concatenated pulse sequences increasingly longer coherence times can be achieved [6][7]. These techniques have not been tested in the laboratory and experimental implementation of these promising techniques motivates current research. There also continues to be extensive work on developing novel pulse sequences which may improve coherence times even further. The goal of this research is to experimentally determine the coherence times which can be achieved with a trapped calcium atom by applying these pulse sequences.