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arXiv:quant-ph/0409135 · IEEE Transactions on Information Theory

Equivalence of Decoupling Schemes and Orthogonal Arrays

Martin Rötteler, Paweł Wocjan

Published Aug 29, 2006• 33 citations
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Abstract

Decoupling schemes are used in quantum information processing to selectively switch off unwanted interactions in a multipartite Hamiltonian. A decoupling scheme consists of a sequence of local unitary operations which are applied to the system's qudits and alternate with the natural time evolution of the Hamiltonian. Several constructions of decoupling schemes have been given in the literature. Here we focus on two such schemes. The first is based on certain triples of submatrices of Hadamard matrices that are closed under pointwise multiplication (see Leung, “Simulation and reversal of$n$-qubit Hamiltonians using Hadamard matrices,” J. Mod. Opt., vol. 49, pp. 1199–1217, 2002), the second uses orthogonal arrays (see Stollsteimer and Mahler, “Suppression of arbitrary internal couplings in a quantum register,” Phys. Rev. A., vol. 64, p. 052301, 2001). We show that both methods lead to the same class of decoupling schemes. We extend the first method to$2$-local qudit Hamiltonians, where$d geq 2$. Furthermore, we extend the second method to$t$-local qudit Hamiltonians, where$t geq 2$and$d geq 2$, by using orthogonal arrays of strength$t$. We also establish a characterization of orthogonal arrays of strength$t$by showing that they are equivalent to decoupling schemes for$t$-local Hamiltonians which have the property that they can be refined to have time-slots of equal length. The methods used to derive efficient decoupling schemes are based on classical error-correcting codes.

DOI
10.1109/tit.2006.880059

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Decoupling schemes are used in quantum information processing to selectively switch off unwanted interactions in a multipartite Hamiltonian. A decoupling scheme consists of a sequence of local unitary operations which are applied to the system's qudits and alternate with the natural time evolution of the Hamiltonian. Several constructions of decoupling schemes have been given in the literature. Here we focus on two such schemes. The first is based on certain triples of submatrices of Hadamard matrices that are closed under pointwise multiplication (see Leung, “Simulation and reversal ofnnn-qubit Hamiltonians using Hadamard matrices,” J. Mod. Opt., vol. 49, pp. 1199–1217, 2002), the second uses orthogonal arrays (see Stollsteimer and Mahler, “Suppression of arbitrary internal couplings in a quantum register,” Phys. Rev. A., vol. 64, p. 052301, 2001). We show that both methods lead to the same class of decoupling schemes. We extend the first method to222-local qudit Hamiltonians, wheredgeq2d geq 2dgeq2. Furthermore, we extend the second method tottt-local qudit Hamiltonians, wheretgeq2t geq 2tgeq2anddgeq2d geq 2dgeq2, by using orthogonal arrays of strengthttt. We also establish a characterization of orthogonal arrays of strengthtttby showing that they are equivalent to decoupling schemes forttt-local Hamiltonians which have the property that they can be refined to have time-slots of equal length. The methods used to derive efficient decoupling schemes are based on classical error-correcting codes.
The task of decoupling, i.e., removing unwanted internal couplings of a quantum system and its couplings to an environment, plays an important role in quantum control theory. There are many efficient decoupling schemes based on combinatorial concepts such as orthogonal arrays, difference schemes, and Hadamard matrices. So far these combinatorial decoupling schemes have relied on the ability to effect sequences of instantaneous, arbitrarily strong control Hamiltonians (bang-bang controls). To overcome the shortcomings of bang-bang control, Viola and Knill proposed a method called ``Eulerian decoupling'' that allows the use of bounded-strength controls for decoupling. However, their method was not directly designed to take advantage of the local structure of internal couplings and couplings to an environment that typically occur in multipartite quantum systems. In this paper we define a combinatorial structure called Eulerian orthogonal array. It merges the desirable properties of orthogonal arrays and Eulerian cycles in Cayley graphs (that are the basis of Eulerian decoupling). We show that this structure gives rise to decoupling schemes with bounded-strength control Hamiltonians that can be used to remove both internal couplings and couplings to an environment of a multipartite quantum system. Furthermore, we show how to construct Eulerian orthogonal arrays having good parameters in order to obtain efficient decoupling schemes.
We address the task of switching off the Hamiltonian of a system by removing all internal and system-environment couplings. We propose dynamical decoupling schemes that use only bounded-strength controls for quantum many-body systems with local system Hamiltonians and local environmental couplings. To do so, we introduce the combinatorial concept of balanced-cycle orthogonal arrays (BOAs) and show how to construct them from classical error-correcting codes. The derived decoupling schemes may be useful as a primitive for more complex schemes, e.g., for Hamiltonian simulation. For the case of n qubits and a two-local Hamiltonian, the length of the resulting decoupling scheme scales as O(n logn), improving over the previously best-known schemes that scaled quadratically with n. More generally, using BOAs constructed from families of Bose-Chaudhuri-Hocquenghem (BCH) codes, we show that bounded-strength decoupling for any ℓ-local Hamiltonian, where ℓ ≥ 2, can be achieved using decoupling schemes of length at most O(nℓ-1log n).
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.
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.
Orthogonal arrays (OAs) are combinatorial configurations with applications in experimental design, error-correcting codes, and quantum information. Local unitary (LU) equivalence provides a natural framework for classifying multipartite entangled states. Using the correspondence between OAs and quantum states, Goyeneche and Życzkowski [Phys. Rev. A, 2014, 90: 022316] posed the problem of determining when OAs are LULULU equivalent. In this paper, we construct OAs from generator matrices and establish conditions for Fourier-based LULULU equivalence of OAs and irredundant orthogonal arrays (IrOAs). For prime alphabets, we identify the Fourier partners of the linear OAs considered here with the arrays of their corresponding dual codes. This gives an explicit connection between LULULU equivalence and coding theory. We also identify families of LULULU equivalent OAs arising from specific classes of linear codes.
Abstract Two orthogonal arrays (OAs) are local unitary (LU) equivalent if they lead to LU-equivalent quantum states. Finding LU-equivalent OAs is an open problem posed by Goyeneche et al ( Phys. Rev. A 90 022316). In this paper, we develop methods of constructing infinite families of LU-equivalent OAs, both fixed-level and mixed-level, by investigating the constructions, structures and properties of OAs. We consequently provide a solution to the open problem. The results significantly advance our understanding of LU-equivalent quantum states and their potential applications in quantum information processing.