Equivalence of Decoupling Schemes and Orthogonal Arrays

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.