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).
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.
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
dgeq2. Furthermore, we extend the second method to
t-local qudit Hamiltonians, where
tgeq2and
dgeq2, by using orthogonal arrays of strength
t. We also establish a characterization of orthogonal arrays of strength
tby 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.
We propose dynamical control schemes for Hamiltonian simulation in many-body quantum systems that avoid instantaneous control operations and rely solely on realistic bounded-strength control Hamiltonians . Each simulation protocol consists of periodic repetitions of a basic control block, constructed as a modification of an 'Eulerian decoupling cycle,' that would otherwise implement a trivial (zero) target Hamiltonian. For an open quantum system coupled to an uncontrollable environment, our approach may be employed to engineer an effective evolution that simulates a target Hamiltonian on the system while suppressing unwanted decoherence to the leading order, thereby allowing for dynamically corrected simulation . We present illustrative applications to both closed- and open-system simulation settings, with emphasis on simulation of non-local (two-body) Hamiltonians using only local (one-body) controls . In particular, we provide simulation schemes applicable to Heisenberg-coupled spin chains exposed to general linear decoherence, and show how to simulate Kitaevʼs honeycomb lattice Hamiltonian starting from Ising-coupled qubits, as potentially relevant to the dynamical generation of a topologically protected quantum memory. Additional implications for quantum information processing are discussed.
We introduce a high-order dynamical decoupling (DD) scheme for arbitrary bounded system-bath interactions in the weak-coupling regime. Given any decoupling group
G that averages the interaction to zero, our construction {guarantees the existence of pulse sequences with at most
(∣G∣−1)K pulses}, while canceling all error terms linear in the system-bath coupling strength up to order
K in the total evolution time. As a corollary, for an
n-qubit system with
k-local system-bath interactions, we obtain an
O(nk−1K)-pulse sequence, a significant improvement over existing schemes with
O(exp(n)) pulses (for
k=O(1)). The construction is obtained via a mapping to the continuous necklace-splitting problem, which asks how to cut a multi-colored interval into pieces that give each party the same share of every color. We provide explicit pulse sequences for suppressing general single-qubit decoherence, prove that the pulse count is asymptotically optimal, and verify the predicted error scaling in numerical simulations. For the same number of pulses, we observe that our sequences outperform the state-of-the-art Quadratic DD in the weak-coupling regime. We also construct explicit high-order sequences for suppressing representative 2-local noise models. Finally, the same construction extends to suppress slow, time-dependent classical noise and to filter-function design.
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.
We address the problem of simulating pair-interaction Hamiltonians in n-node quantum networks where the subsystems have arbitrary, possibly different, dimensions. We show that any pair interaction can be used to simulate any other by applying sequences of appropriate local control sequences. Conditions on time optimal simulation are formulated in terms of spectral majorization of matrices characterizing the coupling parameters. Efficient schemes for decoupling and time reversal can be constructed from orthogonal arrays provided that the dimensions of all nodes are equal to the same prime power. Moreover, we consider a specific system of n harmonic oscillators with bilinear interaction. In this case, decoupling can efficiently be achieved using the combinatorial concept of difference schemes. For this type of interaction we present optimal schemes for inversion.