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.
We consider a quantum computer consisting of n spins with an arbitrary but fixed pair-interaction Hamiltonian and describe how to simulate other pair-interactions by interspersing the natural time evolution with fast local transformations. Calculating the minimal time overhead of such a simulation leads to a convex optimization problem. Lower and upper bounds on the minimal time overhead are derived in terms of chromatic indices of interaction graphs and spectral majorization criteria. These results classify Hamiltonians with respect to their computational power. For a specific Hamiltonian, namely \sigma_z\otimes\sigma_z-interactions between all spins, the optimization is mathematically equivalent to a separability problem of n-qubit density matrices. We compare the complexity defined by such a quantum computer with the usual gate complexity.
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.
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.
In previous papers we have considered mutual simulation of n-partite pair-interaction Hamiltonians. We have focussed on the running time overhead of general simulations, while considering the required number of time steps only for special cases (decoupling and time-reversal). These two complexity measures differ significantly. Here we derive lower bounds on the number of time steps for general simulations. In particular, the simulation of interaction graphs with irrational spectrum requires at least n steps. We discuss as examples graphs that correspond to graph codes and nearest neighbor interactions in 1- and 2-dimensional lattices. In the latter case the lower bounds are almost tight.
The most popular model of quantum computation is the quantum circuit model using single and two qubit gates as elementary transformations. Definitions of quantum complexity usually refer to this model. In contrast, we consider the physical interactions among the qubits as the basic computational resource that allows to carry out complex transformations on the quantum register. One method to compare the computational power of different interactions is to examine the complexity of the so-called mutual simulation of interaction Hamiltonians. The physical interaction can simulate other interactions by interspersing the natural time evolution with external control operations. In previous papers we have considered mutual simulation of n-partite pair-interaction Hamiltonians. We have focussed on the running time overhead of general simulations, while considering the required number of time steps only for special cases (decoupling and time-reversal). These two complexity measures differ significantly. Here we derive lower bounds on the number of time steps for simulations of general interaction graphs. In particular, the simulation of interaction graphs with irrational spectrum requires at least n steps. We discuss as examples graphs that correspond to graph codes and nearest neighbor interactions in 1- and 2-dimensional lattices. In the latter case the lower bounds are almost tight.