The storage efficiency of hash codes and their application in the fast approximate nearest neighbor search, along with the explosion in the size of available labeled image datasets caused an intensive interest in developing learning based hash algorithms recently. In this paper, we present a learning based hash algorithm that utilize ordinal information of feature vectors. We have proposed a novel mathematically differentiable approximation of argmax function for this hash algorithm. It has enabled seamless integration of hash function with deep neural network architecture which can exploit the rich feature vectors generated by convolutional neural networks. We have also proposed a loss function for the case that the hash code is not binary and its entries are digits of arbitrary k-ary base. The resultant model comprised of feature vector generation and hashing layer is amenable to end-to-end training using gradient descent methods. In contrast to the majority of current hashing algorithms that are either not learning based or use hand-crafted feature vectors as input, simultaneous training of the components of our system results in better optimization. Extensive evaluations on NUS-WIDE, CIFAR-10 and MIRFlickr benchmarks show that the proposed algorithm outperforms state-of-art and classical data agnostic, unsupervised and supervised hashing methods by 2.6% to 19.8% mean average precision under various settings.

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).

We advocate a new approach for addressing hidden structure problems and finding efficient quantum algorithms. We introduce and investigate the hidden symmetry subgroup problem (HSSP), which is a generalization of the well-studied hidden subgroup problem (HSP). Given a group acting on a set and an oracle whose level sets define a partition of the set, the task is to recover the subgroup of symmetries of this partition inside the group. The HSSP provides a unifying framework that, besides the HSP, encompasses a wide range of algebraic oracle problems, including quadratic hidden polynomial problems. While the HSSP can have provably exponential quantum query complexity, we obtain efficient quantum algorithms for various interesting cases. To achieve this, we present a general method for reducing the HSSP to the HSP, which works efficiently in several cases related to symmetries of polynomials. The HSSP therefore connects in a rather surprising way certain hidden polynomial problems with the HSP. Using this connection, we obtain the first efficient quantum algorithm for the hidden polynomial problem for multivariate quadratic polynomials over fields of constant characteristic. We also apply the new methods to polynomial function graph problems and present an efficient quantum procedure for constant degree multivariate polynomials over any field. This result improves in several ways the currently known algorithms.

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.