Enhanced quantum searching via entanglement and partial diffusion

Abstract

In this paper, we will define a quantum operator that performs the inversion about the mean only on a subspace of the system (Partial Diffusion Operator). This operator is used in a quantum search algorithm that runs in O( N/M ) for searching an unstructured list of size N with M matches such that 1 ≤ M ≤ N . We will show that the performance of the algorithm is more reliable than known fixed operators quantum search algorithms especially for multiple matches where we can get a solution after a single iteration with probability over 90% if the number of matches is approximately more than one-third of the search space. We will show that the algorithm will be able to handle the case where the number of matches M is unknown in advance such that 1 ≤ M ≤ N in O( N/M ). A performance comparison with Grover's algorithm will be provided.

Quantum search for unknown number of target items by hybridizing fixed-point method with trail-and-error method*

“…In this section, we first discuss the comparisons between our algorithm and the existing quantum search algorithms [27,32,33,[41][42][43] in the case of unknown λ and then give a brief conclusion. Table 1 lists the method, query complexity, and phase(s) of our algorithm and other algorithms.…”

“…Compared with the fixed-point quantum search algorithms [32,33] Compared with the trial-and-error quantum search algorithms [27,[41][42][43], first, with respect to the method, our algorithm does not contain randomness (i.e., random selection of the number of iterations), and enables the success probability always no less than an arbitrary given lower bound between 0 and 1 after the number of iterations exceeds the critical value (i.e., l cri defined by Eq. (20)).…”

“…Another method for the case of unknown λ is the trial-and-error method [27,[41][42][43]. In this regard, the original work was proposed by Boyer et al [27], which trials Grover's algorithm multiple times, where the number of iterations is randomly selected from an exponentially increasing interval along with the number of trials.…”

Enhancing the security of quantum communication by hiding the message in a superposition

“…e l s e v i e r . c o m / l o c a t e / i n s diffusion operator [7]. To completely hide/unhide states in a superposition of N qubits, Oð ffiffiffiffiffiffiffiffiffiffiffiffi ffi N À 1 p Þ iterations of these operators are required.…”

“…The diagonal representation of D p when applied on a 3-qubit system can take the following form [7],…”