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arXiv:1903.04762

Measuring Hamming Distance between Boolean Functions via Entanglement Measure

Khaled El-Wazan

Published Mar 12, 2019• 0 citations
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Abstract

In this paper, we present a fast quantum algorithm to measure the Hamming distance between two or more Boolean functions provided as black-boxes. The proposed algorithm constructs a new black-box with a certain property which is utilized to solve this problem. The introduced algorithm converts measuring the Hamming distance between Boolean functions to measuring entanglement between qubits, using concurrence entanglement measure.

DOI
10.48550/arxiv.1903.04762

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In this paper, we present a fast quantum algorithm to measure the Hamming distance between two or more Boolean functions provided as black-boxes. The proposed algorithm constructs a new black-box with a certain property which is utilized to solve this problem. The introduced algorithm converts measuring the Hamming distance between Boolean functions to measuring entanglement between qubits, using concurrence entanglement measure.
Given a black-box representing an unknown Boolean function fff of nnn variables, in this paper we propose a fast quantum algorithm to test whether or not a certain variable in the function fff is a junta variable. The proposed algorithm creates entanglement between the variable under test and an auxiliary qubit, where the entanglement is measured using concurrence measure to decide if the variable is junta. The paper shows applications to the proposed algorithm in learning and categorization of Boolean functions.
We present a novel quantum algorithm to evaluate the hamming distance between two unknown oracles via measuring the degree of entanglement between two ancillary qubits. In particular, we use the power of the entanglement degree based quantum computing model that preserves at most the locality of interactions within the quantum model structure. This model uses one of two techniques to retrieve the solution of a quantum computing problem at hand. In the first technique, the solution of the problem is obtained based on whether there is an entanglement between the two ancillary qubits or not. In the second, the solution of the quantum computing problem is obtained as a function in the concurrence value, and the number of states that can be generated from the Boolean variables. The proposed algorithm receives two oracles, each oracle represents an unknown Boolean function, then it measures the hamming distance between these two oracles. The hamming distance is evaluated based on the second technique. It is shown that the proposed algorithm provides exponential speedup compared with the classical counterpart for Boolean functions that have large numbers of Boolean variables. The proposed algorithm is explained via a case study. Finally, employing recently developed experimental techniques, the proposed algorithm has been verified using IBM's quantum computer simulator.
Given a Boolean function fff provided as a black-box with nnn variables, this paper will propose a quantum algorithm for testing if a certain variable is junta or εεε-far from being junta. The proposed algorithm constructs another black-box using two copies of the provided black-box. The constructed black-box is used with the partial diffusion operator in an amplitude amplification technique to test whether the variable being tested is junta or not, using O(2n)\mathcal{O}(\sqrt{2^n})O(2n​) queries to the constructed black-box. The proposed algorithm considers a Boolean function of general form, contrary to relevant algorithms proposed by others.
In this paper, we study the following question: given a black box performing some unknown quantum measurement on a multi-qudit system, how do we test whether this measurement has certain property or is far away from having this property. We call this task \textit{property testing} of quantum measurement. We first introduce a metric for quantum measurements, and show that it possesses many nice features. Then we show that, with respect to this metric, the following classes of measurements can be efficiently tested: 1. the stabilizer measurements, which play a crucial role for quantum error correction; 2. the kkk-local measurements, i.e. measurements whose outcomes depend on a subsystem of at most kkk qudits; 3. the permutation-invariant measurements, which include those measurements used in quantum data compression, state estimation and entanglement concentration. In fact, all of them can be tested with query complexity independent of the system's dimension. Furthermore, we also present an algorithm that can test any finite set of measurements. Finally, we consider the following natural question: given two black-box measurement devices, how do we estimate their distance? We give an efficient algorithm for this task, and its query complexity is also independent of the system's dimension. As a consequence, we can easily test whether two unknown measurements are identical or very different.
We consider the quantum query model for computing Boolean functions. The definition of the function is known, but a black box contains the input X = (x1, x2, ..., xn). Black box can be accessed by querying xivalues. The goal is to develop an algorithm, which would compute the function value for arbitrary input using as few queries to the black box as possible. We present two different quantum query algorithms for computing the basic Boolean function - logical AND of two bits. Both algorithms use only one query to determine the function value. Correct answer probability for the first algorithm is 80%, but for the second algorithm it is 90%. To compute this function with the same probability in the classical model, two queries are necessary. We analyze algorithms implementation differences and propose a way to extend the first algorithm to compute AND of two functions.