Optimizing Boolean function components to have the minimum number of inputs in order to reduce the memory space required during these functions in computing devices is a significant demand. This paper proposes a quantum computation approach based on the degree-of-entanglement quantum computation model to estimate the number of junta variables of an unknown Boolean function presented through an oracle. The time complexity of the developed quantum approach is independent of the number of inputs and depends on an allowable assigned error ϵ. Thus, the time complexity of the developed algorithm is O(ϵ−2), compared to O(2n+1) in the traditional approach. Also, the memory space of the developed approach is linear, O(2n+4), in terms of the number of inputs compared to the exponential memory space O(2n+1) using the traditional approach. Therefore, the developed quantum approach has exponential supremacy in comparison to the traditional approach. The developed approach was implemented practically using both the Qiskit simulator and the IBM real quantum computer. The obtained results expose high statistical fidelities between the empirical and theoretical results.
Optimizing Boolean function components to have the minimum number of inputs in order to reduce the memory space required during these functions in computing devices is a significant demand. This paper proposes a quantum computation approach based on the degree-of-entanglement quantum computation model to estimate the number of junta variables of an unknown Boolean function presented through an oracle. The time complexity of the developed quantum approach is independent of the number of inputs and depends on an allowable assigned error ϵ. Thus, the time complexity of the developed algorithm is O(ϵ−2), compared to O(2n+1) in the traditional approach. Also, the memory space of the developed approach is linear, O(2n+4), in terms of the number of inputs compared to the exponential memory space O(2n+1) using the traditional approach. Therefore, the developed quantum approach has exponential supremacy in comparison to the traditional approach. The developed approach was implemented practically using both the Qiskit simulator and the IBM real quantum computer. The obtained results expose high statistical fidelities between the empirical and theoretical results.
Detecting junta variables is a critical issue in Boolean function analysis, circuit design optimization, and machine learning feature selection. In this paper, we investigate a novel quantum computation algorithm based on the Mz operator. The algorithm takes in an unknown oracle concealing a Boolean function with n variables and an unknown input state, which can be quantum or classical. The input state can be complete or incomplete quantum basis states, and it can be a weighted or uniform superposition of basis states. The proposed approach determines whether a given variable is a junta with a time cost ofO(2/ϵ2)and a memory cost of2n+6. The algorithm is analyzed and experimentally implemented using the Qiskit simulator and IBM’s real quantum computer. Experimental results show that the proposed approach achieved a quantum supremacy ratio 6300% higher than that of the classical method when verifying junta variables for Boolean functions with 12 variables. The results suggest that the proposed quantum method can verify junta variables in scenarios beyond the capabilities of current classical or quantum methods.
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.
Junta testing for Boolean functions has sparked a long line of work over recent decades in theoretical computer science, and recently has also been studied for unitary operators in quantum computing. Tolerant junta testing is more general and challenging than the standard version. While optimal tolerant junta testers have been obtained for Boolean functions, there has been no knowledge about tolerant junta testers for unitary operators, which was thus left as an open problem in [Chen, Nadimpalli, and Yuen, SODA2023]. In this paper, we settle this problem by presenting the first algorithm to decide whether a unitary is ε1-close to some quantum k-junta or is ε2-far from any quantum k-junta, where an n-qubit unitary U is called a quantum k-junta if it only non-trivially acts on just k of the n qubits. More specifically, we present a tolerant tester with ε1=8ρε, ε2=ε, and ρ∈(0,1), and the query complexity is O(ε2ρ(1−ρ)kklogk), which demonstrates a trade-off between the amount of tolerance and the query complexity. Note that our algorithm is non-adaptive which is preferred over its adaptive counterparts, due to its simpler as well as highly parallelizable nature. At the same time, our algorithm does not need access to U†, whereas this is usually required in the literature.
In this paper, we study the following variant of the junta learning problem. We are given oracle access to a Boolean function f on n variables that only depends on k variables, and, when restricted to them, equals some predefined function h. The task is to identify the variables the function depends on. When h is the XOR or the OR function, this gives a restricted variant of the Bernstein-Vazirani or the combinatorial group testing problem, respectively. We analyse the general case using the adversary bound, and give an alternative formulation for the quantum query complexity of this problem. We construct optimal quantum query algorithms for the cases when h is the OR function (complexity is Θ(k)) or the exact-half function (complexity is Θ(k1/4)). The first algorithm resolves an open problem from arXiv:1210.1148. For the case when h is the majority function, we prove an upper bound of O(k1/4). All these algorithms can be made exact. We obtain a quartic improvement when compared to the randomised complexity (if h is the exact-half or the majority function), and a quadratic one when compared to the non-adaptive quantum complexity (for all functions considered in the paper).
Given a black-box representing an unknown Boolean function f of n variables, in this paper we propose a fast quantum algorithm to test whether or not a certain variable in the function f 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 propose a quantum algorithm (in the form of a quantum oracle) that estimates the closeness of a given Boolean function to one that satisfies the ``strict avalanche criterion'' (SAC). This algorithm requires n queries of the Boolean function oracle, where n is the number of input variables, this is fewer than the queries required by the classical algorithm to perform the same task. We compare our approach with other quantum algorithms that may be used for estimating the closeness to SAC and it is shown our algorithm verifies SAC with the fewest possible calls to quantum oracle and requires the fewest samples for a given confidence bound.