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.
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.
Given a Boolean function f provided as a black-box with n variables, this paper will propose a quantum algorithm for testing if a certain variable is junta or ε-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) queries to the constructed black-box. The proposed algorithm considers a Boolean function of general form, contrary to relevant algorithms proposed by others.
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 consider the problem of deciding whether an n-qubit unitary (or n-bit Boolean function) is ε1-close to some k-junta or ε2-far from every k-junta, where k-junta unitaries act non-trivially on at most k qubits and as the identity on the rest, and k-junta Boolean functions depend on at most k variables. For constant numbers ε1,ε2 such that 0<ε1<ε2<1, we show the following.1.A non-adaptive O(klogk)-query tolerant (ε1,ε2)-tester for k-junta unitaries when 22ε1<ε2.2.A non-adaptive tolerant (ε1,ε2)-tester for k-junta Boolean functions with O(klogk) quantum queries when 4ε1<ε2.3.A 2O(k)-query tolerant (ε1,ε2)-tester for k-junta unitaries for any ε1,ε2.The first algorithm provides an exponential improvement over the best-known quantum algorithms [CLL24, ADG25]. The second algorithm shows an exponential quantum advantage over any non-adaptive classical algorithm [CDL+24]. The third tester gives the first tolerant junta unitary testing result for an arbitrary gap.
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).