In this article, we introduce an innovative hybrid quantum search algorithm, the Robust Non-oracle Quantum Search (RNQS), which is specifically designed to efficiently identify the minimum value within a large set of random numbers. Distinct from the Grover’s algorithm, the proposed RNQS algorithm circumvents the need for an oracle function that describes the true solution state, a feature often impractical for data science applications. Building on existing non-oracular quantum search algorithms, RNQS enhances robustness while substantially reducing running time. The superior properties of RNQS have been demonstrated through careful analysis and extensive empirical experiments. Our findings underscore the potential of the RNQS algorithm as an effective and efficient solution to combinatorial optimization problems in the realm of quantum computing.
ℓ0 Optimization with Robust Non-Oracular Quantum Search
In this article, we introduce an innovative hybrid quantum search algorithm, the Robust Non-oracle Quantum Search (RNQS), which is specifically designed to efficiently identify the minimum value within a large set of random numbers. Distinct from the Grover’s algorithm, the proposed RNQS algorithm circumvents the need for an oracle function that describes the true solution state, a feature often impractical for data science applications. Building on existing non-oracular quantum search algorithms, RNQS enhances robustness while substantially reducing running time. The superior properties of RNQS have been demonstrated through careful analysis and extensive empirical experiments. Our findings underscore the potential of the RNQS algorithm as an effective and efficient solution to combinatorial optimization problems in the realm of quantum computing.
With the rapid development of quantum computers, quantum algorithms have been studied extensively. However, quantum algorithms tackling statistical problems are still lacking. In this paper, we propose a novel non-oracular quantum adaptive search (QAS) method for the best subset selection problems. QAS performs almost identically to the naive best subset selection method but reduces its computational complexity from O(D) to O(Dlog2D), where D=2p is the total number of subsets over p covariates. Unlike existing quantum search algorithms, QAS does not require the oracle information of the true solution state and hence is applicable to various statistical learning problems with random observations. Theoretically, we prove QAS attains any arbitrary success probability q∈(0.5,1) within O(log2D) iterations. When the underlying regression model is linear, we propose a quantum linear prediction method that is faster than its classical counterpart. We further introduce a hybrid quantum-classical strategy to avoid the capacity bottleneck of existing quantum computing systems and boost the success probability of QAS by majority voting. The effectiveness of this strategy is justified by both theoretical analysis and extensive empirical experiments on quantum and classical computers.
Finding the global minimum of a function with many local extrema is well known to be a challenging computational task. We propose a hybrid quantum-classical algorithm for this optimisation problem, based on a combination of Grover quantum search and the classical Newton-Raphson method. The approach is tailored to situations with a large number of false minima. The main idea is to use quantum search to efficiently locate the vicinity of the true optimum, after which a classical algorithm can solve the problem in only a few iterations. We specifically address the nontrivial question of choosing the discretisation of the function domain, which is critical for quantum approaches to optimisation of continuous functions. While we do not expect that this method will always be superior, we identify a paradigmatic case of a function for which we demonstrate that, remarkably, our hybrid method quadratically outperforms a purely classical approach (measuring performance by the number of function/oracle calls). We evaluate key performance metrics of the algorithm in numerical experiments. We furthermore discuss its possible extensions as well as the potential relevance of hybrid approaches in the context of quantum cognition.
We present a quantum algorithmic routine that extends the realm of Grover-based heuristics for tackling combinatorial optimization problems with arbitrary efficiently computable objective and constraint functions. Building on previously developed quantum methods that were primarily restricted to linear constraints, we generalize the approach to encompass a broader class of problems in discrete domains. To evaluate the potential of our algorithm, we assume the existence of sufficiently advanced logical quantum hardware. With this assumption, we demonstrate that our method has the potential to outperform state-of-the-art classical solvers and heuristics in terms of both runtime scaling and solution quality. The same may be true for more realistic implementations, as the logical quantum algorithm can achieve runtime savings of up to 102−103.
Quantum computing has advanced rapidly in recent years and has shown advantages in a variety of domains. In this paper, we investigate its potential for discrete simulation optimization in the fixed-confidence setting, a fundamental problem in the simulation literature. We first introduce a quantum simulation oracle that prepares a coherent superposition over all candidate solutions and provides the foundation for quantum algorithm design. Based on this oracle, we develop the first Grover-search-based quantum algorithm for discrete simulation optimization, called SOGAS. In particular, SOGAS uses a binary-search framework to progressively eliminate suboptimal solutions while carefully controlling the error probability, and eventually identifies a set of near-optimal solutions. We prove that SOGAS returns a near-optimal solution with probability at least the prescribed confidence level and achieves a quadratic speedup in the dependence of query complexity on the number of candidate solutions. Numerical experiments further show that SOGAS substantially outperforms classical benchmarks and provide empirical evidence for quantum advantage in discrete simulation optimization.
The oracle identification problem (OIP) was introduced by Ambainis et.al. [4], which is given as a set S of M oracles and a hidden oracle f .Our task is to figure out which oracle in S is equal to the hidden f by doing queries to f .OIP includes several problems such as Grover Search as special cases.In this paper, we design robust algorithms, i.e., those which are tolerant against noisy oracles, for OIP.Our results include: (i) For any oracle set S such that ∣S∣ is polynomial in N , O(N) queries are enough to identify the hidden oracle, which is obviously optimal since this OIP includes Grover Search as a special case, (ii) For the case that ∣S∣≤2Nd(d<1) , we design an algorithm whose query complexity is O(NlogM/logN) and matches the lower bound proved in [4].(Hi) We can furthermore design a robust algorithm whose complexity changes smoothly between the complexity of (ii) and the complexity of recovering all information about the hidden oracle whose complexity is O(N) as showed by Buhrman et.al. in [11].Thus our new algorithms are not only robust but also their query complexities are even better than the previous noiseless case [4],
IntroductionFor a search problem associated with a unsorted database, the remarkable Grover's quantum algorithm (1) provides a quadratic speedup over its classical counterpart.The search problem can be described as follows: for a given function f , there exists one unknown element in the set {0, 1, ..., N -1} that satisfies f (x)=-1, say x = τ, whereas the other N -1o n e sg i v e f (x)=1.How many times of evaluations of f are required to determine the element τ for f (τ)=-1?Through a conventional algorithm, one needs O(N) trials to achieve this aim.How about the utility of quantum algorithm for the search?For the scenario in the quantum world, the search problem can be rephrased in the quantum mechanical language: for a given unitary operator I τ , that is sometimes called the oracle operator, and a set of state vectors (orthonormal basis): s = {|0 , |1 , ..., |N -1 }, I τ |x = |x for all states in the set except I τ |x = -|x for x = τ.H owma n yqueriesofI τ are required to determine |τ ?By Grover's algorithm, one needs only O( √ N) quantum mechanical steps to find the marked state |τ out.It has been shown that Grover's algorithm is optimal since it needs minimal oracle calls to perform a quantum search (2).The quantum searching process will be briefly reviewed as follows.The first step of Grover's algorithm is to prepare a superposition state of all elements with uniform probability amplitude:Then apply the Grover kernel G = -I η I τ to |s ,w h er eI η is a unitary operator and contains no bias against the marked state.For large N,a f t e ra b o u tm = π √ N/4 repetitions of G operations, the probability to observe |τ is close to one, i.e., G m |s ∼ |τ .( 2 ) Since every single G involves one query of I τ ,onlyO( √ N) searching steps are required for a quantum search task.In what follows, we will first investigate on the general SU(2) formulation for the kernel of Grover's searching operator G.The discussions of quantum searching certainty, robustness, and the analog analogue version of the Grover's algorithm will be given afterwards.