Grover’s quantum search algorithm achieves a quadratic speedup over classical search methods but requires an exact oracle that perfectly distinguishes marked states. We propose a modification that replaces the exact oracle with an approximate oracle constructed from a local similarity operator. This operator applies phase shifts based on the similarity between candidate states and the marked state, enabling approximate search while improving robustness to noise. Our analysis shows that the algorithm reliably amplifies marked states even under oracle imperfections, with a tunable trade-off between optimality and error resilience. We validate the method with analytical results and simulations, demonstrating efficient performance in both single- and multi-object search scenarios. Potential applications include error-prone quantum systems and quantum machine learning tasks involving similarity-based search over feature vectors.
Grover’s quantum search algorithm achieves a quadratic speedup over classical search methods but requires an exact oracle that perfectly distinguishes marked states. We propose a modification that replaces the exact oracle with an approximate oracle constructed from a local similarity operator. This operator applies phase shifts based on the similarity between candidate states and the marked state, enabling approximate search while improving robustness to noise. Our analysis shows that the algorithm reliably amplifies marked states even under oracle imperfections, with a tunable trade-off between optimality and error resilience. We validate the method with analytical results and simulations, demonstrating efficient performance in both single- and multi-object search scenarios. Potential applications include error-prone quantum systems and quantum machine learning tasks involving similarity-based search over feature vectors.
Abstract Grover’s quantum algorithm can find a marked item from an unstructured database faster than any classical algorithm, and hence it has been used for several applications such as cryptanalysis and optimization. When there exist multiple marked items, Grover’s algorithm has the property of finding one of them uniformly at random. To further broaden the application range, it was generalized so that it finds marked items with probabilities according to their priority by encoding the priority into amplitudes applied by Grover’s oracle operator. In this paper, to achieve a similar generalization, we examine a different encoding that incorporates the priority into phases applied by the oracle operator. We compare the previous and our oracle operators and observe that which one performs better depends on priority parameters. Since the priority parameters can be considered as the magnitude of the correlated phase error on Grover’s oracle operator, the analysis of our oracle operator also reveals the robustness of the original Grover’s algorithm against correlated noises. We further numerically show that the coherence between multiple marked items tends to increase the probability of finding the most prioritized one in Grover’s algorithm with our oracle operator.
The Grover algorithm is a fundamental quantum algorithm that achieves a quadratic speedup for unstructured search problems, requiring O(√N) queries instead of O(N) classically. It works by repeatedly applying an oracle and a diffusion operator to amplify the probability of marked states. This advantage makes it relevant to cryptography, optimization, constraint satisfaction, and as a general primitive via amplitude amplification in areas like quantum machine learning and simulation. However, practical implementations are severely constrained by current Noisy Intermediate Scale Qualtum machines with limited coherence, deep oracle circuits and lack of scalable QRAM, restricting demonstrations to small-scale experiments with reproducibility challenges.
Abstract Grover's algorithm provides a quadratic speedup over classical algorithms to search for marked elements in an unstructured database. The original algorithm is probabilistic, returning a marked element with bounded error. When it is known beforehand there are M marked elements in N elements, there exist several schemes to achieve the exact version that returns a marked element with certainty, by using the generalized Grover's iteration G(α, β) := Sr(β) So(α), where the phase oracle So(α) multiplies a marked state by eiα, and the phase rotation Sr(β) multiplies the initial state |ψ0⟩ (an equal-superposition of all states) by e-iβ.However, in all the existing schemes the value range of α and β is limited; for instance, in the three early schemes α and β are determined by M/N. Thus, a natural question arises: Given the phase oracle So(α) with an arbitrary angle α, or the phase rotation Sr(β) with an arbitrary angle β, can we always construct an exact search algorithm without sacrificing the quadratic speedup advantage? The significance of this problem lies not only in the expansion of mathematical form, but also in its application value. We answer the question affirmatively, by presenting a search framework with adjustable parameters. Technically, we propose the fixed-axis-rotation (FXR) method where each iteration of the search algorithm can be geometrically seen as rotating about a fixed axis on the Bloch sphere. Furthermore, two applications of the proposed search framework are developed: the first is to learn exactly the secret string hidden by the Hamming distance oracle, and the other to solve the element distinctness promise problem deterministically. The two applications correspond to the two different cases where α or β is fixed respectively.
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.
Grover's algorithm stands as a cornerstone of quantum computing, offering a theoretical quadratic speedup for unstructured search problems. This allows for finding a target item in an unsorted database of N entries in approximately the square root of N steps, a significant improvement over classical algorithms which require a number of operations on the order of N in the worst case. This speedup becomes particularly compelling for very large datasets, where classical exhaustive search becomes computationally intractable. The algorithm achieves this by leveraging fundamental quantum mechanical principles, primarily superposition and interference. It operates by preparing qubits in a uniform superposition of all possible states, then iteratively applying a quantum oracle to mark the desired state by flipping its phase, followed by a diffusion operator that amplifies the probability of the marked state. Despite its theoretical promise, real-world implementation faces substantial challenges stemming from Noisy Intermediate-Scale Quantum (NISQ) hardware, which is characterized by high error rates and short coherence times. Consequently, achieving a true, scalable quantum advantage remains a future endeavor, contingent on significant hardware advancements. Nevertheless, Grover's algorithm holds significant potential across diverse fields, including enhancing brute-force attacks on cryptographic systems, accelerating solutions for optimization problems, and speeding up search processes in machine learning and unstructured database queries.
Quantum search is among the most important algorithms in quantum computing. At its core is quantum amplitude amplification, a technique that achieves a quadratic speedup over classical search by combining two global reflections: the oracle, which marks the target, and the diffusion operator, which reflects about the initial state. We show that this speedup can be preserved when the oracle is the only global operator, with all other operations acting locally on non-overlapping partitions of the search register. We present a recursive construction that, when the initial and target states both decompose as tensor products over these chosen partitions, admits an exact closed-form solution for the algorithm's dynamics. This is enabled by an intriguing degeneracy in the principal angles between successive reflections, which collapse to just two distinct values governed by a single recursively defined angle.
Applied to unstructured search, a problem that naturally satisfies the tensor decomposition, the approach retains the O(N) oracle complexity of Grover search when each partition contains at least log2(log2N) qubits. On an 18-qubit search problem, partitioning into two stages reduces the non-oracle circuit depth by as much as 51%-96% relative to Grover, requiring up to 9% additional oracle calls. For larger problem sizes this oracle overhead rapidly diminishes, and valuable depth reductions persist when the oracle circuit is substantially deeper than the diffusion operator. More broadly, these results show that a global diffusion operator is not necessary to achieve the quadratic speedup in quantum search, offering a new perspective on this foundational algorithm. Moreover, the scalar reduction at the heart of our analysis inspires and motivates new directions and innovations in quantum algorithm design and evaluation.