This paper presents the Single-Rotation Quantum Search Algorithm, which achieves amplitude-amplification with exactly one oracle call and one rotation operation. We prove that for unstructured search over N items, a single rotation by π/4 radians is both necessary and sufficient to amplify the marked state probability from 1/N to 1/2. This represents the theoretical minimum: no algorithm can succeed with fewer than one oracle call, and no additional rotations are needed beyond the single π/4 rotation. We provide explicit closed forms for the rotation operator with numerical values, demonstrating complete physical realizability. The algorithm is implemented by one oracle call, one diffusion call, and log(N) reflections. For N = 2n search spaces, the algorithm achieves success probability exactly 1/2 with one iteration, compared to O( N ) iterations for Grover’s algorithm.
This paper presents the Single-Rotation Quantum Search Algorithm, which achieves amplitude-amplification with exactly one oracle call and one rotation operation. We prove that for unstructured search over N items, a single rotation by π/4 radians is both necessary and sufficient to amplify the marked state probability from 1/N to 1/2. This represents the theoretical minimum: no algorithm can succeed with fewer than one oracle call, and no additional rotations are needed beyond the single π/4 rotation. We provide explicit closed forms for the rotation operator with numerical values, demonstrating complete physical realizability. The algorithm is implemented by one oracle call, one diffusion call, and log(N) reflections. For N = 2n search spaces, the algorithm achieves success probability exactly 1/2 with one iteration, compared to O( N ) iterations for Grover’s algorithm.
The author has recently presented the Single-Iteration Quantum Search Algorithm, which achieves amplitude-amplification with exactly one oracle call and one π/4 rotation [1]. For N = 2n search spaces, the algorithm achieves success probability exactly 1/2 with one iteration of oracle operator and one π/4 rotation with log(N) different diffusion calls, compared to O( N ) iterations (O( N ) oracle calls and O( N ) same diffusion calls) for Grover’s algorithm. This work presents a simpler and cleaner derivation for one oracle call and log(N) diffusion calls, while the original proof was based on Cartan-Dieudonné theorem [1]. The standard lower bounds for unstructured quantum search still apply even though the number of oracle calls is reduced from O( N ) to O(1) and the number of diffusion calls is reduced from O( N ) to O(logN).
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.
Grover’s algorithm achieves O( N ) query complexity for unstructured search, a result proven optimal by Zalka for algorithms using a fixed oracle operator. This paper presents the Exponential Speedup Algorithm, a modified amplitude amplification algorithm that escapes Zalka’s optimality bound by using a sequence of iteration-dependent rotation operators U0, U1, ..., UK−1, where each operator implements a different rotation angle that depends explicitly on the iteration number k, rather than repeatedly applying a single fixed operator. The algorithm achieves geometric series convergence in the amplitude ratio, where r(k) = βk for a parameter β > 1, compared to the arithmetic series r(k) ≈ 2k + 1 in standard Grover. This geometric growth reduces the iteration count from O( N ) to O(logN) = O(n), where N = 2n. The mathematical framework for this approach was established in the author’s previous work, which proved that K = O(logN) iterations suffice to amplify the marked state probability from 1/N to ≥1/2. This paper addresses three questions left open in that work, rigorously proving that: 1) Zalka’s O( N ) lower bound applies only to algorithms using fixed operators; 2) the iteration-dependent rotation operators Uk are unitary and physically realizable; and 3) the explicit N × N unitary matrix for Uk can be derived with closed-form expressions for all matrix elements.
Oracle-based quantum computation reframes efficiency from classical time/space costs to query complexity, i.e., the number of accesses to a black-box unitary oracle Uf needed to reveal a hidden property or target state. This study comparatively analyzes query complexity across four canonical oracle-driven algorithms—Deutsch’s, Bernstein–Vazirani, Grover’s search, and Amplitude Amplification—to clarify how superposition, entanglement, interference, and phase kickback translate into provable speedups independent of hardware details. The objective is to systematically connect oracle structure to achievable quantum advantage by contrasting exact, single-query interference algorithms with iterative, probabilistic rotation-based algorithms, and by benchmarking classical deterministic/probabilistic query requirements against quantum bounds. Methodologically, we (i) mathematically model Deutsch and Bernstein–Vazirani via stepwise state-vector evolution to expose phase kickback and phase-encoded answer extraction in one oracle call, (ii) geometrically track Grover and Amplitude Amplification as two-dimensional state-space rotations, quantifying how repeated oracle-driven reflections amplify success amplitude, and (iii) compute classical query baselines and compare them directly with quantum query scaling. Results show that Deutsch reduces classical 2 queries to exactly 1 quantum query for constant-versus-balanced testing, while Bernstein–Vazirani deterministically recovers an n-bit hidden string s with (O(1)) quantum queries versus (O(n)) classical queries. For unstructured search over (N=2n) items, Grover achieves a bounded-error quadratic improvement from (O(N)) to (O(√{N}) queries, and Amplitude Amplification generalizes this to boost success probability from (p) to near certainty in O (1/√p) oracle applications. The central implication is a structural divide: rigid algebraic structure permits maximal, single-query speedups, whereas unstructured “chaos” necessitates iterative geometric amplification, which is more broadly applicable but demands highly stable, error-corrected hardware; consequently, quantum systems are best viewed as specialized co-processors activated when oracle structure warrants a quantum shortcut.
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.
This paper concerns the Grover algorithm that permits to make amplification of quantum states previously tagged by an Oracle. Grover's algorithm allows searches in an unstructure database of n entries finding a marked element with a quadratic speedup. The algorithm requires a predefined number of runs to succeed with probability close to one.This article provides a description of the amplitude amplification quantum algorithm mechanism in a very short computational way, based on tensor products and provides a geometric presentation of the successive system states. All the basis changes are fully described to provide an alternative to the wide spread Grover description based only on matrices and complex tensor computation. Our experiments encompass numerical evaluations of circuit using the Qiskit library of IBM that meet the theoretical considerations