The ability to perform quantum error correction (QEC) and robust gate operations on encoded qubits opens the door to demonstrations of quantum algorithms. Contemporary QEC schemes typically require mid-circuit measurements with feed-forward control, which are challenging for qubit control, often slow, and susceptible to relatively high error rates. In this work, we propose and experimentally demonstrate a universal toolbox of fault-tolerant logical operations on error-detecting codes without mid-circuit measurements on a trapped-ion quantum processor. We present modular logical state teleportation between two four-qubit error-detecting codes without measurements during algorithm execution. Moreover, we realize a fault-tolerant universal gate set on an eight-qubit error-detecting code hosting three logical qubits, based on state injection, which can be executed by coherent gate operations only. We apply this toolbox to experimentally realize Grover’s quantum search algorithm fault-tolerantly on three logical qubits encoded in eight physical qubits, with the implementation displaying clear identification of the desired solution states. Our work demonstrates the practical feasibility and provides first steps into the largely unexplored direction of measurement-free quantum computation.

This paper firstly demonstrates that by appropriately specifying the evolution interval in quantum partial adiabatic evolution, the time complexity of the resulting quantum search algorithm can achieve the well-known quadratic speedup bound of quantum computation. Then this result is used as a reference to remedy invalid quantum partial adiabatic search schemes, which means its time complexity doesn't align with the optimal quantum speedup. And, it is found that the two versions of quantum partial adiabatic evolutions, i.e., "global" and "local" ones must be treated separately, despite previous studies indicating only minor differences in algorithmic performance between the two cases. Additionally, for the "local" variant of the quantum partial adiabatic search algorithm, it can be seen that there are other possible choices for the evolution interval besides the referred one. Our findings further show that significant variations in the resulting single-round success probabilities occur between valid and invalid quantum partial adiabatic evolutions.

Quantum computing offers promising capabilities to improve Intrusion Detection Systems (IDS) against increasingly complex cyberattacks. The Grover’s search algorithm is simulated in a classical environment using Microsoft Excel to evaluate its effectiveness in detecting anomalous network activity. The quantum-inspired model applies amplitude amplification to identify malicious packets more efficiently than a standard rule-based detection approach. Feasibility is assessed through performance indicators that compare detection precision and search effort between both methods. Results show improvements in structured search performance, underscoring the potential for exponential gains when deployed on quantum hardware. These findings provide a practical foundation for future quantum-enhanced IDS architectures.

The work presents a demonstration of functional understanding and practical implementation of quantum computing processes. The project considers both quantum operations (for example, superposition of qubits using Hadamard gates) and complex algorithms, such as Grover's quantum search algorithm. This algorithm is used to optimize the search for a specifically labeled element in an n-qubit space and is one of the main demonstrations of the quantum speedup that gives quantum computing an advantage over classical models. The presented codes are divided into independent, modular functions, which reflects the observance of functional programming principles in quantum algorithms. It is these systematic calculations that improve the transparency, accessibility, and reusability of the code, which is critically important in a quantum development environment where experimentation and multiple testing of algorithms are widely used. The code was simulated using Qiskit's "Aer", which provides multishot execution and statistical analysis of the results. In addition, the implementation of Grover's algorithm used oracle modeling.

We investigate optimizing quantum tree search algorithms by employing a nested Grover Algorithm. This approach seeks to enhance results compared to previous Grover-based methods by expanding the tree of partial assignments to a specific depth and conducting a quantum search within the subset of remaining assignments. The study explores the implications and constraints of this approach, providing a foundation for quantum artificial intelligence applications. Instead of utilizing conventional heuristic functions that are incompatible with quantum tree search, we introduce the partial candidate solution, which indicates a node at a specific depth of the tree. By employing such a function, we define the concatenated oracle, which enables us to decompose the quantum tree search using Grover’s algorithm. With a branching factor of 2 and a depth of m, the costs of Grover’s algorithm are . The concatenated oracle allows us to reduce the cost to for m partial candidate solutions.

Abstract We propose an iterative algorithm to investigate the cooperative evolution dominated by information encoded within state spaces in a random quantum cellular automaton. Inspired by the 2-gram model in statistical linguistics, the updates of quantum states are determined by a given corpus, which serves as the interactions to induce quantum correlations. These local cooperative interactions lead to block-diagonal evolution quantified by the entanglement asymmetry. We evaluate the influence of two-site gates on the iteration process and reveal the associated search speedup. Crucially, we demonstrate the information scrambling dependent of gate sampling, uncovering an intriguing bond percolation. Our results provide an adaptive paradigm for quantum search algorithms and random many-body dynamics.

The advent of quantum computing poses substantial risks to conventional cryptographic mechanisms, particularly hash-based authentication and zero-knowledge identification (ZKI) protocols, which are susceptible to quantum algorithms such as Grover's and Shor's. This study presents a comprehensive benchmarking framework for evaluating the quantum resistance of cryptographic hash functions through performance metrics (execution time, memory utilization), statistical characteristics (entropy, bit-level randomness, avalanche effect), and security attributes (collision and preimage resistance) across diverse input sizes and edge conditions. A novel hybrid hashing strategy, integrating SHA-512 and BLAKE3 in a defense-in-depth configuration, is introduced to enhance post-quantum resilience. Its efficacy is validated via Grover's algorithm simulations, demonstrating a methodology for evaluating the increased computational workload for quantum search algorithms relative to conventional hash functions. The framework incorporates visualization utilities and structured reporting modules, enabling systematic assessment and practical implementation of quantum-resistant cryptographic solutions within ZKI systems. Findings indicate that while classical hashes such as SHA-256 and SHA-512 exhibit theoretically diminished security in quantum threat scenarios, the proposed hybrid method acts as a practical risk mitigation strategy with acceptable computational overhead, providing a viable pathway for safeguarding authentication systems against quantum-capable adversaries.

Optimization is central to urban mobility, where ride-sharing platforms must continuously match passengers with vehicles under complex constraints such as destination compatibility, cost, route efficiency, and acceptable waiting time. Classical exact methods like integer programming scale poorly as the problem size grows and struggle with highly dynamic inputs such as fluctuating demand, travel times, and vehicle capacities. Quantum computing offers an alternative paradigm: by exploiting superposition and interference, quantum search algorithms can explore large combinatorial spaces more efficiently than classical brute-force search. This study presents a proof-of-concept quantum formulation of a foundational ride-sharing assignment problem and investigates the use of Grover’s search algorithm. The model considers five unique passengers (Mehmet, Aytuğ, Filiz, Alara, and Ayliz) and two vehicles. Each passenger must be assigned to one of the two vehicles, subject to a balanced distribution objective (either a 3–2 or 2–3 split) and individual feasibility constraints. The problem is encoded on a 5-qubit quantum register, where the state of each qubit (|0⟩ or |1⟩) denotes assignment to Vehicle A or Vehicle B, mapping the full 2⁵ = 32-state Hilbert space to all possible allocations. A Grover oracle is designed to “mark” all assignments that satisfy the balanced-split and feasibility conditions by applying a phase flip to those computational basis states. The algorithm is implemented and simulated using IBM’s Qiskit framework. The results show that, after two Grover iterations, the total probability of measuring one of the ten valid “balanced” states increases from 31.25% to over 95%, in line with the expected quadratic speed-up of Grover’s algorithm over classical exhaustive search, which requires O(N) evaluations for N = 32 states. Although intentionally simplified—excluding realistic factors such as travel times, traffic, capacities, and time windows—this work demonstrates that a vehicle-sharing assignment problem can be cast as a quantum search task. Future work will focus on scaling to larger instances, incorporating richer operational constraints, and exploring quantum-classical hybrid approaches for real-world logistics optimization.

Optimal number of queries for phase-matching quantum search

This article investigates the performance impact of five classical optimization approximation algorithms on our previously introduced quantum search algorithm, termed the Boolean–Hamiltonians Transform for Quantum Approximate Optimization Algorithm (BHT-QAOA), to effectively search for all best-approximated solutions for Boolean-based problems. These optimization approximation algorithms are BFGS, L-BFGS-B, SLSQP, COBYLA, and COBYQA. Their performance impact is evaluated and compared using two proposed performance metrics—(i) the final number of function evaluations (the lower numbers denote the best optimization approximation algorithms) and (ii) the final quality of qubit measurements (the higher values indicate all best-approximated solutions were found for a problem). Arbitrary classical Boolean problems in various logical structures were examined and evaluated using the BHT-QAOA, these five optimization approximation algorithms, and a simulated noisy model of an IBM quantum computer. Broadly, the BHT-QAOA, with these five algorithms, successfully finds all optimized approximated solutions for these problems. Specifically, both the BFGS and SLSQP algorithms successfully find all best-approximated solutions for these problems, in the context of fewer function evaluations and higher quality of qubit measurements.

This paper investigates nonlinear Diophantine equations as a foundation for post-quantum cryptography. Unlike RSA and ECC, which rely on factorization and discrete logarithms vulnerable to Shor’s algorithm, nonlinear systems with mixed degrees (quadratic, cubic, quartic) are NP-hard and lack efficient solutions under classical or quantum computation. We outline a framework where public keys are defined by equation coefficients and private keys exploit trapdoor knowledge of solutions. Encryption embeds plaintext into disguised equations, while decryption applies the trapdoor efficiently. Security analysis shows resistance to Gröbner basis attacks, lattice reductions, and quantum search, positioning these equations as a strong basis for post-quantum cryptographic schemes.

Abstract The objective of Grover's quantum algorithm is to efficiently identify a single solution within a search space. However, there are search problems that require finding all possible solutions.A direct, repeated application of Grover's algorithm is unsuitable for these problems, as it lacks an internal mechanism to avoid rediscovering previous solutions. Here we introduce a quantum algorithm that efficiently and deterministically discovers all solutions by preparing special initial states of the index register, a step accomplished through the leverage of knowledge from solutions obtained during execution. We experimentally demonstrate our algorithm on a superconducting quantum processor. This work provides quantum examples for search scenarios with some knowledge about solutions.

Abstract In this paper, two novel quantum identity authentication schemes with a third party (TP) and without a TP are respectively suggested by using Grover’s quantum searching algorithm. Each scheme only needs two-particle product states as the initial quantum resource and single-particle measurements, and does not use quantum entangled states, quantum entangled state measurements, quantum entanglement swapping or quantum teleportation. Hence, these two schemes are easy to realize in practical.

Abstract Searching for an unknown marked vertex on a given graph (also known as spatial search) is an extensively discussed topic in the area of quantum algorithms, with a plethora of results based on different quantum walk models and targeting various types of graphs. Most of these algorithms have a non‐zero probability of failure. In recent years, there have been some efforts to design quantum spatial search algorithms with success probability. However, these works either only work for very special graphs or only for the case where there is only one marked vertex. In this work, a different and elegant approach to quantum spatial search is proposed, obtaining deterministic quantum search algorithms that can find a marked vertex with certainty on any Laplacian integral graph with any predetermined proportion of marked vertices. Thus, this work discovers the largest class of graphs that allow deterministic quantum search, making it easy to design deterministic quantum search algorithms for many graphs, subsuming different graphs discussed in previous works, in a unified framework.

Mathematica notebook to generate Figures 3, 5, and 6 from "Quantum Search with a Generalized Laplacian," https://doi.org/10.48550/arXiv.2506.22013.

The famous Goldbach conjecture states that any even natural number N N greater than 2 2 can be written as the sum of two prime numbers p^{\text{(I)}} p (I) and p^{\text{(II)}} p (II) . In this article we propose a quantum analogue device that solves the following problem: given a small prime p^{\text{(I)}} p (I) , identify a member N N of a \mathcal{N} 𝒩 -strong set even numbers for which N-p^{\text{(I)}} N − p (I) is also a prime. A table of suitable large primes p^{\text{(II)}} p (II) is assumed to be known a priori. The device realizes the Grover quantum search protocol and as such ensures a \sqrt{\mathcal{N}} 𝒩 quantum advantage. Our numerical example involves a set of 51 even numbers just above the highest even classical-numerically explored so far [T. O. e Silva, S. Herzog, and S. Pardi, Mathematics of Computation 83, 2033 (2013)]. For a given small prime number p^{\text{(I)}}=223 p (I) = 223 , it took our quantum algorithm 5 steps to identify the number N=4× 10^{18}+14 N = 4 × 10 18 + 14 as featuring a Goldbach partition involving 223 223 and another prime, namely p^{\text{(II)}}=4× 10^{18}-239 p (II) = 4 × 10 18 − 239 . Currently, our algorithm limits the number of evens to be tested simultaneously to \mathcal{N} \sim \ln(N) 𝒩 ∼ ln ( N ) : larger samples will typically contain more than one even that can be partitioned with the help of a given p^{\text{(I)}} p (I) , thus leading to a departure from the Grover paradigm.

A powerful strategy to accelerate quantum-walk-based search algorithms leverages on resetting protocols, where a detector monitors a target site and the evolution of the walker is restarted if no detection occurs within a fixed time interval. The optimal resetting rate can be extracted from the time evolution of the probability S(t) that the detector has not clicked up to time t. We analyze S(t) for a quantum walk on a one-dimensional lattice when the coupling between sites decays algebraically as d^{-α} with the distance d, for α∈(0,∞). At long times, S(t) decays with a universal power-law exponent that is independent of α. At short times, S(t) exhibits a plethora of phase transitions as a function of α. From this, we provide a strategy to determine the optimal resetting rate. We identify two regimes: for α>1, the resetting rate r is bounded from below by the velocity with which information propagates causally across the lattice; for α<1, instead, the long-range hopping tends to localize the walker: The optimal resetting rate depends on the size of the lattice and diverges as α→0. Our strategy directly connects local measurement outcomes with the global dynamics encoded in S(t). We derive simple models explaining our numerical results, shedding light on the interplay of long-range coherent dynamics, symmetries, and local quantum measurement processes in determining equilibrium. Our findings offer experimentally testable predictions and provide new physical insights on optimizing quantum search through resetting.

We study Grover's quantum walk on a new class of graphs, termed segmented complete graphs, which combine high symmetry with detailed spectral properties. Using these graphs, we implement Grover's search algorithm and investigate its performance, focussing on the relationship between graph volume, optimal search time and success probabilities. Our results generalize classical findings for directed weighted graphs and provide new insights into enhancing quantum search algorithms on complex graph structures.This article is part of the theme issue 'Numerical analysis, spectral graph theory, orthogonal polynomials and quantum algorithms'.

A quantum walk on a lattice is a paradigm of a quantum search in a database. The database qubit strings are the lattice sites, qubit rotations are tunneling events, and the target site is tagged by an energy shift. For quantum walks on a continuous time, the walker diffuses across the lattice and the search ends when it localizes at the target site. The search time T can exhibit Grover's optimal scaling with the lattice size N, namely, T∼N, on an all-connected, complete lattice. For finite-range tunneling between sites, instead, Grover's optimal scaling is warranted when the lattice is a hypercube of d&gt;4 dimensions. Here, we show that Grover's optimum can be reached in lower dimensions on lattices of long-range interacting particles, when the interaction strength scales algebraically with the distance r as 1/rα and 0&lt;α&lt;3d/2. For α&lt;d the dynamics mimics the one of a globally connected graph. For d&lt;α&lt;d+2, the quantum search on the graph can be mapped to a short-range model on a hypercube with spatial dimension ds=2d/(α−d), indicating that the search is optimal for ds&gt;4. Our work identifies an exact relation between criticality of long-range and short-range systems, it provides a quantitative demonstration of the resources that long-range interactions provide for quantum technologies, and indicates when existing experimental platforms can implement efficient analog quantum search algorithms.

Quantum search for gravitational wave of massive black hole binaries