Unstructured search by random and quantum walk

The task of finding an entry in an unsorted list of $N$ elements famously takes $O(N)$ queries to an oracle for a classical computer and $O(\sqrt{N})$ queries for a quantum computer using Grover's algorithm. Reformulated as a spatial search problem, this corresponds to searching the complete graph, or all-to-all network, for a marked vertex by querying an oracle. In this tutorial, we derive how discrete- and continuous-time (classical) random walks and quantum walks solve this problem in a thorough and pedagogical manner, providing an accessible introduction to how random and quantum walks can be used to search spatial regions. Some of the results are already known, but many are new. For large $N$, the random walks converge to the same evolution, both taking $N \ln(1/\epsilon)$ time to reach a success probability of $1-\epsilon$. In contrast, the discrete-time quantum walk asymptotically takes $\pi\sqrt{N}/2\sqrt{2}$ timesteps to reach a success probability of $1/2$, while the continuous-time quantum walk takes $\pi\sqrt{N}/2$ time to reach a success probability of $1$.

Quantum Walks: Schur Functions Meet Symmetry Protected Topological Phases

This paper uncovers and exploits a link between a central object in harmonic analysis, the so-called Schur functions, and the very hot topic of symmetry protected topological phases of quantum matter. This connection is found in the setting of quantum walks, i.e. quantum analogs of classical random walks. We prove that topological indices classifying symmetry protected topological phases of quantum walks are encoded by matrix Schur functions built out of the walk. This main result of the paper reduces the calculation of these topological indices to a linear algebra problem: calculating symmetry indices of finite-dimensional unitaries obtained by evaluating such matrix Schur functions at the symmetry protected points $$\pm 1$$ ± 1 . The Schur representation fully covers the complete set of symmetry indices for 1D quantum walks with a group of symmetries realizing any of the symmetry types of the tenfold way. The main advantage of the Schur approach is its validity in the absence of translation invariance, which allows us to go beyond standard Fourier methods, leading to the complete classification of non-translation invariant phases for typical examples.

Integrated photonics enables continuous-beam electron phase modulation

Integrated photonics facilitates extensive control over fundamental light–matter interactions in manifold quantum systems including atoms1, trapped ions2,3, quantum dots4 and defect centres5. Ultrafast electron microscopy has recently made free-electron beams the subject of laser-based quantum manipulation and characterization6–11, enabling the observation of free-electron quantum walks12–14, attosecond electron pulses10,15–17 and holographic electromagnetic imaging18. Chip-based photonics19,20 promises unique applications in nanoscale quantum control and sensing but remains to be realized in electron microscopy. Here we merge integrated photonics with electron microscopy, demonstrating coherent phase modulation of a continuous electron beam using a silicon nitride microresonator. The high-finesse (Q0 ≈ 106) cavity enhancement and a waveguide designed for phase matching lead to efficient electron–light scattering at extremely low, continuous-wave optical powers. Specifically, we fully deplete the initial electron state at a cavity-coupled power of only 5.35 microwatts and generate >500 electron energy sidebands for several milliwatts. Moreover, we probe unidirectional intracavity fields with microelectronvolt resolution in electron-energy-gain spectroscopy21. The fibre-coupled photonic structures feature single-optical-mode electron–light interaction with full control over the input and output light. This approach establishes a versatile and highly efficient framework for enhanced electron beam control in the context of laser phase plates22, beam modulators and continuous-wave attosecond pulse trains23, resonantly enhanced spectroscopy24–26 and dielectric laser acceleration19,20,27. Our work introduces a universal platform for exploring free-electron quantum optics28–31, with potential future developments in strong coupling, local quantum probing and electron–photon entanglement.

An Efficient Visually Meaningful Quantum Walks-Based Encryption Scheme for Secure Data Transmission on IoT and Smart Applications

Smart systems and technologies have become integral parts of modern society. Their ubiquity makes it paramount to prioritise securing the privacy of data transferred between smart devices. Visual encryption is a technique employed to obscure images by rendering them meaningless to evade attention during transmission. However, the astounding computing power ascribed to quantum technology implies that even the best visually encrypted systems can be effortlessly violated. Consequently, the physical realisation quantum hardware portends great danger for visually encrypted date on smart systems. To circumvent this, our study proposes the integration of quantum walks (QWs) as a cryptographic mechanism to forestall violation of the integrity of images on smart systems. Specifically, we use QW first to substitute the original image and to subsequently permutate and embed it onto the reference image. Based on this structure, our proposed quantum walks visually meaningful cryptosystem facilities confidential transmission of visual information. Simulation-based experiments validate the performance of the proposed system in terms of visual quality, efficiency, robustness, and key space sensitivity, and by that, its potential to safeguard smart systems now and as we transition to the quantum era.

Two component quantum walk in one-dimensional lattice with hopping imbalance

We investigate the two-component quantum walk in one-dimensional lattice. We show that the inter-component interaction strength together with the hopping imbalance between the components exhibit distinct features in the quantum walk for different initial states. When the walkers are initially on the same site, both the slow and fast particles perform independent particle quantum walks when the interaction between them is weak. However, stronger inter-particle interactions result in quantum walks by the repulsively bound pair formed between the two particles. For different initial states when the walkers are on different sites initially, the quantum walk performed by the slow particle is almost independent of that of the fast particle, which exhibits reflected and transmitted components across the particle with large hopping strength for weak interactions. Beyond a critical value of the interaction strength, the wave function of the fast particle ceases to penetrate through the slow particle signalling a spatial phase separation. However, when the two particles are initially at the two opposite edges of the lattice, then the interaction facilitates the complete reflection of both of them from each other. We analyze the above mentioned features by examining various physical quantities such as the on-site density evolution, two-particle correlation functions and transmission coefficients.