Quantum Walks Research Articles

Implementation of a continuous-time quantum walk on a sparse graph

Implementation of a continuous-time quantum walk on a sparse graph

Universal scaling hypothesis of quantum spatial search in complex networks

Since quantum spatial searches on networks have a strong network dependence, the question arises whether the universal perspective exists in this quantum algorithm for various networks. Here, we propose a universal scaling hypothesis of the quantum spatial search on complex networks such as small-world and scale-free networks. The average path length, a key quantity in the complex network science, is useful to expose this universal feature, where the collapse plot can be generated for the optimal time, the maximal finding probability, and the optimal hopping parameter. Based on the path integral method, we also clarify that the probability amplitude in the continuous-time quantum walk can be determined by the path length distribution. Published by the American Physical Society 2024

Non-Bloch parity-time-symmetric phase transition in quantum walks

Non-Bloch parity-time-symmetric phase transition in quantum walks

Anomalous quantum dynamics in a lossy nonlocal system

Abstract The non-Hermitian skin effect and non-Hermitian edge burst reflect the vital role of the spatial boundary in non-Hermitian systems from both static and dynamic perspectives. Here, we study a non-Hermitian dissipative lattice with nonlocal coupling and demonstrate many interesting static and dynamic phenomena. In the case of a global coupling with all sites coupled with each other, we observe an anomalous hopping resonance, at which a quantum walker initially placed at one single site almost completely escape from the boundary of the system regardless of its initial position. In the case of the non-global, but still infinite-range coupling, the interplay between nonlocal hopping and non-Hermitian skin effect leads to the emergence of local bulk modes presenting multi-partite density distribution. The presence of local bulk modes induce nontrivial dynamics of a quantum walker, featuring multiple peaks of lost probability in spatially separated domains. Our findings demonstrate unique properties induced by nonlocal coupling in non-Hermitian systems.

Information scrambling in quantum walks: Discrete-time formulation of Krylov complexity

Information scrambling in quantum walks: Discrete-time formulation of Krylov complexity

Quantum walk option pricing model based on binary tree

Quantum walk option pricing model based on binary tree

Quantum phenomena in biological systems

Quantum biology is a modern field of research that aims to understand how quantum effects can affect the chemistry underlying various biological processes. This paper reviews several examples of biological processes where quantum effects might play a notable role. Initially, the photon capture mechanism present in vision is discussed, where the energy of the photon is used to cause conformational changes to chromophoric proteins. The second example elaborates the highly efficient energy transfer process present in photosynthesis and discusses, in particular, how the random quantum walk process may enhance the performance drastically. Subsequently, the vertebrate magnetoreception, and the possible associated role of the radical pair mechanism in the process is considered. The review concludes with the discussion of some speculative ideas of putative quantum effects arising in neural processes.

Dynamic quantum secret sharing with identity verification based on quantum walks

Abstract A dynamic quantum secret sharing protocol with authentication is proposed based on quantum walks. The protocol can share specific secrets and also update the secret information periodically based on the existing quantum walks structure. Participants can be added or removed while keeping the original secret unchanged. This is more flexible in practical applications. Before the secrets are shared, each participant is authenticated to ensure the legitimacy of all participants' identities. In addition, the protocol uses single photons as quantum resources, avoiding the complex preparation of entangled states, which makes the protocol more advantageous and potential in practical applications. Security analyses show that the protocol is resistant to collusion attack and other common attacks.

Necessity of orthogonal basis vectors for the two-anyon problem in a one-dimensional lattice* *Supported by National Science Foundation of China (Grant Nos. 12074340 and 12474492) and the Science Foundation of Zhejiang Sci-Tech University (Grants No. 20062098-Y, 19062463-Y and 22062344-Y).

Few-body physics for anyons has been intensively studied within the anyon-Hubbard model, including the quantum walk and Bloch oscillations of two-anyon states. Recently, theoretical and experimental simulations of two-anyon states in a one-dimensional lattice have been carried out by expanding the wavefunction in terms of non-orthogonal basis vectors, resulting in non-physical degrees of freedom. In the present work, we deduce finite difference equations for the two-anyon state in a one-dimensional lattice by solving the Schrödinger equation with orthogonal and complete basis vectors. Such an orthogonal scheme gives all the orthogonal physical eigenstates, while the conventional (non-orthogonal) method produces many non-physical redundant eigensolutions whose components violate the anyonic commutation relations. The dynamical property of the two-anyon states in a sufficiently large lattice is investigated and compared in both the orthogonal and conventional schemes. For initial states with two anyons at the same site or two (next-)neighboring sites, we observe the same dynamical behavior in both schemes, including the revival probability, probability density function and two-body correlation. For other initial states, the conventional scheme produces erroneous states that no longer obey the anyonic relations. The period of Bloch oscillations in the pseudo-fermionic limit has been found to be twice that in the bosonic limit, while these oscillations disappear at other statistical parameters. Our findings are vital for quantum simulations of few-body anyonic physics in lattice models.

Multi-particle quantum walks on 3D integrated photonic chip

Quantum walks provide a speed-up in computational power for various quantum algorithms and serve as inspiration for the construction of complex graph representations. Many pioneering works have been dedicated to expanding the experimental state space and the complexity of graphs. However, these experiments are mostly limited to small experimental scale, which do not reach a many-body level and fail to reflect the multi-particle quantum interference effects among non-adjacent modes. Here, we present a quantum walk with three photons on a two-dimensional triangular lattice, which is mapped to a 19 × 19 × 19 high-dimensional state space and constructs a complex graph with 6859 nodes and 45,486 edges. By utilizing the statistical signatures of the output combinations and incorporating machine learning techniques, we successfully validate the nonclassical properties of the experiment. Our implementation provides a paradigm for exponentially expanding the state space and graph complexity of quantum walks, paving the way for surmounting the classical regime in large-scale quantum simulations.

First Hitting Times on a Quantum Computer: Tracking vs. Local Monitoring, Topological Effects, and Dark States.

We investigate a quantum walk on a ring represented by a directed triangle graph with complex edge weights and monitored at a constant rate until the quantum walker is detected. To this end, the first hitting time statistics are recorded using unitary dynamics interspersed stroboscopically by measurements, which are implemented on IBM quantum computers with a midcircuit readout option. Unlike classical hitting times, the statistical aspect of the problem depends on the way we construct the measured path, an effect that we quantify experimentally. First, we experimentally verify the theoretical prediction that the mean return time to a target state is quantized, with abrupt discontinuities found for specific sampling times and other control parameters, which has a well-known topological interpretation. Second, depending on the initial state, system parameters, and measurement protocol, the detection probability can be less than one or even zero, which is related to dark-state physics. Both return-time quantization and the appearance of the dark states are related to degeneracies in the eigenvalues of the unitary time evolution operator. We conclude that, for the IBM quantum computer under study, the first hitting times of monitored quantum walks are resilient to noise. However, a finite number of measurements leads to broadening effects, which modify the topological quantization and chiral effects of the asymptotic theory with an infinite number of measurements.

Dirac quantum walk on tetrahedra

Dirac quantum walk on tetrahedra

Searching weighted barbell graphs with Laplacian and adjacency quantum walks

Searching weighted barbell graphs with Laplacian and adjacency quantum walks

Quantum state processing through controllable synthetic temporal photonic lattices

AbstractQuantum walks on photonic platforms represent a physics-rich framework for quantum measurements, simulations and universal computing. Dynamic reconfigurability of photonic circuitry is key to controlling the walk and retrieving its full operation potential. Universal quantum processing schemes based on time-bin encoding in gated fibre loops have been proposed but not demonstrated yet, mainly due to gate inefficiencies. Here we present a scalable quantum processor based on the discrete-time quantum walk of time-bin-entangled photon pairs on synthetic temporal photonic lattices implemented on a coupled fibre-loop system. We utilize this scheme to path-optimize quantum state operations, including the generation of two- and four-level time-bin entanglement and the respective two-photon interference. The design of the programmable temporal photonic lattice enabled us to control the dynamic of the walk, leading to an increase in the coincidence counts and quantum interference measurements without recurring to post-selection. Our results show how temporal synthetic dimensions can pave the way towards efficient quantum information processing, including quantum phase estimation, Boson sampling and the realization of topological phases of matter for high-dimensional quantum systems in a cost-effective, scalable and robust fibre-based setup.

Absence of bound states for quantum walks and CMV matrices via reflections

We give a criterion based on reflection symmetries in the spirit of Jitomirskaya–Simon to show absence of point spectrum for (split-step) quantum walks and Cantero–Moral–Velázquez (CMV) matrices. To accomplish this, we use some ideas from a recent paper by the authors and collaborators to implement suitable reflection symmetries for such operators. We give several applications. For instance, we deduce arithmetic delocalization in the phase for the unitary almost-Mathieu operator and singular continuous spectrum for generic CMV matrices generated by the Thue–Morse subshift.

Randomized SearchRank: A semiclassical approach to a quantum search engine

The quantum SearchRank algorithm is a promising tool for a future quantum search engine based on PageRank quantization. However, this algorithm loses its functionality when the N/M ratio between the network size N and the number of marked nodes M is sufficiently large. We propose a modification of the algorithm, replacing the underlying Szegedy quantum walk with a semiclassical walk. To maintain the same time complexity as the quantum SearchRank algorithm we propose a simplification of the algorithm. This new algorithm is called randomized SearchRank, since it corresponds to a quantum walk over a randomized mixed state. The performance of the SearchRank algorithms is first analyzed on an example network and then statistically on a set of different networks of increasing size and different number of marked nodes. On the one hand, to test the search ability of the algorithms, it is computed how the probability of measuring the marked nodes decreases with N/M for the quantum SearchRank, but remarkably it remains at a high value around 0.9 for our semiclassical algorithms, solving the quantum SearchRank problem. The time complexity of the algorithms is also analyzed, obtaining a quadratic speedup with respect to the classical ones. On the other hand, the ranking functionality of the algorithms has been investigated, obtaining a good agreement with the classical PageRank distribution. Finally, the dependence of these algorithms on the intrinsic PageRank damping parameter has been clarified. Our results suggest that this parameter should be below a threshold so that the execution time does not increase drastically. Published by the American Physical Society 2024

Optimizing topology for quantum probing with discrete-time quantum walks

Optimizing topology for quantum probing with discrete-time quantum walks

Subdiffusive dynamics in photonic random walks probed with classical light.

The random walk of photons in a tight-binding lattice is known to exhibit diffusive motion similar to classical random walks under decoherence, clearly illustrating the quantum-to-classical transition. In this study, we reveal that the random walk of intense classical light under dephasing dynamics can disentangle quantum and ensemble averaging, making it possible to observe subdiffusive walker dynamics, i.e., a behavior very distinct from both a classical and a quantum walker. These findings are demonstrated through proposing photonic random walk in synthetic temporal lattices, based on pulse dynamics in coupled fiber loops.

Absolute Zeta functions and periodicity of quantum walks on cycles

The quantum walk is a quantum counterpart of the classical random walk. On the other hand, absolute zeta functions can be considered as zeta functions over $\mathbf{F}_1$. This study presents a connection between quantum walks and absolute zeta functions. In this paper, we focus on Hadamard walks and $3$-state Grover walks on cycle graphs. The Hadamard walks and the Grover walks are typical models of the quantum walks. We consider the periods and zeta functions of such quantum walks. Moreover, we derive the explicit forms of the absolute zeta functions of corresponding zeta functions. Also, it is shown that our zeta functions of quantum walks are absolute automorphic forms.

Repeated measurements and random scattering in quantum walks

Repeated measurements and random scattering in quantum walks