We report on a discrete-time quantum walk that uses the momentum of ultra-cold rubidium-87 atoms as the walk space and two internal atomic states as the coin degree of freedom. Each step of the walk consists of a coin toss (a microwave pulse) followed by a unitary shift operator (a resonant ratchet pulse). We carry out a comprehensive experimental study on the effects of various parameters, including the strength of the shift operation, coin parameters, noise, and initialization of the system on the behavior of the walk. The walk dynamics can be well controlled in our experiment; potential applications include atom interferometry and engineering asymmetric walks.
We implement a quantum walk in phase space with a new mechanism based on the superconducting resonator-assisted double quantum dots. By analyzing the hybrid system, we obtain the necessary factors of realization of a quantum walk in phase space: the walker, coin, coin flipping and conditional phase shift. In order to implement the coin flipping operator, we add a driving field to the resonator. The interaction between the quantum dots and resonator field is used to implement conditional phase shift. Furthermore, we show with different driving fields the quantum walk in phase space exhibits a b…
We present a discrete-time, one-dimensional quantum walk based on the entanglement between the momentum of ultracold rubidium atoms (the walk space) and two internal atomic states (the "coin" degree of freedom). Our scheme is highly flexible and can provide a platform for a wide range of applications such as quantum search algorithms, the observation of topological phases, and the realization of walks with higher dimensionality. Along with the investigation of the quantum-to-classical transition, we demonstrate the distinctive features of a quantum walk and contrast them to those of its classi…
We perform generalized measurements of a qubit by realizing the qubit as a coin in a photonic quantum walk and subjecting the walker to projective measurements. Our experimental technique can be used to realize photonically any rank-1 single-qubit positive operator-valued measure via constructing an appropriate interferometric quantum-walk network and then projectively measuring the walker's position at the final step.
We demonstrate an implementation of unambiguous state discrimination of two equally probable single-qubit states via a one-dimensional photonic quantum walk experimentally. Furthermore we experimentally realize a quantum walk algorithm for implementing a generalized measurement in terms of positive operator value measurement on a single qubit. The measurement of the single-photons' positions corresponds to a measurement of an element of the positive operator value measurement on the polarizations of the single-photons.
We consider the time dependent dynamics of an atom in a two-color pumped cavity, longitudinally through a side mirror and transversally via direct driving of the atomic dipole. The beating of the two driving frequencies leads to a time dependent effective optical potential that forces the atom into a non-trivial motion, strongly resembling a discrete random walk behavior between lattice sites. We provide both numerical and analytical analysis of such a quasi-random walk behavior.
Quantum walks are a promising framework for developing quantum algorithms and quantum simulations. They represent an important test case for the application of quantum computers. Here we present different forms of discrete-time quantum walks (DTQWs) and show their equivalence for physical realizations. Using an appropriate digital mapping of the position space on which a walker evolves to the multiqubit states of a quantum processor, we present different configurations of quantum circuits for the implementation of DTQWs in one-dimensional position space. We provide example circuits for a five-…
Quantum walks are more than tools for building quantum algorithms. They have been used effectively to model and simulate quantum dynamics in many complex physical processes. Particularly, a variant of discrete-time quantum walk known as split-step quantum walk is closely related to Dirac cellular automata and topological insulators whose realizations rely on position-dependent control of evolution operators. Owing to the ease of manipulating multiple degrees of freedom of photons, we provide an optical setup of split-step operators which in combination with position-dependent coin (PDC) operat…
We discuss an efficient physical realization of topological quantum walks on a finite lattice. The $N$-point lattice is realized with $\log_2 N$ qubits, and the quantum circuit utilizes a number of quantum gates which is polynomial in the number of qubits. In a certain scaling limit, we show that a large number of steps is implemented with a number of quantum gates which is independent of the number of steps. We ran the quantum algorithm on the IBM-Q five-qubit quantum computer, thus experimentally demonstrating topological features, such as boundary bound states, on a lattice with $N=4$ point…
We experimentally demonstrate a quantum walk on a line in phase space using one and two trapped ion. A walk with up to 23 steps is realized by subjecting an ion to state-dependent displacement operations interleaved with quantum coin tossing operations. To analyze the ion's motional state after each step we apply a technique that directly maps the probability density distribution onto the ion's internal state. The measured probability distributions and the position's second moment clearly show the non-classical character of the quantum walk. To further highlight the difference between the clas…
The atom-optics kicked rotor can be used to prepare specific momentum distributions on a discrete basis set. We implement a continuous-time quantum walk and a quantum search protocol in this momentum basis. In particular we propose ways to identify a specific marked state from the final momentum distribution after the walker's evolution. Our protocol is guided by current experimental possibilities making it accessible to experimentally implemented quantum walks with Bose-Einstein condensates.
Quantum state transfer between different sites is a significant problem for quantum networks and quantum computers. By selecting quantum walks with two coins as the basic model and two coin spaces as the communication carriers, we successfully implement quantum state transfer on various graphs (EPL, \textbf{124} (2018) 60009) \cite{Shang_2019}. Here, we demonstrate the experimental implementation of this scheme using IBM quantum experience platform. In particular, we show the transfer of Bell state, GHZ state and W state on complete graph on the quantum device. Also, we observe that our protoc…
Symmetric informationally complete positive operator-valued measurement (SIC-POVM) is one important class of quantum measurement which is crucial for various quantum information processing tasks. SIC-POVMs have the advantage of providing an unbiased estimator for quantum states with the minimal number of outcomes needed for full tomography. We present an experimental approach on a photonic quantum walk which can be used to implement SIC-POVMs on a single-qbubit. The projection measurements of single-photons' positions correspond to elements of SIC-POVM on the polarization of single-photons.
We investigate the possibility of implementing a sequence of quantum walks whose probability distributions give an overall positive winning probability, while it is negative for the single walks (Parrondo's paradox). In particular, we have in mind an experimental realisation with a Bose-Einstein condensate in which the walker's space is momentum space. Experimental problems in the precise implementation of the coin operations for our discrete-time quantum walks are analysed in detail. We study time-dependent phase fluctuations of the coins as well as perturbations arising from the finite momen…
Network centrality has important implications well beyond its role in physical and information transport analysis; as such, various quantum walk-based algorithms have been proposed for measuring network vertex centrality. In this work, we propose a continuous-time quantum walk algorithm for determining vertex centrality, and show that it generalizes to arbitrary graphs via a statistical analysis of randomly generated scale-free and Erdős-Rényi networks. As a proof of concept, the algorithm is detailed on a 4-vertex star graph and physically implemented via linear optics, using spatial and pola…
Since a limit distribution of a discrete-time quantum walk on the line was derived in 2002, a lot of limit theorems for quantum walks with a localized initial state have been reported. On the other hand, in quantum probability theory, there are four notions of independence (free, monotone, commuting, and boolean independence) and quantum central limit theorems associated to each independence have been investigated. The relation between quantum walks and quantum probability theory is still unknown. As random walks are fundamental models in the Kolmogorov probability theory, can the quantum walk…
The goal of teleportation is to transfer the state of one particle to another particle. In coined quantum walks, conditional shift operators can introduce entanglement between position space and coin space. This entanglement resource can be used as a quantum channel for teleportation, as proposed by Wang, Shang and Xue [Quantum Inf. Process. 16, 221 (2017)]. Here, we demonstrate the implementation of quantum teleportation using quantum walks on a five-qubit quantum computer and a 32-qubit simulator provided by IBM quantum experience beta platform. We show the teleportation of single-qubit, two…
We propose a scheme to implement the quantum walk for SU(1,1) in the phase space, which generalizes those associated with the Heisenberg-Weyl group. The movement of the walker described by the SU(1,1) coherent states can be visualized on the hyperboloid or the Poincaré disk. In both one-mode and two-mode realizations, we introduce the corresponding coin-flip and conditional-shift operators for the SU(1,1) group, whose relations with those for Heisenberg-Weyl group are analyzed. The probability distribution, standard deviation and the von Neumann entropy are employed to describe the walking pro…
We build interacting Fock spaces from association schemes and set up quantum walks on the resulting regular graphs (distance-regular and distance-transitive). The construction is valid for growing graphs and the interacting Fock space is well defined asymptotically for the growing graph. To realize the quantum walks defined on the graphs in terms of anyons we switch to the dual view of the association schemes and identify the corresponding modular tensor categories from the Bose-Mesner algebra. Informally, the fusion ring induced by the association scheme and a topological twist can be the bas…
Collective measurements on identically prepared quantum systems can extract more information than local measurements, thereby enhancing information-processing efficiency. Although this nonclassical phenomenon has been known for two decades, it has remained a challenging task to demonstrate the advantage of collective measurements in experiments. Here we introduce a general recipe for performing deterministic collective measurements on two identically prepared qubits based on quantum walks. Using photonic quantum walks, we realize experimentally an optimized collective measurement with fidelity…
We present an implementation scheme for a quantum walk in the orbital angular momentum space of a laser beam. The scheme makes use of a ring interferometer, containing a quarter-wave plate and a q plate. This setup enables one to perform an arbitrary number of quantum walk steps. In addition, the classical nature of the implementation scheme makes it possible to observe the quantum walk evolution in real time. We use nonquantum entanglement of the laser beam's polarization with its orbital angular momentum to implement the quantum walk.
Over the last decade there have been many advances in studies of quantum walks (QWs) including a momentum-space QW recently realized in our spinor Bose-Einstein condensate system. This QW possessed behaviors that generally agreed with theoretical predictions; however, it also showed momentum distributions that were not adequately explained by the theory. We present a theoretical model which proves that the coherent dynamics of the spinor condensate is sufficient to explain the experimental data without invoking the presence of a thermal cloud of atoms as in the original theory. Our numerical f…
Photons can carry spin angular momentum (SAM) and orbital angular momentum (OAM), which can be used to realize a qubit system and a high-dimension system respectively. This spin-orbital system is very suitable for implementing one-dimensional discrete-time quantum random walks. We propose a simple scheme of quantum walks on the spin-orbital angular momentum space of photons, where photons walk on the infinity OAM space controlled by their SAM. By employing the recent invention of an optical device, the so-called 'q-plate', our scheme is more simple and efficient than others because there is no…
Each step in a quantum random walk is typically understood to have two basic components; a `coin-toss' which produces a random superposition of two states, and a displacement which moves each component of the superposition by different amounts. Here we suggest the realization of a walk in momentum space with a spinor Bose-Einstein condensate subject to a quantum ratchet realized with a pulsed, off-resonant optical lattice. By an appropriate choice of the lattice detuning, we show how the atomic momentum can be entangled with the internal spin states of the atoms. For the coin-toss, we propose …
We analytically investigate the recently proposed and implemented discrete-time quantum walk based on kicked ultra-cold atoms. We show how the internal level structure of the kicked atoms leads to the emergence of a relative light-shift phase immediately relevant for the experimental realization. Analytical solutions are provided for the momentum distribution for both the case of quantum resonance and the near-resonant quasimomenta.
A Quantum Walk (QW) simulating the flat $(1 + 2)$D Dirac Eq.\ on a spatial polar grid is constructed. Because fermions are represented by spinors, which do not constitute a representation of the rotation group, but rather of its double cover, the QW can only be defined globally on an extended spacetime where the polar angle extends from $0$ to $4 π$. The coupling of the QW with arbitrary electromagnetic fields is also presented. Finally, the cylindrical relativistic Landau levels of the Dirac Eq.\ are computed explicitly and simulated by the QW.
Quantum walk has been regarded as a primitive to universal quantum computation. By using the operations required to describe the single particle discrete-time quantum walk on a position space we demonstrate the realization of the universal set of quantum gates on two- and three-qubit systems. The idea is to utilize the effective Hilbert space of the single qubit and the position space on which it evolves in order to realize multi-qubit states and universal set of quantum gates on them. Realization of many non-trivial gates and engineering arbitrary states is simpler in the proposed quantum wal…
Through introducing controlled alternative quantum walks, we present controlled alternate quantum walks (CAQW) based quantum hash function. CAQW based quantum hash function have excellent security, outstanding statistical performance and splendid expansibility. Furthermore, due to the structure of alternative quantum walks, implementing CAQW based quantum hash function significantly reduces the resources necessary for its feasible experimental realization than implementing other quantum hash functions. Besides, CAQW based quantum hash function has expansibility.
We construct a decomposition procedure for converting split-step quantum walks into ordinary quantum walks with alternating coins, and we show that this decomposition enables a feasible linear optical realization of split-step quantum walks by eliminating quantum-control requirements. As salient applications, we show how our scheme will simulate Majorana modes and edge states.
Quantum random walks have received much interest due to their non-intuitive dynamics, which may hold the key to a new generation of quantum algorithms. What remains a major challenge is a physical realization that is experimentally viable and not limited to special connectivity criteria. We present a scheme for walking on arbitrarily complex graphs, which can be realized using a variety of quantum systems such as a BEC trapped inside an optical lattice. This scheme is particularly elegant since the walker is not required to physically step between the nodes; only flipping coins is sufficient.
Quantum walks are powerful tools for quantum applications and for designing topological systems. Although they are simulated in a variety of platforms, genuine two-dimensional realizations are still challenging. Here we present an innovative approach to the photonic simulation of a quantum walk in two dimensions, where walker positions are encoded in the transverse wavevector components of a single light beam. The desired dynamics is obtained by means of a sequence of liquid-crystal devices, which apply polarization-dependent transverse "kicks" to the photons in the beam. We engineer our quant…
We propose and experimentally demonstrate that a Mach-Zehnder interferometer composed of polarized beam splitters and a pentaprism in the place of one of the mirrors works as a linear optical quantum controlled-NOT (CNOT) gate. To perform the information processing, the polarization and orbital angular momentum (OAM) of the photons act as the control and target qubits, respectively. The readout process is simple, requiring only a linear polarizer and a triangular diffractive aperture before detection. The viability and stability of the experiment makes the present proposal a valuable candidate…
We investigate continuous-time quantum walks of two indistinguishable anyons in one-dimensional lattices with both on-site and nearest-neighbor interactions based on the fractional Jordan-Wigner transformation. It is shown that the two-body correlations in position space are symmetric about the initial sites of two quantum walkers in the Bose limit ($χ=0$ ) and Fermi limit ( $χ=1$), while in momentum space this happens only in the Bose limit. An interesting asymmetry arises in the correlation for most cases with the statistical parameter $χ$ varying in between. It turns out that the origin of …
We propose a scheme for the realization of a quantum walker and a quantum simulator for the Dirac equation with ultracold spinor atoms in driven optical lattices. A precise control of the dynamics of the atomic matter wave can be realized using time-dependent external forces. If the force depends on the spin state of the atoms, the dynamics will entangle the inner and outer degrees of freedom which offers unique opportunities for quantum information and quantum simulation. Here, we introduce a method to realize a quantum walker based on the state-dependent transport of spinor atoms and a coher…
Quantum walks play an important role for developing quantum algorithms and quantum simulations. Here we present one dimensional three-state quantum walk(lazy quantum walk) and show its equivalence for circuit realization in ternary quantum logic for the first of its kind. Using an appropriate logical mapping of the position space on which a walker evolves onto the multi-qutrit states, we present efficient quantum circuits considering the nearest neighbour position space for the implementation of lazy quantum walks in one-dimensional position space in ternary quantum system. We also address sca…
We report on an experimental realization of unidirectional transporting island structures in an otherwise chaotic phase space of the delta-kicked rotor system. Using a Bose-Einstein Condensate as a source of ultracold atoms, we employ asymmetric phase modulation in the kicks, with the narrow momentum distribution of the atoms allowing us to address individual island structures. We observe quantum ratchet behavior in this system, with clear directed momentum current in the absence of a directional force, which we characterize and connect to ε-classical theory.
We investigate a system of two atoms in an optical lattice, performing a quantum walk by state-dependent shift operations and a coin operation acting on the internal states. The atoms interact, e.g., by cold collisions, whenever they are in the same potential well of the lattice. Under such conditions they typically develop a bound state, so that the two atoms effectively perform a quantum walk together, rarely moving further from each other than a few lattice sites. The theoretical analysis is based on a theory of quantum walks with a point defect, applied to the difference variable. We also …
We have realized a quantum walk in momentum space with a rubidium spinor Bose-Einstein condensate by applying a periodic kicking potential as a walk operator and a resonant microwave pulse as a coin toss operator. The generated quantum walks appear to be stable for up to ten steps and then quickly transit to classical walks due to spontaneous emissions induced by laser beams of the walk operator. We investigate these quantum to classical walk transitions by introducing well controlled spontaneous emissions with an external light source during quantum walks. Our findings demonstrate a scheme to…
Classical optics can be used to efficiently implement certain quantum information processing tasks with a high degree of control, for example, one-dimensional quantum walks through the space of orbital angular momentum of light directed by its polarization. To explore the potential of quantum information processing with classical light, we here suggest a method to realize d-dimensional quantum walks with classical optics---an important step towards robust implementation of certain quantum algorithms. In this scheme, different degrees of freedom of light, such as frequency, orbital angular mome…
We study the propagation of a hole in degenerate (paramagnetic) spin environments. This canonical problem has important connections to a number of physical systems, and is perfectly suited for experimental realization with ultra-cold atoms in an optical lattice. At the short-to-intermediate timescale that we can access using a stochastic-series-type numeric scheme, the propagation turns out to be distinctly non-diffusive with the proba- bility distribution featuring minima in both space and time due to quantum interference, yet the motion is not ballistic, except at the beginning. We discuss p…
Discrete-time quantum walks are known to exhibit exotic topological states and phases. Physical realization of quantum walks in a noisy environment may destroy these phases. We investigate the behavior of topological states in quantum walks in the presence of a lossy environment. The environmental effects in the quantum walk dynamics are addressed using the non-Hermitian Hamiltonian approach. We show that the topological phases of the quantum walks are robust against moderate losses. The topological order in one-dimensional split-step quantum walk persists as long as the Hamiltonian is $\mathc…
Free electron beams that carry high values of orbital angular momentum (OAM) possess large magnetic moments along the propagation direction. This makes them an ideal probe for measuring the electronic and magnetic properties of materials, and for fundamental experiments in magnetism. However, their generation requires the use of complex diffractive elements, which usually take the form of nano-fabricated holograms. Here, we show how the limitations of focused ion beam milling in the fabrication of such holograms can be overcome by using electron beam lithography. We demonstrate experimentally …
We propose a quantum-electrodynamics scheme for implementing the discrete-time, coined quantum walk with the walker corresponding to the phase degree of freedom for a quasi-magnon field realized in an ensemble of nitrogen-vacancy centres in diamond. The coin is realized as a superconducting flux qubit. Our scheme improves on an existing proposal for implementing quantum walks in cavity quantum electrodynamics by removing the cumbersome requirement of varying drive-pulse durations according to mean quasiparticle number. Our improvement is relevant to all indirect-coin-flip cavity quantum-electr…
We report on the experimental realization of electric quantum walks, which mimic the effect of an electric field on a charged particle in a lattice. Starting from a textbook implementation of discrete-time quantum walks, we introduce an extra operation in each step to implement the effect of the field. The recorded dynamics of such a quantum particle exhibits features closely related to Bloch oscillations and interband tunneling. In particular, we explore the regime of strong fields, demonstrating contrasting quantum behaviors: quantum resonances vs. dynamical localization depending on whether…
The quantum walk was originally proposed as a quantum mechanical analogue of the classical random walk, and has since become a powerful tool in quantum information science. In this paper, we show that discrete time quantum walks provide a versatile platform for studying topological phases, which are currently the subject of intense theoretical and experimental investigation. In particular, we demonstrate that recent experimental realizations of quantum walks simulate a non-trivial one dimensional topological phase. With simple modifications, the quantum walk can be engineered to realize all of…
One of the proposals for the exploitation of two-dimensional quantum walks has been the efficient generation of entanglement. Unfortunately, the technological effort required for the experimental realization of standard two-dimensional quantum walks is significantly demanding. In this respect, an alternative scheme with less challenging conditions has been recently studied, particularly in terms of spatial-entanglement generation [C. Di Franco, M. Mc Gettrick, and Th. Busch, Phys. Rev. Lett. 106, 080502 (2011)]. Here, we extend the investigation to a scenario where a measurement is performed o…
We consider one-dimensional quantum walks in optical linear networks with synthetically introduced disorder and tunable system parameters allowing for the engineered realization of distinct topological phases. The option to directly monitor the walker's probability distribution makes this optical platform ideally suited for the experimental observation of the unique signatures of the one-dimensional topological Anderson transition. We analytically calculate the probability distribution describing the quantum critical walk in terms of a (time staggered) spin polarization signal and propose a co…
In this article we investigate the effects of shifting position decoherence, arisen from the tunneling effect in the experimental realization of the quantum walk, on the one-dimensional discreet time quantum walk. We show that in the regime of this type of noise the quantum behavior of the walker does not fade, in contrary to the coin decoherence for which the walker undergos the quantum-to-classical transition even for weak noise. Particularly, we show that the quadratic dependency of the variance on the time and also the coin-position entanglement, i.e. two important quantum aspects of the c…
