Two-dimensional Quantum Walk Research Articles

Experimental Proposal on Non-Hermitian Skin Effect by Two-dimensional Quantum Walk with a Single Trapped Ion

Non-Hermitian Hamiltonians are widely used in describing open systems with gain and loss, among which a key phenomenon is the non-Hermitian skin effect. Here we report an experimental scheme to realize a two-dimensional (2D) discrete-time quantum walk with non-Hermitian skin effect in a single trapped ion. It is shown that the coin and 2D walker states can be labeled in the spin of the ion and the coherent-state lattice of the ion motion, respectively. We numerically observe a directional bulk flow, whose orientations are controlled by dissipative parameters, showing the emergence of the non-Hermitian skin effect. We then discuss an experimental implementation of our scheme in a laser-controlled trapped Ca+ ion. Our experimental proposal may be applicable to research of dissipative quantum walk systems and may be able to generalize to other platforms, such as superconducting circuits and atoms in cavity.

Generation of high-dimensional qudit quantum states via two-dimensional quantum walks

Generation of high-dimensional qudit quantum states via two-dimensional quantum walks

Manipulating directional flow in a two-dimensional photonic quantum walk under a synthetic magnetic field

Matter transport is a fundamental process in nature. Understanding and manipulating flow in a synthetic media often have rich implications for modern device design. Here we experimentally demonstrate directional transport of photons in a two-dimensional quantum walk, where the light propagation is highly tunable through dissipation and synthetic magnetic flux. The directional flow hereof underlies the emergence of the non-Hermitian skin effect, with its orientation continuously adjustable through the photon-loss parameters. By contrast, the synthetic magnetic flux originates from an engineered geometric phase, which, by inducing localized cyclotron orbits, suppresses the bulk flow through magnetic confinement. We further demonstrate how the directional flow and synthetic flux impact the dynamics of the Floquet topological edge modes along an engineered boundary. Our results exemplify an intriguing strategy for engineering directed light transport, highlighting the interplay of non-Hermiticity and gauge fields in synthetic systems of higher dimensions.

Eigenvalues and threshold resonances of a two-dimensional split-step quantum walk with strong shift

Eigenvalues and threshold resonances of a two-dimensional split-step quantum walk with strong shift

A hybrid NEQR image encryption cryptosystem using two-dimensional quantum walks and quantum coding

Quantum mechanisms can provide strong security in image processing against the disadvantages of pseudorandom and periodic of traditional chaotic systems. Based on its uncertainty and measurement collapse principle, this study proposes an image encryption system based on two-dimensional quantum walks and quantum coding. First, a pseudorandom number generator (PRNG) based on quantum entanglement and coherence principles is designed to generate a secret code stream inconsistent with the original image. Second, using the new enhanced quantum representation (NEQR) model, a quantum right cyclic shift operator and a quantum XOR operator are designed to obtain the final encrypted image. Experimental results and analysis show that the image encryption scheme has high security performance.

Topological Spin Texture of Chiral Edge States in Photonic Two-Dimensional Quantum Walks.

Topological insulators host topology-linked boundary states, whose spin and charge degrees of freedom could be exploited to design topological devices with enhanced functionality. We experimentally observe that dissipationless chiral edge states in a spin-orbit coupled anomalous Floquet topological phase exhibit topological spin texture on boundaries, realized via a two-dimensional quantum walk. Our experiment shows that, for a walker traveling around a closed loop along the boundary in real space, its spin evolves and winds through a great circle on the Bloch sphere, which implies that edge-spin texture has nontrivial winding. This topological spin winding is protected by a chiral-like symmetry emerging for the low-energy Hamiltonian. Our experiment confirms that two-dimensional anomalous Floquet topological systems exhibit topological spin texture on the boundary, which could inspire novel topology-based spintronic phenomena and devices.

FeynmanPAQS: a graphical interface program for photonic analog quantum computing

We present a user-friendly software for photonic analog quantum computing with a graphical user interface (GUI) that allows for convenient operation without requiring programming skills. Hamiltonians can be flexibly set by either importing the waveguide position files or manually plotting the configuration on the interactive board of the GUI. Our software provides a powerful approach to theoretical studies of two-dimensional quantum walks, quantum stochastic walks, multiparticle quantum walks, and boson sampling, which may all be feasibly implemented in the physical experimental system on photonic chips, and it will inspire a rich diversity of applications for photonic quantum computing and quantum simulation. We have improved algorithms to ensure the efficiency of permanent calculation and provided case studies on educational uses, which bring users easier access to the studies of photonic quantum simulation.

Topological Corner States in Non-Unitary Coinless Discrete-Time Quantum Walks

The discrete-time quantum walk provides a versatile platform for exploring abundant topological phenomena due to its intrinsic spin-orbit coupling. In this work, we study the non-Hermitian second-order topology in a two-dimensional non-unitary coinless discrete-time quantum walk, which is realizable in the three-dimensional photonic waveguides. By adding the non-unitary gain-loss substep operators into the one-step operator of the coinless discrete-time quantum walk, we find the appearance of the four-degenerate zero-dimensional corner states at ReE = 0 when the gain-loss parameter of the system is larger than a critical value. This intriguing phenomenon originates from the nontrivial second-order topology of the system, which can be characterized by a second-order topological invariant of polarizations. Finally, we show that the exotic corner states can be observed experimentally through the probability distributions during the multistep non-unitary coinless discrete-time quantum walks. Our work potentially pave the way for exploring exotic non-Hermitian higher-order topological states of matter in coinless discrete-time quantum walks.

Generating Haar-Uniform Randomness Using Stochastic Quantum Walks on a Photonic Chip

As random operations for quantum systems are intensively used in various quantum information tasks, a trustworthy measure of the randomness in quantum operations is highly demanded. The Haar measure of randomness is a useful tool with wide applications, such as boson sampling. Recently, a theoretical protocol was proposed to combine quantum control theory and driven stochastic quantum walks to generate Haar-uniform random operations. This opens up a promising route to converting classical randomness to quantum randomness. Here, we implement a two-dimensional stochastic quantum walk on the integrated photonic chip and demonstrate that the average of all distribution profiles converges to the even distribution when the evolution length increases, suggesting the 1-pad Haar-uniform randomness. We further show that our two-dimensional array outperforms the one-dimensional array of the same number of waveguide for the speed of convergence. Our Letter demonstrates a scalable and robust way to generate Haar-uniform randomness that can provide useful building blocks to boost future quantum information techniques.

Generation of hyperentangled states and two-dimensional quantum walks using J or q plates and polarization beam splitters

A single photon can be made to entangle simultaneously in its different internal degrees of freedom (DoF)---polarization, orbital angular momentum (OAM), and frequency---as well as in its external DoF---path. Such entanglement in multiple DoF is known as hyperentanglement and provides additional advantage for quantum information processing. We propose a passive optical setup using $q$ plates and polarization beam splitters to hyperentangle an incoming single photon in polarization, OAM, and path DoF. By mapping polarization DoF to a two-dimensional coin state, and path and OAM DoF to two spatial dimensions, $x$ and $y$, we present a scheme for realization of a two-dimensional discrete-time quantum walk using only polarization beam splitters and $q$ plates ensuing the generation of hyperentangled states. The amount of hyperentanglement generated is quantified by measuring the entanglement negativity between any two DoF. We further show that hyperentanglement generation can be controlled by using an additional coin operation or by replacing the $q$ plate with a $J$ plate.

Collective unitary evolution with linear optics by Cartan decomposition

Unitary operation is an essential step for quantum information processing. We first propose an iterative procedure for decomposing a general unitary operation without resorting to controlled-NOT gate and single-qubit rotation library. Based on the results of decomposition, we design two compact architectures to deterministically implement arbitrary two-qubit polarization-spatial and spatial-polarization collective unitary operations, respectively. The involved linear optical elements are reduced from 25 to 20 and 21 to 20, respectively. Moreover, the parameterized quantum computation can be flexibly manipulated by wave plates and phase shifters. As an application, we construct the specific quantum circuits to realize two-dimensional quantum walk and quantum Fourier transformation. Our schemes are simple and feasible with the current technology.

Decoherence in two-dimensional quantum walks with two- and four-state coins

Two-dimensional quantum walks using a two-state coin have simpler experimental implementation than two-dimensional quantum walks using a four-state coin. However, decoherence occurs inevitably during the evolution of quantum walks due to the coupling between the quantum systems and their environment. Thus, it is interesting to investigate the robustness against decoherence for two- and four-state two-dimensional quantum walks. Here, we investigate the effects of the decoherence on two- and four-state two-dimensional quantum walks produced by the broken-link-type noise and compare their robustness against the broken-link-type noise. Specifically, we analyze the quantum correlation between the two spatial dimensions x and y by using measurement-induced disturbance for the two-state quantum walks, i.e. the alternate walk and the Pauli walk, and the four-state quantum walks, i.e. the Grover, Hadamard and Fourier walks, respectively. Our analysis shows that the two-state walks are more robust against the broken-link-type noise than the four-state walks.

Two-dimensional quantum walks of correlated photons

Quantum walks in an elaborately designed graph are a powerful tool for simulating physical and topological phenomena, constructing novel quantum algorithms, and realizing universal quantum computing. Integrated photonics technology has emerged as a versatile platform for implementing a variety of quantum information tasks and as a promising candidate for performing large-scale quantum walks. Both extending physical dimensions and involving more particles will increase the complexity of the evolving systems. Pioneering studies have demonstrated a single particle walking on two-dimensional lattices and multiple walkers interfering on a one-dimensional structure. However, multiple particles evolving in a genuine two-dimensional space in a scalable fashion has remained a vacancy for nearly 10 years. We present a genuine two-dimensional quantum walk with correlated photons on a triangular photonic lattice, which is mapped to a 37 × 37 high-dimensional state space. The genuine two-dimensional quantum walk breaks through the physical restrictions of single-particle evolution, allowing for the encoding of information in large spaces and construction of high-dimensional graphs, which are beneficial for quantum information processing. Between the chip and the two-dimensional fanout interface, site-by-site addressing enables simultaneous detection of over 600 nonclassical interferences and observation of quantum correlations that violate a classical limit by 57 standard deviations. Our implementation provides a paradigm for multi-photon quantum walks in a two-dimensional configuration on a large scale, paving the way for practical quantum simulation and computation beyond the classical regime.

Persistence of topological phases in non-Hermitian quantum walks

Discrete-time quantum walks are known to exhibit exotic topological states and phases. Physical realization of quantum walks in a lossy environment may destroy these phases. We investigate the behaviour 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 respects exact {{mathcal {P}}}{{mathcal {T}}}-symmetry. Although the topological nature persists in two-dimensional quantum walks as well, the {{mathcal {P}}}{{mathcal {T}}}-symmetry has no role to play there. Furthermore, we observe topological phase transition in two-dimensional quantum walks that is induced by losses in the system.

Two-Dimensional Quantum Walk with Non-Hermitian Skin EffectsSupported by the National Natural Science Foundation of China (Grant Nos. 11974331, 11674306, and 61590932), and the National Key R&D Program (Grant Nos. 2016YFA0301700 and 2017YFA0304100).

We construct a two-dimensional, discrete-time quantum walk, exhibiting non-Hermitian skin effects under open-boundary conditions. As a confirmation of the non-Hermitian bulk-boundary correspondence, we show that the emergence of topological edge states is consistent with the Floquet winding number, calculated using a non-Bloch band theory, invoking time-dependent generalized Brillouin zones. Further, the non-Bloch topological invariants associated with quasienergy bands are captured by a non-Hermitian local Chern marker in real space, defined via the local biorthogonal eigenwave functions of a non-unitary Floquet operator. Our work aims to stimulate further studies of non-Hermitian Floquet topological phases where skin effects play a key role.

Topological delocalization in the completely disordered two-dimensional quantum walk

We investigate numerically and theoretically the effect of spatial disorder on two-dimensional split-step discrete-time quantum walks with two internal "coin" states. Spatial disorder can lead to Anderson localization, inhibiting the spread of quantum walks, putting them at a disadvantage against their diffusively spreading classical counterparts. We find that spatial disorder of the most general type, i.e., position-dependent Haar random coin operators, does not lead to Anderson localization but to a diffusive spread instead. This is a delocalization, which happens because disorder places the quantum walk to a critical point between different anomalous Floquet-Anderson insulating topological phases. We base this explanation on the relationship of this general quantum walk to a simpler case more studied in the literature and for which disorder-induced delocalization of a topological origin has been observed. We review topological delocalization for the simpler quantum walk, using time evolution of the wave functions and level spacing statistics. We apply scattering theory to two-dimensional quantum walks and thus calculate the topological invariants of disordered quantum walks, substantiating the topological interpretation of the delocalization and finding signatures of the delocalization in the finite-size scaling of transmission. We show criticality of the Haar random quantum walk by calculating the critical exponent $\eta$ in three different ways and find $\eta$ $\approx$ 0.52 as in the integer quantum Hall effect. Our results showcase how theoretical ideas and numerical tools from solid-state physics can help us understand spatially random quantum walks.

Overview: recent development and applications of reduction and lackadaisicalness techniques for spatial search quantum walk in the near term

The adjacency matrices of graphs provide the foundation for constructing the Hamiltonians of Continuous-Time Quantum Walks (CTQWs). Various classes of graphs have been identified to be highly reducible and the reduced Hamiltonian preserves the dynamics of the original system. This makes the CTQW implementation feasible in the near term for search problems of large size. Highly reducible Hamiltonians are desirable because existing quantum devices are of limited size in terms of the number of qubits. In this work, we review the recent developments of dimensionality reduction and coupling factor value finding techniques. The CTQWs based on a reduced Hamiltonian can search optimally when the correctly calculated coupling factor is used. We list identified highly reducible graphs and include their optimality proofs when correct coupling factors are used. In addition, we discuss the recent developments on Lackadaisical Quantum Walkers (LQW) (a type of coin-based discrete-time quantum walk) for one- and two-dimensional spatial search. The optimal lower upper bound remains open in one- and two-dimensional Discrete-Time Quantum Walk.

A two-dimensional quantum walk driven by a single two-side coin* *Project supported by the National Natural Science Foundation of China (Grant Nos. 11674056 and U1930402) and the startup fund from Beijing Computational Science Research Center. KKW acknowledges support from the Project funded by China Postdoctoral Science Foundation (Grant No. 2019M660016).

We study a two-dimensional quantum walk with only one walker alternatively walking along the horizontal and vertical directions driven by a single two-side coin. We find the analytical expressions of the first two moments of the walker’s position distribution in the long-time limit, which indicates that the variance of the position distribution grows quadratically with walking steps, showing a ballistic spreading typically for quantum walks. Besides, we analyze the correlation by calculating the quantum mutual information and the measurement-induced disturbance respectively as the outcome of the walk in one dimension is correlated to the other with the coin as a bridge. It is shown that the quantum correlation between walker spaces increases gradually with the walking steps.

Topological quantum walk with discrete time-glide symmetry

Discrete quantum walks are periodically driven systems with discrete time evolution. In contrast to ordinary Floquet systems, no microscopic Hamiltonian exists, and the one-period time evolution is given directly by a series of unitary operators. Regarding each constituent unitary operator as a discrete time step, we formulate discrete space-time symmetry in quantum walks and evaluate the corresponding symmetry protected topological phases. In particular, we study chiral and/or time-glide symmetric topological quantum walks in this formalism. Due to discrete nature of time evolution,the topological classification is found to be different from that in conventional Floquet systems. As a concrete example, we study a two-dimensional quantum walk having both chiral and time-glide symmetries, and identify the anomalous edge states protected by these symmetries.

Second-order topological insulator in a coinless discrete-time quantum walk

Higher-order topological insulators not only exhibit exotic bulk-boundary correspondence principle, but also have an important application in quantum computing. However, they have never been achieved in quantum walk. In this paper, we construct a two-dimensional coinless discrete-time quantum walk to simulate second-order topological insulator with zero-dimensional corner states. We show that both of the corner and edge states can be observed through the probability distribution of the walker after multi-step discrete-time quantum walks. Furthermore, we demonstrate the robustness of the topological corner states by introducing the static disorder. Finally, we propose a possible experimental implementation to realize this discrete-time quantum walk in a three-dimensional integrated photonic circuits. Our work offers a new route to explore exotic higher-order topological matters using discrete-time quantum walks.