The Simplest 2D Quantum Walk Detects Chaoticity

Discrete-time quantum walks, quantum generalizations of classical random walks, provide a framework for quantum information processing, quantum algorithms and quantum simulation of condensed matter systems. The key property of quantum walks, which lies at the heart of their quantum information applications, is the possibility for a parametric quantum speed-up in propagation compared to classical random walks. In this work we study propagation of quantum walks on percolation-generated two-dimensional random lattices. In large-scale simulations of topological and trivial split-step walks, we identify distinct pre-diffusive and diffusive behaviors at different time scales. Importantly, we show that even arbitrarily weak concentrations of randomly removed lattice sites give rise to a complete breakdown of the superdiffusive quantum speed-up, reducing the motion to ordinary diffusion. By increasing the randomness, quantum walks eventually stop spreading due to Anderson localization. Near the localization threshold, we find that the quantum walks become subdiffusive. The fragility of quantum speed-up implies dramatic limitations for quantum information applications of quantum walks on random geometries and graphs.

Studies on two-particle quantum walks show that the spatial interaction between walkers will dynamically generate complex entanglement. However, those entanglement states are usually on a large state space and their evolutions are complex. It makes the entanglement states generated by quantum walk difficult to be applied directly in many applications of quantum information, such as quantum teleportation and quantum cryptography. In this paper, we firstly analyse a localization phenomena of two-particle quantum walk and then introduce how to use it to generate a Bell state. We will show that one special superposition component of the walkers' state is localized on the root vertex if a certain interaction exists between walkers. This localization is interesting because it is contrary to our knowledge that quantum walk spreads faster than its classical counterpart. More interestingly, the localized component is a Bell state in the coin space of two walkers. By this method, we can obtain a Bell state easily from the quantum walk with spatial interaction by a local measurement, which is required in many applications. Through simulations, we verify that this method is able to generate the Bell state $$\frac{1}{\sqrt{2}}(|A \rangle _1|A\rangle _2 \pm |B\rangle _1|B\rangle _2)$$12(|Aź1|Aź2±|Bź1|Bź2) in the coin space of two walkers with fidelity greater than $$99.99999\,\%$$99.99999% in theory, and we have at least a $$50\,\%$$50% probability to obtain the expected Bell state after a proper local measurement.

A quantum walk, that is, a synthetic quantum system mainly impremented by photonic systems, whose dynamics is described by a time-evolution operator, provides potential applications for quantum computing and information. It is further interesting that the quantum walk possesses novel topological phases akin to those of Floquet topological insulators, which are topological insulators driven by a time-periodic field[1,2]. Recently, a one-dimensional quantum walk dynamics associated with gain and loss is experimentally implemented by optical fiber loops [3]. The experiment shows that the energy of the system is kept to be real as a manifestation of PT symmetry (combined parity and time-reversal symmetry) in spite of the open quantum system.

In the age of the post-Moore era, the next-generation computing model would be a hybrid architecture consisting of different physical components, such as photonic chips. In 2008, it was proposed that the solving of the NAND-tree problem can be sped up by quantum walk. This scheme is groundbreaking due to the universality of the NAND gate. However, experimental demonstration has not been achieved so far, mostly due to the challenge in preparing the propagating initial state. Here we propose an alternative solution by including a structure called a "quantum slide," where a propagating Gaussian wave packet can be generated deterministically along a properly engineered chain. In our experimental demonstration, the optical NAND tree is capable of solving computational problems with a total of four input bits, based on the femtosecond laser 3D direct-writing technique on a photonic chip. These results remove one main roadblock to photonic NAND-tree computation, and the construction of a quantum slide may find other interesting applications in quantum information and quantum optics.

Open quantum walks (OQWs) (also known as open quantum random walks) are quantum analogs of classical Markov chains in probability theory, and have potential application in quantum information and quantum computation. Quantum Bernoulli noises (QBNs) are annihilation and creation operators acting on Bernoulli functionals, and can be used as the environment of an open quantum system. In this paper, by using QBNs as the environment, we introduce an OQW on a general higher-dimensional integer lattice. We obtain a quantum channel representation of the walk, which shows that the walk is indeed an OQW. We prove that all the states of the walk are separable provided its initial state is separable. We also prove that, for some initial states, the walk has a limit probability distribution of higher-dimensional Gauss type. Finally, we show links between the walk and a unitary quantum walk recently introduced in terms of QBNs.

Motivated by applications in quantum information and quantum control, a new type of C-numerical range, the relative C-numerical range denoted as WK (C, A), is introduced. It arises upon replacing the unitary group U(N) in the definition of the classical C-numerical range by any of its compact and connected subgroups K ⊂ U(N). The geometric properties of the relative C-numerical range are analyzed in detail. Counterexamples prove that its geometry is more intricate than in the classical case: e.g., W K (C, A) is neither star-shaped nor simply connected. Yet, a well-known result on the rotational symmetry of the classical C-numerical range extends to WK (C, A), as shown by a new approach based on Lie theory. Furthermore, we concentrate on the subgroup , i.e., the n-fold tensor product of SU(2), which is of particular interest in applications. In this case, sufficient conditions are derived for WK (C, A) being a circular disc centered at the origin of the complex plane. Finally, the previous results are illustrated in detail for SU(2) ⊗ SU(2).

Protein-DNA target search relies on quantum walk

A large number of scientific proposals made in the last few years are based on transport and manipulation of information using single quantum objects. Some of them make use of entanglement in pairs of particles such as twin photons. Although theoretical proposals have demonstrated highly interesting perspectives in the quantum information domain, experimental realizations and applications still suffer from the complexity of experimental set-ups and technological limitations. This paper presents various approaches aiming at efficient twin photon semiconductor sources. The emergence of these compact and integrated devices would be an important technological breakthrough in quantum information applications.

Photon polarization serves as an essential quantum information carrier in quantum information and measurement applications. Routing of arbitrarily polarized single photons and polarization‐entangled photons is a crucial technology for scaling up quantum information applications. Here, a low‐loss, noiseless, polarization‐maintaining routing of arbitrarily polarized single photons and, crucially, multi‐photon entangled states is demonstrated where the entanglement is encoded in orthogonal polarization bases, at the telecom L‐band. The interferometer‐based router is constructed by optics with a low angle of incidence and cross‐aligned electro‐optic crystals, achieving the polarization‐maintaining operation with a minimal number of optical components. The routing of arbitrarily‐polarized heralded single photons with a 0.057 dB (1.3%) loss, a 22 dB switching extinction ratio, and 99% polarization process fidelity to ideal identity operation is demonstrated. Moreover, the high‐quality router achieves the routing of two‐photon N00N‐type entangled states with a highly maintained interference visibility of 97%. The demonstrated router scheme preserving multi‐photon polarization state paves the way toward polarization‐encoded photonic quantum network as well as multi‐photon entanglement synthesis via spatial‐ and time‐multiplexing techniques.

Understanding quantum dynamical features seems to be needed to open the next step of quantum information science. One strong candidate to understand quantum dynamical phenomena is a quantumwalk. Quantumwalks are defined as the quantummechanical analogue of the classical random walk. However, those are not the quantization of the classical random walks. Like the classical random walks, there may be many applications of the quantum walks. Therefore, the community of the quantum walk is an interdisciplinary field from mathematics to the experimental physics. The quantum walks have two types; the discrete-time and the continuous-time quantum walks. Both common and unique features are the followings; (i) the ballistic transport system, which corresponds to quantum transportation and (ii) non-normal probability distribution under the uniform system, and (iii) the Anderson-like localization. By combining these features, quantum walks show an intriguing phenomena in mathematics, physics, and computer science. The community of quantum walks is gradually increased. They held some workshops, e.g., at Tokyo Institute of Technology, Tokyo, Japan in 2011 (cancellation due to the Japan big earthquakes and aftermath) and at Instituto de Fisica Corpuscular, Valencia, Spain in 2011. A workshop will be held at the Institute for Molecular Science, Okazaki, Japan on November, 2012 to collect the quantum walkers. Many scientists have a potential to study the quantum walks. The aim of this special issue is to know what is the role of quantum walks for the other potentially related fields and to reveal many varieties of unique aspects and open problems on the quantum walks. Therefore, we have prepared three comprehensive reviews from mathematics, theoretical physics, and quantum information, and seven contributed articles from theoretical viewpoints. I hope that the quantum walks

Quantum walks are the quantum counterpart of classical random walks and have various applications in quantum information science. Polar molecules have rich internal energy structure and long coherence time and thus are considered as a promising candidate for quantum information processing. In this paper, we propose a theoretical scheme for implementing discrete-time quantum walks on a circle with dipole-dipole coupled SrO molecules. The states of the walker and the coin are encoded in the pendular states of polar molecules induced by an external electric field. We design the optimal microwave pulses for implementing quantum walks on a four-node circle and a three-node circle by multi-target optimal control theory. To reduce the accumulation of decoherence and improve the fidelity, we successfully realize a step of quantum walk with only one optimal pulse. Moreover, we also encode the walker into a three-level molecular qutrit and a four-level molecular ququart and design the corresponding optimal pulses for quantum walks, which can reduce the number of molecules used. It is found that all the quantum walks on a circle in our scheme can be achieved via optimal control fields with high fidelities. Our results could shed some light on the implementation of discrete-time quantum walks and high-dimensional quantum information processing with polar molecules.

Quantum random walks have attracted special interest because they could lead to new quantum algorithms. Photons can carry orbital angular momentum (OAM) thereby offering a practical realization of a high-dimensional quantum information carrier. By employing OAM of photons, we experimentally realized the one-dimensional discrete-time quantum random walk. Three steps of a one-dimensional quantum random walk were implemented in our protocol showing the obvious difference between quantum and classical random walks.

Quantum walks are fundamentally different from random walks due to the quantum superposition property of quantum objects. Quantum walk process was found to be very useful for quantum information and quantum computation applications. In this paper we demonstrate how to use quantum walks as a tool to generate high-dimensional two-particle fermionic entanglement. The generated entanglement can survive longer in the presence of depolorazing noise due to the periodicity of quantum walk dynamics. The possibility to create two distinguishable qudits in a system of tunnel-coupled semiconductor quantum dots is discussed.

There are some problems that quantum computers seem to be exponentially faster than classical computers, like factoring large numbers, machine learning, and simulation of quantum systems. Constructing an appropriate quantum algorithm becomes more important for solving these specific problems. In principle, any quantum algorithm can recast by a quantum random walk algorithm. Although quantum random walk with a few qubits has been implemented in a variety of systems, the experimental demonstration of solid-state quantum random walk remains elusive. Here we report the experimental implementation of the quantum continuous-time random walk algorithm by a two-qubit quantum processor in a nitrogen–vacancy center in diamond. We found that quantum random walk on a circle does not converge to any stationary distribution and exhibit a reversible property. Our results represent a further investigation of quantum walking dynamics in solid spin platforms, may also lead to other practical applications by the use of quantum continuous-time random walk for quantum algorithm design and quantum coherence transport.

The concept of quantum steering was originally introduced by Schrödinger to describe the "spooky action-at-a-distance" effect noted in the Einstein-Podolsky-Rosen (EPR) paradox, whereby local measurements performed on one party apparently adjust (steer) the state of another distant party. In this talk, I will give an introduction about the advances of the EPR steering and its advantage as quantum resource. Then I will present our efforts on characterizing bipartite and multipartite steering and developing its unique applications in quantum information processing. I will give an overview of our recent developments on quantum steering and its applications in quantum information. I will share our view about the current challenges, opportunities and the future directions for this topic.