Path-integral solution of the one-dimensional Dirac quantum cellular automaton

Quantum cellular automata, which describe the discrete and exactly causal unitary evolution of a lattice of quantum systems, have been recently considered as a fundamental approach to quantum field theory and a linear automaton for the Dirac equation in one dimension has been derived. In the linear case a quantum cellular automaton is isomorphic to a quantum walk and its evolution is conveniently formulated in terms of transition matrices. The semigroup structure of the matrices leads to a new kind of discrete path-integral, different from the well known Feynman checkerboard one, that is solved analytically in terms of Jacobi polynomials of the arbitrary mass parameter.     

Universally programmable quantum cellular automaton

We discuss the role of classical control in the context of reversible quantum cellular automata. Employing the structure theorem for quantum cellular automata, we give a general construction scheme to turn an arbitrary cellular automaton with external classical control into an autonomous one, thereby proving the computational equivalence of these two models. We use this technique to construct a universally programmable cellular automaton on a one-dimensional lattice with single cell dimension 12.     

Zeno''s paradox in quantum cellular automata

The effect of Zeno's paradox in quantum theory is demonstrated with the aid of quantum mechanical cellular automata. It is shown that the degree of non-unitarity of the cellular automaton evolution and the frequency of consecutive measurements of cellular automaton states are operationally indistinguishable.     

Equivalent programmable quantum processors

Programmable quantum circuits, or processors, have the advantage over single-purpose quantum circuits that they can be used to perform more than one function. The inputs of a quantum processor consist of two quantum states, the first, the data register, is a state on which an operation is to be performed, and the second, the program, determines the operation to be performed on the data. In this paper we study how to determine whether two different quantum processors perform the same set of operations on the data. We define an equivalence between quantum processors that is quite natural in a circuit model of quantum information processing. Two processors are equivalent if one can be converted into the other by inserting fixed unitary gates at the input and the output of the program register. We then use this definition to find a necessary condition for two processors to be equivalent. We also study the beam splitter as an example of a quantum processor and find that this example suggests that as well as there being an equivalence relation on processors, there may also be a partial ordering.     

Lattices of Quantum Automata

We defined and studied three different types of lattice-valued finite state quantum automata (LQA) and four different kinds of LQA operations, discussed their advantages, disadvantages, and various properties. There are four major results obtained in this paper. First, no one of the above mentioned LQA follows the law of lattice value conservation. Second, the theorem of classical automata theory, that each nondeterministic finite state automaton has an equivalent deterministic one, is not necessarily valid for finite state quantum automata. Third, we proved the existence of semilattices and also lattices formed by different types of LQA. Fourth, there are tight relations between properties of the original lattice l and those of the l-valued lattice formed by LQA.     

Sequential implementation of global quantum operations

We study the possibility for a global unitary applied on an arbitrary number of qubits to be decomposed in a sequential unitary procedure, where an ancillary system is allowed to interact only once with each qubit. We prove that sequential unitary decompositions are in general impossible for genuine entangling operations, even with an infinite-dimensional ancilla, being the controlled-NOT gate a paradigmatic example. Nevertheless, we find particular nontrivial operations in quantum information that can be performed in a sequential unitary manner, as is the case of quantum error correction and quantum cloning.     

Quantum field as a quantum cellular automaton: The Dirac free evolution in one dimension

We present a quantum cellular automaton model in one space-dimension which has the Dirac equation as emergent. This model, a discrete-time and causal unitary evolution of a lattice of quantum systems, is derived from the assumptions of homogeneity, parity and time-reversal invariance.     

Unifying variational methods for simulating quantum many-body systems

We introduce a unified formulation of variational methods for simulating ground state properties of quantum many-body systems. The key feature is a novel variational method over quantum circuits via infinitesimal unitary transformations, inspired by flow equation methods. Variational classes are represented as efficiently contractible unitary networks, including the matrix-product states of density matrix renormalization, multiscale entanglement renormalization (MERA) states, weighted graph states, and quantum cellular automata. In particular, this provides a tool for varying over classes of states, such as MERA, for which so far no efficient way of variation has been known. The scheme is flexible when it comes to hybridizing methods or formulating new ones. We demonstrate the functioning by numerical implementations of MERA, matrix-product states, and a new variational set on benchmarks.     

On the absence of homogeneous scalar unitary cellular automata

Failure to find homogeneous scalar unitary cellular automata (CA) in one dimension led to consideration of only “approximately unitary” CA - which motivated our recent proof of a No-go Lemma in one dimension. In this note we extend the one dimensional result to prove the absence of nontrivial homogeneous scalar unitary CA on Euclidean lattices in any dimension.     

Directional quantum random walk induced by coherence

Quantum walk (QW), which is considered as the quantum counterpart of the classical random walk (CRW), is actually the quantum extension of CRW from the single-coin interpretation. The sequential unitary evolution engenders correlation between different steps in QW and leads to a non-binomial position distribution. In this paper, we propose an alternative quantum extension of CRW from the ensemble interpretation, named quantum random walk (QRW), where the walker has many unrelated coins, modeled as two-level systems, initially prepared in the same state. We calculate the walker's position distribution in QRW for different initial coin states with the coin operator chosen as Hadamard matrix. In one-dimensional case, the walker's position is the asymmetric binomial distribution. We further demonstrate that in QRW, coherence leads the walker to perform directional movement. For an initially decoherenced coin state, the walker's position distribution is exactly the same as that of CRW. Moreover, we study QRW in 2D lattice, where the coherence plays a more diversified role in the walker's position distribution.     

基于非对称量子通道受控QOT量子投票协议

提出一种量子投票协议, 协议基于非对称量子通道受控量子局域幺正操作隐形传输(quantum operation teleportation, QOT). 由公正机构CA提供的零知识证明的量子身份认证, 保证选民身份认证的匿名性. 计票机构Bob制造高维Greenberger-Horne-Zeilinger 纠缠态建立一个高维量子通信信道. 选民对低维的量子选票进行局域幺正操作的量子投票, 是通过非对称基的测量和监票机构Charlie的辅助测量隐形传输的. Bob在Charlie帮助下可以通过幺正操作结果得到投票结果. 与其他一般的QOT量子投票协议相比, 该协议利用量子信息与传输的量子信道不同维, 使单粒子信息不能被窃取、防止伪造.选举过程由于有Charlie的监督, 使得投票公正和不可抵赖.由于量子局域幺正操作隐形传输的成功概率是1, 使量子投票的可靠性得以保证. 关键词： 量子投票 高维GHZ纠缠态 非对称基测量 量子操作隐形传输     

Ergodicity of quantum cellular automata

We define a class of dynamical maps on the quasi-local algebra of a quantum spin system, which are quantum analoges of probabilistic cellular automata. We develop criteria for such a system to be ergodic, i.e., to posses a unique invariant state. Intuitively, ergodicity obtains if the local transition operators exhibit sufficiently large disorder. The ergodicity criteria also imply bounds for the exponential decay of correlations in the unique invariant state. The main technical tool is a quantum version of oscillation norms, defined in the classical case as the sum over all sites of the variations of an observable with respect to local spin flips.     

Entanglement generation of Clifford quantum cellular automata

Clifford quantum cellular automata (CQCAs) are a special kind of quantum cellular automata (QCAs) that incorporate Clifford group operations for the time evolution. Despite being classically simulable, they can be used as basic building blocks for universal quantum computation. This is due to the connection to translation-invariant stabilizer states and their entanglement properties. We will give a self-contained introduction to CQCAs and investigate the generation of entanglement under CQCA action. Furthermore, we will discuss finite configurations and applications of CQCAs.     

Quantum ω-Automata over Infinite Words and Their Relationships

Inspired by the results of finite automata working on infinite words, we studied the quantum ω-automata with Büchi, Muller, Rabin and Streett acceptance condition. Quantum finite automata play a pivotal part in quantum information and computational theory. Investigation of the power of quantum finite automata over infinite words is a natural goal. We have investigated the classes of quantum ω-automata from two aspects: the language recognition and their closure properties. It has been shown that quantum Muller automaton is more dominant than quantum Büchi automaton. Furthermore, we have demonstrated the languages recognized by one-way quantum finite automata with different quantum acceptance conditions. Finally, we have proved the closure properties of quantum ω-automata.

     

Quantifying quantum correlation via quantum coherence

Resource theory is applied to quantify the quantum correlation of a bipartite state and a computable measure is proposed. Since this measure is based on quantum coherence, we present another possible physical meaning for quantum correlation, i.e., the minimum quantum coherence achieved under local unitary transformations. This measure satisfies the basic requirements for quantifying quantum correlation and coincides with concurrence for pure states. Since no optimization is involved in the final definition, this measure is easy to compute irrespective of the Hilbert space dimension of the bipartite state.     

Several teleportation schemes of an arbitrary unknown multi-particle state via different quantum channels

We first provide four new schemes for two-party quantum teleportation of an arbitrary unknown multi-particle state by using three-, four-, and five-particle states as the quantum channel, respectively. The successful probability and fidelity of the four schemes reach 1. In the first two schemes, the receiver can only apply one of the unitary transformations to reconstruct the original state, making it easier for these two schemes to be directly realized. In the third and fourth schemes, the sender can preform Bell-state measurements instead of multipartite entanglement measurements of the existing similar schemes, which makes real experiments more suitable. It is found that the last three schemes may become tripartite controlled teleportation schemes of teleporting an arbitrary multi-particle state after a simple modification. Finally, we present a new scheme for three-party sharing an arbitrary unknown multi-particle state. In this scheme, the sender first shares three three-particle GHZ states with two agents. After setting up the secure quantum channel, an arbitrary unknown multi-particle state can be perfectly teleported if the sender performs three Bell-state measurements, and either of two receivers operates an appropriate unitary transformation to obtain the original state with the help of other receiver's three single-particle measurements. The successful probability and fidelity of this scheme also reach 1. It is demonstrated that this scheme can be generalized easily to the case of sharing an arbitrary unknown multi-particle state among several agents.     

Maximally causal quantum mechanics

We present a new causal quantum mechanics in one and two dimensions developed recently at TIFR by this author and V Singh. In this theory both position and momentum for a system point have Hamiltonian evolution in such a way that the ensemble of system points leads to position and momentum probability densities agreeing exactly with ordinary quantum mechanics