Criticality, the area law, and the computational power of projected entangled pair states

The projected entangled pair state (PEPS) representation of quantum states on two-dimensional lattices induces an entanglement based hierarchy in state space. We show that the lowest levels of this hierarchy exhibit a very rich structure including states with critical and topological properties. We prove, in particular, that coherent versions of thermal states of any local 2D classical spin model correspond to such PEPS, which are in turn ground states of local 2D quantum Hamiltonians. This correspondence maps thermal onto quantum fluctuations, and it allows us to analytically construct critical quantum models exhibiting a strict area law scaling of the entanglement entropy in the face of power law decaying correlations. Moreover, it enables us to show that there exist PEPS which can serve as computational resources for the solution of NP-hard problems.     

Final state effects of deep impurities in semiconductors

Current theoretical and computational models for optical excitation processes of deep level impurities in semiconductors tend to concentrate on the impurity itself, largely ignoring the effects the impurity has no final states. Using a spherical band model, we show these effects can be included in calculations on optical absorption processes. The band state is modified by the presence of the impurity and this modified state is used in the calculation of optical matrix elements. We show that final state effects can cause significant changes in the local density of states and optical matrix elements.     

Computational difficulty of computing the density of states

We study the computational difficulty of computing the ground state degeneracy and the density of states for local Hamiltonians. We show that the difficulty of both problems is exactly captured by a class which we call #BQP, which is the counting version of the quantum complexity class quantum Merlin Arthur. We show that #BQP is not harder than its classical counting counterpart #P, which in turn implies that computing the ground state degeneracy or the density of states for classical Hamiltonians is just as hard as it is for quantum Hamiltonians.     

Low depth quantum circuits for Ising models

A scheme for measuring complex temperature partition functions of Ising models is introduced. Two applications of this scheme are presented. First, through appropriate Wick rotations, those amplitudes can be analytically continued to yield estimates for partition functions of Ising models. Bounds on the estimated error are provided through a central-limit theorem whose validity extends beyond the present context; it holds for example for estimations of the Jones polynomial. The kind of state preparations and measurements involved in this application can be made independent of the system size or the parameters of the system being simulated. Second, the scheme allows to accurately estimate non-trivial invariants of links. Another result concerns the computational power of estimations of partition functions for real temperature classical ferromagnetic Ising models. We provide conditions under which estimating such partition functions allows to reconstruct scattering amplitudes of quantum circuits, making the problem BQP-hard. We also show fidelity overlaps for ground states of quantum Hamiltonians, which serve as a witness to quantum phase transitions, can be estimated from classical Ising model partition functions. Finally, we discuss how accurate corner magnetisation measurements on thermal states of two-dimensional Ising models lead to fully polynomial random approximation schemes (FPRAS) for the partition function.     

Entanglement or separability: the choice of how to factorize the algebra of a density matrix

Quantum entanglement has become a resource for the fascinating developments in quantum information and quantum communication during the last decades. It quantifies a certain nonclassical correlation property of a density matrix representing the quantum state of a composite system. We discuss the concept of how entanglement changes with respect to different factorizations of the algebra which describes the total quantum system. Depending on the considered factorization a quantum state appears either entangled or separable. For pure states we always can switch unitarily between separability and entanglement, however, for mixed states a minimal amount of mixedness is needed. We discuss our general statements in detail for the familiar case of qubits, the GHZ states, Werner states and Gisin states, emphasizing their geometric features. As theorists we use and play with this free choice of factorization, which for an experimentalist is often naturally fixed. For theorists it offers an extension of the interpretations and is adequate to generalizations, as we point out in the examples of quantum teleportation and entanglement swapping.     

Universal resources for measurement-based quantum computation

We investigate which entanglement resources allow universal measurement-based quantum computation via single-qubit operations. We find that any entanglement feature exhibited by the 2D cluster state must also be present in any other universal resource. We obtain a powerful criterion to assess the universality of graph states by introducing an entanglement measure which necessarily grows unboundedly with the system size for all universal resource states. Furthermore, we prove that graph states associated with 2D lattices such as the hexagonal and triangular lattice are universal, and obtain the first example of a universal nongraph state.     

Entanglement Dynamics of Electrons and Photons

Entanglement is a fundamental feature of quantum theory as well as a key resource for quantum computing and quantum communication, but the entanglement mechanism has not been found at present. We think when the two subsystems exist interaction directly or indirectly, they can be in entanglement state. such as, in the Jaynes-Cummings model, the entanglement between the atom and the light field comes from their interaction. In this paper, we have studied the entanglement mechanism of electron-electron and photon-photon, which are from the spin-spin interaction. We found their total entanglement states are relevant both space state and spin state. When two electrons or two photons are far away, their entanglement states should be disappeared even if their spin state is entangled.     

Joint remote state preparation of a W-type state via W-type states

An all W-type state task is put forward: joint remote state preparation of a W-type state via W-type states. We propose two probabilistic yet faithful schemes for the task. The first scheme uses two arbitrary W-type states as the shared quantum resource and the second scheme exploits three such states. We show that, while the first scheme requires some additional quantum resource and technical operations from the receiver, the second scheme allows any completely unequipped party to play the role of receiver. In both schemes the classical communication cost is one bit per preparer.     

Measurement-Based Quantum Computation and Undecidable Logic

We establish a connection between measurement-based quantum computation and the field of mathematical logic. We show that the computational power of an important class of quantum states called graph states, representing resources for measurement-based quantum computation, is reflected in the expressive power of (classical) formal logic languages defined on the underlying mathematical graphs. In particular, we show that for all graph state resources which can yield a computational speed-up with respect to classical computation, the underlying graphs—describing the quantum correlations of the states—are associated with undecidable logic theories. Here undecidability is to be interpreted in a sense similar to Gödel’s incompleteness results, meaning that there exist propositions, expressible in the above classical formal logic, which cannot be proven or disproven.     

Chaos in social learning with multiple true states

Most existing social learning models assume that there is only one underlying true state. In this work, we consider a social learning model with multiple true states, in which agents in different groups receive different signal sequences generated by their corresponding underlying true states. Each agent updates his belief by combining his rational self-adjustment based on the external signals he received and the influence of his neighbors according to their communication. We observe chaotic oscillation in the belief evolution, which implies that neither true state could be learnt correctly by calculating the largest Lyapunov exponents and Hurst exponents.     

Necessary and sufficient condition for nonzero quantum discord

Quantum discord characterizes "nonclassicality" of correlations in quantum mechanics. It has been proposed as the key resource present in certain quantum communication tasks and quantum computational models without containing much entanglement. We obtain a necessary and sufficient condition for the existence of nonzero quantum discord for any dimensional bipartite states. This condition is easily experimentally implementable. Based on this, we propose a geometrical way of quantifying quantum discord. For two qubits this results in a closed form of expression for discord. We apply our results to the model of deterministic quantum computation with one qubit, showing that quantum discord is unlikely to be the reason behind its speedup.     

Hexagonal discrete Boltzmann models with and without rest particles

We consider seven different hexagonal discrete Boltzmann models corresponding to one, two, three, and five hexagons with or without rest particles. In the microscopic collisions the number of particles associated with a given speed is not necessarily conserved, except for two models without rest particles. We compare different behaviors for the macroscopic quantities between models with and without rest particles and when the number of velocities (or hexagons) increases. We study similarity waves with two asymptotic states and consider two classes of solutions at one asymptotic state: either isotropic (densities associated with the same speed are equal) or anisotropic. Two macroscopic quantities seem useful for such studies: internal energy and mass ratio across the asymptotic states, which satisfy a relation deduced from continuous theory. Here we report results for the isotropic solutions, whoch only exist, for both models, in the subdomains where the propagation speed is larger than some well-defined value. Outside these subdomains, modifications occur when the rest particle desity becomes large. For both models we find a monotonic internal energy and subdomains with a mass ratio equal to the one in continuous theory.     

Quantum Computation of Fuzzy Numbers

We study the possibility of performing fuzzy set operations on a quantum computer. After giving a brief overview of the necessary quantum computational and fuzzy set theoretical concepts we demonstrate how to encode the membership function of a digitized fuzzy number in the state space of a quantum register by using a suitable superposition of tensor product states that form a computational basis. We show that a standard quantum adder is capable to perform Kaufmann's addition of fuzzy numbers in the course of only one run by acting at once on all states in the superposition, which leads to a considerable gain in the number of required operations with respect to performing such addition on a classical computer.     

Computational difficulty of finding matrix product ground states

We determine the computational difficulty of finding ground states of one-dimensional (1D) Hamiltonians, which are known to be matrix product states (MPS). To this end, we construct a class of 1D frustration-free Hamiltonians with unique MPS ground states and a polynomial gap above, for which finding the ground state is at least as hard as factoring. Without the uniqueness of the ground state, the problem becomes NP complete, and thus for these Hamiltonians it cannot even be certified that the ground state has been found. This poses new bounds on convergence proofs for variational methods that use MPS.     

Partially entangled states bridge in quantum teleportation

The traditional method for information transfer in a quantum communication system using partially entangled state resource is quantum distillation or direct teleportation. In order to reduce the waiting time cost in hop-by-hop transmission and execute independently in each node, we propose a quantum bridging method with partially entangled states to teleport quantum states from source node to destination node. We also prove that the designed specific quantum bridging circuit is feasible for partially entangled states teleportation across multiple intermediate nodes. Compared to two traditional ways, our partially entanglement quantum bridging method uses simpler logic gates, has better security, and can be used in less quantum resource situation.     

Implementation of the Deutsch-Jozsa algorithm violates nonlocal realism

The theoretical resource state for the implementation of the Deutsch-Jozsa algorithm is a multiqubit pure uncorrelated state. We show that N-qubit pure uncorrelated quantum states cannot admit rotationally invariant nonlocal realistic theories with a violation factor of 3^N. We find the violation factor 3^Nwhen the measurement setup is entire range of settings for each of the observers, that is, considering rotationally invariant nonlocal realistic theories along with the property of a correlation function in the quantum theory. The implementation of the Deutsch-Jozsa algorithm theoretically relying on N-qubit pure uncorrelated states rules out rotationally invariant nonlocal realism with a violation factor of 3^Nin an ideal case. Our analysis relies on the property of theoretical resource states for the algorithm. We cannot simulate the Deutsch-Jozsa algorithm by using rotationally invariant nonlocal realistic theories due to the property of theoretical resource states for the algorithm.     

Kaleidoscope of life: A 24-neighbourhood outer-totalistic cellular automaton

One of the challenges of cellular automaton research is finding models with a low complexity and at the same time a rich dynamics. A measure of low complexity is the number of states in the model and the number of transition rules to switch between those states. In this paper, we propose a 2-dimensional 2-state cellular automaton that-though governed by a single simple transition rule-has a sufficiently rich dynamics to be computationally universal. According to the transition rule, a cell’s state is determined by the sum of the states of the cells at orthogonal or diagonal distances one or two from the cell (distance-2 Moore neighbourhood), but not by the previous state of the cell itself. Notwithstanding its simplicity, this model is able to generate a great variety of patterns, including several types of stable configurations, oscillators and patterns that move over cellular space (gliders). We prove the computational universality of the model by constructing a universal set of logic gates (NOT and AND) from these patterns. A key element in this proof is the shifting of phases and positions of signals such that they meet the input requirements of the logic gates. Similarities of the model with classical spin systems are also discussed.     

Variational quantum eigensolvers by variance minimization

The original variational quantum eigensolver (VQE) typically minimizes energy with hybrid quantum-classical optimization that aims to find the ground state. Here, we propose a VQE based on minimizing energy variance and call it the variance-VQE, which treats the ground state and excited states on the same footing, since an arbitrary eigenstate for a Hamiltonian should have zero energy variance. We demonstrate the properties of the variance-VQE for solving a set of excited states in quantum chemistry problems. Remarkably, we show that optimization of a combination of energy and variance may be more efficient to find low-energy excited states than those of minimizing energy or variance alone. We further reveal that the optimization can be boosted with stochastic gradient descent by Hamiltonian sampling, which uses only a few terms of the Hamiltonian and thus significantly reduces the quantum resource for evaluating variance and its gradients.     

Computing with a single qubit faster than the computation quantum speed limit

The possibility to save and process information in fundamentally indistinguishable states is the quantum mechanical resource that is not encountered in classical computing. I demonstrate that, if energy constraints are imposed, this resource can be used to accelerate information-processing without relying on entanglement or any other type of quantum correlations. In fact, there are computational problems that can be solved much faster, in comparison to currently used classical schemes, by saving intermediate information in nonorthogonal states of just a single qubit. There are also error correction strategies that protect such computations.     

Variational quantum simulation of thermal statistical states on a superconducting quantum processer

Quantum computers promise to solve finite-temperature properties of quantum many-body systems, which is generally challenging for classical computers due to high computational complexities. Here, we report experimental preparations of Gibbs states and excited states of Heisenberg $XX$ and $XXZ$ models by using a 5-qubit programmable superconducting processor. In the experiments, we apply a hybrid quantum-classical algorithm to generate finite temperature states with classical probability models and variational quantum circuits. We reveal that the Hamiltonians can be fully diagonalized with optimized quantum circuits, which enable us to prepare excited states at arbitrary energy density. We demonstrate that the approach has a self-verifying feature and can estimate fundamental thermal observables with a small statistical error. Based on numerical results, we further show that the time complexity of our approach scales polynomially in the number of qubits, revealing its potential in solving large-scale problems.