scholarly journals Matchgate and space-bounded quantum computations are equivalent

2009 ◽  
Vol 466 (2115) ◽  
pp. 809-830 ◽  
Author(s):  
Richard Jozsa ◽  
Barbara Kraus ◽  
Akimasa Miyake ◽  
John Watrous
Keyword(s):  
Quantum Computation ◽  
Polynomial Time ◽  
Perfect Matchings ◽  
Spin Chains ◽  
Quantum Gates ◽  
Nearest Neighbour ◽  
Computational Power ◽  
One Dimensional ◽  
Logarithmic Space ◽  
Algebraic Constraints

Matchgates are an especially multiflorous class of two-qubit nearest-neighbour quantum gates, defined by a set of algebraic constraints. They occur for example in the theory of perfect matchings of graphs, non-interacting fermions and one-dimensional spin chains. We show that the computational power of circuits of matchgates is equivalent to that of space-bounded quantum computation with unitary gates, with space restricted to being logarithmic in the width of the matchgate circuit. In particular, for the conventional setting of polynomial-sized (logarithmic-space generated) families of matchgate circuits, known to be classically simulatable, we characterize their power as coinciding with polynomial-time and logarithmic-space-bounded universal unitary quantum computation.

QUANTUM COMPUTATION WITH RESTRICTED AMPLITUDES

2003 ◽  
Vol 14 (05) ◽  
pp. 853-870 ◽  
Author(s):  
HARUMICHI NISHIMURA
Keyword(s):  
Quantum Computation ◽  
Polynomial Time ◽  
Exact Algorithms ◽  
Quantum Computers ◽  
Turing Machines ◽  
Computational Power ◽  
Polynomial Time Algorithms ◽  
Exact Quantum ◽  
Bounded Error ◽  
Transition Amplitudes

In this paper, we explore the power of quantum computers with restricted transition amplitudes. In 1997 Adleman, DeMarrais, and Huang showed that quantum Turing machines (QTMs) with the amplitudes from [Formula: see text] are computationally equivalent to ones with the polynomial-time computable amplitudes as machines implementing bounded-error polynomial-time algorithms. We show that QTMs with the amplitudes from [Formula: see text] is polynomial-time equivalent to deterministic Turing machines as machines implementing exact algorithms, i.e., algorithms that output correct answers with certainty. By extending this result, it is shown that exact quantum computers with rational biased coins are equivalent to classical computers. Moreover, we discuss the computational power of exact quantum computers with multiple types of coins. We also show that, from the viewpoint of zero-error polynomial-time algorithms, [Formula: see text] is not more powerful than [Formula: see text] as the set of amplitudes taken by QTMs; however, it is sufficient to solve the factoring problem.

scholarly journals Generation of elementary gates and Bell’s states using controlled adiabatic evolutions

2019 ◽  
Vol 17 (03) ◽  
pp. 1950020
Author(s):  
Abderrahim Benmachiche ◽  
Ali Sellami ◽  
Sherzod Turaev ◽  
Derradji Bahloul ◽  
Azeddine Messikh ◽  
...  
Keyword(s):  
Quantum Computation ◽  
Open Systems ◽  
Circuit Model ◽  
Quantum Gates ◽  
Single Quantum ◽  
New Model ◽  
Adiabatic Quantum Computation ◽  
Usual Quantum

Fundamental quantum gates can be implemented effectively using adiabatic quantum computation or circuit model. Recently, Hen combined the two approaches to introduce a new model called controlled adiabatic evolutions [I. Hen, Phys. Rev. A, 91(2) (2015) 022309]. This model was specifically designed to implement one and two-qubit controlled gates. Later, Santos extended Hen’s work to implement [Formula: see text]-qubit controlled gates [A. C. Santos and M. S. Sarandy, Sci. Rep., 5 (2015) 15775]. In this paper, we discuss the implementation of each of the usual quantum gates, as well as demonstrate the possibility of preparing Bell’s states using the controlled adiabatic evolutions approach. We conclude by presenting the fidelity results of implementing single quantum gates and Bell’s states in open systems.

scholarly journals Undecidability in quantum thermalization

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Naoto Shiraishi ◽  
Keiji Matsumoto
Keyword(s):  
Turing Machine ◽  
Thermal Equilibrium ◽  
General Theorem ◽  
Nearest Neighbour ◽  
Many Body ◽  
Initial State ◽  
One Dimensional ◽  
Universal Turing Machine ◽  
Many Body Systems ◽  
Neighbour Interaction

AbstractThe investigation of thermalization in isolated quantum many-body systems has a long history, dating back to the time of developing statistical mechanics. Most quantum many-body systems in nature are considered to thermalize, while some never achieve thermal equilibrium. The central problem is to clarify whether a given system thermalizes, which has been addressed previously, but not resolved. Here, we show that this problem is undecidable. The resulting undecidability even applies when the system is restricted to one-dimensional shift-invariant systems with nearest-neighbour interaction, and the initial state is a fixed product state. We construct a family of Hamiltonians encoding dynamics of a reversible universal Turing machine, where the fate of a relaxation process changes considerably depending on whether the Turing machine halts. Our result indicates that there is no general theorem, algorithm, or systematic procedure determining the presence or absence of thermalization in any given Hamiltonian.

Magnetic properties of manganese based one-dimensional spin chains

2015 ◽  
Vol 44 (46) ◽  
pp. 19812-19819 ◽  
Author(s):  
K. S. Asha ◽  
K. M. Ranjith ◽  
Arvind Yogi ◽  
R. Nath ◽  
Sukhendu Mandal
Keyword(s):  
Heat Capacity ◽  
Magnetic Properties ◽  
Magnetic Susceptibility ◽  
Spin Chains ◽  
One Dimensional

Magnetic susceptibility and heat capacity of three manganese based structures are measured and modeled with one-dimensional antiferromagnetic chains.

scholarly journals A computationally efficient estimator for mutual information

2008 ◽  
Vol 464 (2093) ◽  
pp. 1203-1215 ◽  
Author(s):  
Dafydd Evans
Keyword(s):  
Data Analysis ◽  
Mutual Information ◽  
Time Complexity ◽  
Exploratory Data Analysis ◽  
Nearest Neighbour ◽  
Computationally Efficient ◽  
One Dimensional ◽  
Exploratory Data ◽  
Efficient Alternative ◽  
Computationally Expensive

Mutual information quantifies the determinism that exists in a relationship between random variables, and thus plays an important role in exploratory data analysis. We investigate a class of non-parametric estimators for mutual information, based on the nearest neighbour structure of observations in both the joint and marginal spaces. Unless both marginal spaces are one-dimensional, we demonstrate that a well-known estimator of this type can be computationally expensive under certain conditions, and propose a computationally efficient alternative that has a time complexity of order ( N  log  N ) as the number of observations N →∞.

scholarly journals Assessing Bound States in a One-Dimensional Topological Superconductor: Majorana versus Tamm

2021 ◽  
Author(s):  
Niccolò Traverso Ziani ◽  
Lucia Vigliotti ◽  
Matteo Carrega ◽  
Fabio Cavaliere
Keyword(s):  
Quantum Computation ◽  
Bound States ◽  
Research Activity ◽  
One Dimensional ◽  
Majorana Bound States ◽  
Topological Superconductors ◽  
Topological Quantum ◽  
Tamm State ◽  
Topological Quantum Computation ◽  
Intense Research

Majorana bound states in topological superconductors have attracted intense research activity in view of applications in topological quantum computation. However, they are not the only example of topological bound states that can occur in such systems. We here study a model in which both Majorana and Tamm bound states compete. We show both numerically and analytically that, surprisingly, the Tamm state remains partially localized even when the spectrum becomes gapless. Despite this fact, we demonstrate that the Majorana polarization shows a clear transition between the two regimes.

scholarly journals Face-Width of Pfaffian Braces and Polyhex Graphs on Surfaces

10.37236/3540 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Dong Ye ◽  
Heping Zhang
Keyword(s):  
Bipartite Graph ◽  
Polynomial Time ◽  
Perfect Matching ◽  
Cartesian Product ◽  
Perfect Matchings ◽  
Face Width ◽  
Graphs On Surfaces ◽  
Positive Genus ◽  
Pfaffian Graph

A graph $G$ with a perfect matching is Pfaffian if it admits an orientation $D$ such that every central cycle $C$ (i.e. $C$ is of even size and $G-V(C)$ has a perfect matching) has an odd number of edges oriented in either direction of the cycle. It is known that the number of perfect matchings of a Pfaffian graph can be computed in polynomial time. In this paper, we show that every embedding of a Pfaffian brace (i.e. 2-extendable bipartite graph)  on a surface with a positive genus has face-width at most 3.  Further, we study Pfaffian cubic braces and obtain a characterization of Pfaffian polyhex graphs: a polyhex graph is Pfaffian if and only if it is either non-bipartite or isomorphic to the cube, or the Heawood graph, or the Cartesian product $C_k\times K_2$ for even integers $k\ge 6$.

scholarly journals Preferences Single-Peaked on a Circle

2020 ◽  
Vol 68 ◽  
pp. 463-502 ◽  
Author(s):  
Dominik Peters ◽  
Martin Lackner
Keyword(s):  
Polynomial Time ◽  
Choice Theory ◽  
Social Choice Theory ◽  
Recognition Algorithm ◽  
Approval Voting ◽  
Voting Rules ◽  
One Dimensional ◽  
Facility Location Problems ◽  
Impossibility Results ◽  
Fast Recognition

We introduce the domain of preferences that are single-peaked on a circle, which is a generalization of the well-studied single-peaked domain. This preference restriction is useful, e.g., for scheduling decisions, certain facility location problems, and for one-dimensional decisions in the presence of extremist preferences. We give a fast recognition algorithm of this domain, provide a characterisation by finitely many forbidden subprofiles, and show that many popular single- and multi-winner voting rules are polynomial-time computable on this domain. In particular, we prove that Proportional Approval Voting can be computed in polynomial time for profiles that are single-peaked on a circle. In contrast, Kemeny's rule remains hard to evaluate, and several impossibility results from social choice theory can be proved using only profiles in this domain.

scholarly journals The computational power of matchgates and the XY interaction on arbitrary graphs

2014 ◽  
Vol 14 (11&12) ◽  
pp. 901-916
Author(s):  
Daniel J. Brod ◽  
Andrew M. Childs
Keyword(s):  
Quantum Computing ◽  
Quantum Computation ◽  
Connected Graph ◽  
Nearest Neighbor ◽  
Proper Subset ◽  
Computational Power ◽  
Arbitrary Graphs

Matchgates are a restricted set of two-qubit gates known to be classically simulable when acting on nearest-neighbor qubits on a path, but universal for quantum computation when the qubits are arranged on certain other graphs. Here we characterize the power of matchgates acting on arbitrary graphs. Specifically, we show that they are universal on any connected graph other than a path or a cycle, and that they are classically simulable on a cycle. We also prove the same dichotomy for the XY interaction, a proper subset of matchgates related to some implementations of quantum computing.

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