10 research outputs found

    A single-shot measurement of the energy of product states in a translation invariant spin chain can replace any quantum computation

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    In measurement-based quantum computation, quantum algorithms are implemented via sequences of measurements. We describe a translationally invariant finite-range interaction on a one-dimensional qudit chain and prove that a single-shot measurement of the energy of an appropriate computational basis state with respect to this Hamiltonian provides the output of any quantum circuit. The required measurement accuracy scales inverse polynomially with the size of the simulated quantum circuit. This shows that the implementation of energy measurements on generic qudit chains is as hard as the realization of quantum computation. Here a ''measurement'' is any procedure that samples from the spectral measure induced by the observable and the state under consideration. As opposed to measurement-based quantum computation, the post-measurement state is irrelevant.Comment: 19 pages, transition rules for the CA correcte

    Quantum computation beyond the circuit model

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    Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2008.Includes bibliographical references (p. 133-144).The quantum circuit model is the most widely used model of quantum computation. It provides both a framework for formulating quantum algorithms and an architecture for the physical construction of quantum computers. However, several other models of quantum computation exist which provide useful alternative frameworks for both discovering new quantum algorithms and devising new physical implementations of quantum computers. In this thesis, I first present necessary background material for a general physics audience and discuss existing models of quantum computation. Then, I present three new results relating to various models of quantum computation: a scheme for improving the intrinsic fault tolerance of adiabatic quantum computers using quantum error detecting codes, a proof that a certain problem of estimating Jones polynomials is complete for the one clean qubit complexity class, and a generalization of perturbative gadgets which allows k-body interactions to be directly simulated using 2-body interactions. Lastly, I discuss general principles regarding quantum computation that I learned in the course of my research, and using these principles I propose directions for future research.by Stephen Paul Jordan.Ph.D

    A Promisebqp-Complete String Rewriting Problem

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    We consider the following combinatorial problem. We are given three strings s, t, and t′ of length L over some fixed finite alphabet and an integer m that is polylogarithmic in L. We have a symmetric relation on substrings of constant length that specifies which substrings are allowed to be replaced with each other. Let Δ(n) denote the difference between the numbers of possibilities to obtain t from s and t′ from s after n ∈ N replacements. The problem is to determine the sign of Δ(m). As promises we have a gap condition and a growth condition. The former states that |Δ(m)| ≥ ∈ cm where ∈ is inverse polylogarithmic in L and c \u3e 0 is a constant. The latter is given by Δ(n) ≤ cn for all n. We show that this problem is PromiseBQP-complete, i.e., it represents the class of problems that can be solved efficiently on a quantum computer. © Rinton Press

    A PROMISEBQP-COMPLETE STRING REWRITING PROBLEM

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    We consider the following combinatorial problem. We are given three strings s, t, and t\u27 of length L over some fixed finite alphabet and an integer in that is polylogarithmic in L. We have a symmetric relation on substrings of constant length that specifies which substrings are allowed to be replaced with each other. Let Delta(n) denote the difference between the numbers of possibilities to obtain t from s and t\u27 from s after n is an element of N replacements. The problem is to determine the sign of A(m). As promises we have a gap condition and a growth condition. The former states that Delta(m) \u3e = is an element of c(m) where is inverse polylogarithmic in L and c \u3e 0 is a constant. The latter is given by Delta(n) \u3c = c(n) for all n. We show that this problem is PromiseBQP-complete, i.e., it represents the class of problems that can be solved efficiently on a quantum computer
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