Fast Universal Quantum Computation with Railroad-switch Local Hamiltonians

We present two universal models of quantum computation with a time-independent, frustration-free Hamiltonian. The first construction uses 3-local (qubit) projectors, and the second one requires only 2-local qubit-qutrit projectors. We build on Feynman's Hamiltonian computer idea and use a railroad-switch type clock register. The resources required to simulate a quantum circuit with L gates in this model are O(L) small-dimensional quantum systems (qubits or qutrits), a time-independent Hamiltonian composed of O(L) local, constant norm, projector terms, the possibility to prepare computational basis product states, a running time O(L log^2 L), and the possibility to measure a few qubits in the computational basis. Our models also give a simplified proof of the universality of 3-local Adiabatic Quantum Computation.Comment: Added references to work by de Falco et al., and realized that Feynman's '85 paper already contained the idea of a switch in i

IS THERE A CLASSICAL ANALOG OF A QUANTUM TIME-TRANSLATION MACHINE?

In a recent article [D. Suter, Phys. Rev. {\bf A 51}, 45 (1995)] Suter has claimed to present an optical implementation of the quantum time-translation machine which ``shows all the features that the general concept predicts and also allows, besides the quantum mechanical, a classical description.'' It is argued that the experiment proposed and performed by Suter does not have the features of the quantum time-translation machine and that the latter has no classical analog.Comment: 7 pages, LaTe

Variance Control in Weak Value Measurement Pointers

The variance of an arbitrary pointer observable is considered for the general case that a complex weak value is measured using a complex valued pointer state. For the typical cases where the pointer observable is either its position or momentum, the associated expressions for the pointer's variance after the measurement contain a term proportional to the product of the weak value's imaginary part with the rate of change of the third central moment of position relative to the initial pointer state just prior to the time of the measurement interaction when position is the observable - or with the initial pointer state's third central moment of momentum when momentum is the observable. These terms provide a means for controlling pointer position and momentum variance and identify control conditions which - when satisfied - can yield variances that are smaller after the measurement than they were before the measurement. Measurement sensitivities which are useful for estimating weak value measurement accuracies are also briefly discussed.Comment: submitted to Phys Rev

Fault-tolerant quantum computation with long-range correlated noise

We prove a new version of the quantum accuracy threshold theorem that applies to non-Markovian noise with algebraically decaying spatial correlations. We consider noise in a quantum computer arising from a perturbation that acts collectively on pairs of qubits and on the environment, and we show that an arbitrarily long quantum computation can be executed with high reliability in D spatial dimensions, if the perturbation is sufficiently weak and decays with the distance r between the qubits faster than 1/r^D.Comment: 4 page

Finite automata for caching in matrix product algorithms

A diagram is introduced for visualizing matrix product states which makes transparent a connection between matrix product factorizations of states and operators, and complex weighted finite state automata. It is then shown how one can proceed in the opposite direction: writing an automaton that ``generates'' an operator gives one an immediate matrix product factorization of it. Matrix product factorizations have the advantage of reducing the cost of computing expectation values by facilitating caching of intermediate calculations. Thus our connection to complex weighted finite state automata yields insight into what allows for efficient caching in matrix product algorithms. Finally, these techniques are generalized to the case of multiple dimensions.Comment: 18 pages, 19 figures, LaTeX; numerous improvements have been made to the manuscript in response to referee feedbac

Nonlocal Aspects of a Quantum Wave

Various aspects of nonlocality of a quantum wave are discussed. In particular, the question of the possibility of extracting information about the relative phase in a quantum wave is analyzed. It is argued that there is a profound difference in the nonlocal properties of the quantum wave between fermion and boson particles. The phase of the boson quantum state can be found from correlations between results of measurements in separate regions. These correlations are identical to the Einstein-Podolsky-Rosen (EPR) correlations between two entangled systems. An ensemble of results of measurements performed on fermion quantum waves does not exhibit the EPR correlations and the relative phase of fermion quantum waves cannot be found from these results. The existence of a physical variable (the relative phase) which cannot be measured locally is the nonlocality aspect of the quantum wave of a fermion.Comment: 12 page

Sequential weak measurement

The notion of weak measurement provides a formalism for extracting information from a quantum system in the limit of vanishing disturbance to its state. Here we extend this formalism to the measurement of sequences of observables. When these observables do not commute, we may obtain information about joint properties of a quantum system that would be forbidden in the usual strong measurement scenario. As an application, we provide a physically compelling characterisation of the notion of counterfactual quantum computation

PR-box correlations have no classical limit

One of Yakir Aharonov's endlessly captivating physics ideas is the conjecture that two axioms, namely relativistic causality ("no superluminal signalling") and nonlocality, so nearly contradict each other that a unique theory - quantum mechanics - reconciles them. But superquantum (or "PR-box") correlations imply that quantum mechanics is not the most nonlocal theory (in the sense of nonlocal correlations) consistent with relativistic causality. Let us consider supplementing these two axioms with a minimal third axiom: there exists a classical limit in which macroscopic observables commute. That is, just as quantum mechanics has a classical limit, so must any generalization of quantum mechanics. In this classical limit, PR-box correlations violate relativistic causality. Generalized to all stronger-than-quantum bipartite correlations, this result is a derivation of Tsirelson's bound without assuming quantum mechanics.Comment: for a video of this talk at the Aharonov-80 Conference in 2012 at Chapman University, see quantum.chapman.edu/talk-10, published in Quantum Theory: A Two-Time Success Story (Yakir Aharonov Festschrift), eds. D. C. Struppa and J. M. Tollaksen (New York: Springer), 2013, pp. 205-21