Classical simulation of limited-width cluster-state quantum computation

We present a classical protocol, using the matrix product state representation, to simulate cluster-state quantum computation at a cost polynomial in the number of qubits in the cluster and exponential in d -- the width of the cluster. We use this result to show that any log-depth quantum computation in the gate array model, with gates linking only nearby qubits, can be simulated efficiently on a classical computer.Comment: 4 pages, 1 figur

Efficient classical simulation of the approximate quantum Fourier transform

We present a method for classically simulating quantum circuits based on the tensor contraction model of Markov and Shi (quant-ph/0511069). Using this method we are able to classically simulate the approximate quantum Fourier transform in polynomial time. Moreover, our approach allows us to formulate a condition for the composability of simulable quantum circuits. We use this condition to show that any circuit composed of a constant number of approximate quantum Fourier transform circuits and log-depth circuits with limited interaction range can also be efficiently simulated.Comment: 5 pages, 3 figure

No purification for two copies of a noisy entangled state

We consider whether two copies of a noisy entangled state can be transformed into a single copy of greater purity using local operations and classical communication. We show that it is never possible to achieve such a purification with certainty when the family of noisy states is twirlable (i.e. when there exists a local transformation that maps all states into the family, yet leaves the family itself invariant). This implies that two copies of a Werner state cannot be deterministically purified. Furthermore, due to the construction of the proof, it will hold not only in quantum theory, but in any generalised probabilistic theory. We use this to show that two copies of a noisy PR-box (a hypothetical device more non-local than is allowed by quantum theory) cannot be purified.Comment: 4 pages, 2 figure

Simulating all non-signalling correlations via classical or quantum theory with negative probabilities

Many-party correlations between measurement outcomes in general probabilistic theories are given by conditional probability distributions obeying the non-signalling condition. We show that any such distribution can be obtained from classical or quantum theory, by relaxing positivity constraints on either the mixed state shared by the parties, or the local functions which generate measurement outcomes. Our results apply to generic non-signalling correlations, but in particular they yield two distinct quasi-classical models for quantum correlations.Comment: 6 page

Work extraction and thermodynamics for individual quantum systems

Thermodynamics is traditionally concerned with systems comprised of a large number of particles. Here we present a framework for extending thermodynamics to individual quantum systems, including explicitly a thermal bath and work-storage device (essentially a `weight' that can be raised or lowered). We prove that the second law of thermodynamics holds in our framework, and give a simple protocol to extract the optimal amount of work from the system, equal to its change in free energy. Our results apply to any quantum system in an arbitrary initial state, in particular including non-equilibrium situations. The optimal protocol is essentially reversible, similar to classical Carnot cycles, and indeed, we show that it can be used it to construct a quantum Carnot engine.Comment: 11 pages, no figures. v2: published version. arXiv admin note: substantial text overlap with arXiv:1302.281

Causal Fermions in Discrete Spacetime

In this paper, we consider fermionic systems in discrete spacetime evolving with a strict notion of causality, meaning they evolve unitarily and with a bounded propagation speed. First, we show that the evolution of these systems has a natural decomposition into a product of local unitaries, which also holds if we include bosons. Next, we show that causal evolution of fermions in discrete spacetime can also be viewed as the causal evolution of a lattice of qubits, meaning these systems can be viewed as quantum cellular automata. Following this, we discuss some examples of causal fermionic models in discrete spacetime that become interesting physical systems in the continuum limit: Dirac fermions in one and three spatial dimensions, Dirac fields and briefly the Thirring model. Finally, we show that the dynamics of causal fermions in discrete spacetime can be efficiently simulated on a quantum computer.Comment: 16 pages, 1 figur

Discrete Spacetime and Relativistic Quantum Particles

We study a single quantum particle in discrete spacetime evolving in a causal way. We see that in the continuum limit any massless particle with a two dimensional internal degree of freedom obeys the Weyl equation, provided that we perform a simple relabeling of the coordinate axes or demand rotational symmetry in the continuum limit. It is surprising that this occurs regardless of the specific details of the evolution: it would be natural to assume that discrete evolutions giving rise to relativistic dynamics in the continuum limit would be very special cases. We also see that the same is not true for particles with larger internal degrees of freedom, by looking at an example with a three dimensional internal degree of freedom that is not relativistic in the continuum limit. In the process we give a formula for the Hamiltonian arising from the continuum limit of massless and massive particles in discrete spacetime.Comment: 6 page

No quantum advantage for nonlocal computation

We investigate the problem of "nonlocal" computation, in which separated parties must compute a function with nonlocally encoded inputs and output, such that each party individually learns nothing, yet together they compute the correct function output. We show that the best that can be done classically is a trivial linear approximation. Surprisingly, we also show that quantum entanglement provides no advantage over the classical case. On the other hand, generalized (i.e. super-quantum) nonlocal correlations allow perfect nonlocal computation. This gives new insights into the nature of quantum nonlocality and its relationship to generalised nonlocal correlations.Comment: 4 page