2,953 research outputs found
Is there a physically universal cellular automaton or Hamiltonian?
It is known that both quantum and classical cellular automata (CA) exist that
are computationally universal in the sense that they can simulate, after
appropriate initialization, any quantum or classical computation, respectively.
Here we introduce a different notion of universality: a CA is called physically
universal if every transformation on any finite region can be (approximately)
implemented by the autonomous time evolution of the system after the complement
of the region has been initialized in an appropriate way. We pose the question
of whether physically universal CAs exist. Such CAs would provide a model of
the world where the boundary between a physical system and its controller can
be consistently shifted, in analogy to the Heisenberg cut for the quantum
measurement problem. We propose to study the thermodynamic cost of computation
and control within such a model because implementing a cyclic process on a
microsystem may require a non-cyclic process for its controller, whereas
implementing a cyclic process on system and controller may require the
implementation of a non-cyclic process on a "meta"-controller, and so on.
Physically universal CAs avoid this infinite hierarchy of controllers and the
cost of implementing cycles on a subsystem can be described by mixing
properties of the CA dynamics. We define a physical prior on the CA
configurations by applying the dynamics to an initial state where half of the
CA is in the maximum entropy state and half of it is in the all-zero state
(thus reflecting the fact that life requires non-equilibrium states like the
boundary between a hold and a cold reservoir). As opposed to Solomonoff's
prior, our prior does not only account for the Kolmogorov complexity but also
for the cost of isolating the system during the state preparation if the
preparation process is not robust.Comment: 27 pages, 1 figur
Simple digital quantum algorithm for symmetric first order linear hyperbolic systems
This paper is devoted to the derivation of a digital quantum algorithm for
the Cauchy problem for symmetric first order linear hyperbolic systems, thanks
to the reservoir technique. The reservoir technique is a method designed to
avoid artificial diffusion generated by first order finite volume methods
approximating hyperbolic systems of conservation laws. For some class of
hyperbolic systems, namely those with constant matrices in several dimensions,
we show that the combination of i) the reservoir method and ii) the alternate
direction iteration operator splitting approximation, allows for the derivation
of algorithms only based on simple unitary transformations, thus perfectly
suitable for an implementation on a quantum computer. The same approach can
also be adapted to scalar one-dimensional systems with non-constant velocity by
combining with a non-uniform mesh. The asymptotic computational complexity for
the time evolution is determined and it is demonstrated that the quantum
algorithm is more efficient than the classical version. However, in the quantum
case, the solution is encoded in probability amplitudes of the quantum
register. As a consequence, as with other similar quantum algorithms, a
post-processing mechanism has to be used to obtain general properties of the
solution because a direct reading cannot be performed as efficiently as the
time evolution.Comment: 28 pages, 12 figures, major rewriting of the section describing the
numerical method, simplified the presentation and notation, reorganized the
sections, comments are welcome
Classical computing, quantum computing, and Shor's factoring algorithm
This is an expository talk written for the Bourbaki Seminar. After a brief
introduction, Section 1 discusses in the categorical language the structure of
the classical deterministic computations. Basic notions of complexity icluding
the P/NP problem are reviewed. Section 2 introduces the notion of quantum
parallelism and explains the main issues of quantum computing. Section 3 is
devoted to four quantum subroutines: initialization, quantum computing of
classical Boolean functions, quantum Fourier transform, and Grover's search
algorithm. The central Section 4 explains Shor's factoring algorithm. Section 5
relates Kolmogorov's complexity to the spectral properties of computable
function. Appendix contributes to the prehistory of quantum computing.Comment: 27 pp., no figures, amste
Undecidable word problem in subshift automorphism groups
This article studies the complexity of the word problem in groups of
automorphisms of subshifts. We show in particular that for any Turing degree,
there exists a subshift whose automorphism group contains a subgroup whose word
problem has exactly this degree
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