686 research outputs found
Index theory of one dimensional quantum walks and cellular automata
If a one-dimensional quantum lattice system is subject to one step of a
reversible discrete-time dynamics, it is intuitive that as much "quantum
information" as moves into any given block of cells from the left, has to exit
that block to the right. For two types of such systems - namely quantum walks
and cellular automata - we make this intuition precise by defining an index, a
quantity that measures the "net flow of quantum information" through the
system. The index supplies a complete characterization of two properties of the
discrete dynamics. First, two systems S_1, S_2 can be pieced together, in the
sense that there is a system S which locally acts like S_1 in one region and
like S_2 in some other region, if and only if S_1 and S_2 have the same index.
Second, the index labels connected components of such systems: equality of the
index is necessary and sufficient for the existence of a continuous deformation
of S_1 into S_2. In the case of quantum walks, the index is integer-valued,
whereas for cellular automata, it takes values in the group of positive
rationals. In both cases, the map S -> ind S is a group homomorphism if
composition of the discrete dynamics is taken as the group law of the quantum
systems. Systems with trivial index are precisely those which can be realized
by partitioned unitaries, and the prototypes of systems with non-trivial index
are shifts.Comment: 38 pages. v2: added examples, terminology clarifie
Physics as Quantum Information Processing: Quantum Fields as Quantum Automata
Can we reduce Quantum Field Theory (QFT) to a quantum computation? Can
physics be simulated by a quantum computer? Do we believe that a quantum field
is ultimately made of a numerable set of quantum systems that are unitarily
interacting? A positive answer to these questions corresponds to substituting
QFT with a theory of quantum cellular automata (QCA), and the present work is
examining this hypothesis. These investigations are part of a large research
program on a "quantum-digitalization" of physics, with Quantum Theory as a
special theory of information, and Physics as emergent from the same
quantum-information processing. A QCA-based QFT has tremendous potential
advantages compared to QFT, being quantum "ab-initio" and free from the
problems plaguing QFT due to the continuum hypothesis. Here I will show how
dynamics emerges from the quantum processing, how the QCA can reproduce the
Dirac-field phenomenology at large scales, and the kind of departures from QFT
that that should be expected at a Planck-scale discreteness. I will introduce
the notions of linear field quantum automaton and local-matrix quantum
automaton, in terms of which I will provide the solution to the Feynman's
problem about the possibility of simulating a Fermi field with a quantum
computer.Comment: This version: further improvements in notation. Added reference. Work
presented at the conference "Foundations of Probability and Physics-6" (FPP6)
held on 12-15 June 2011 at the Linnaeus University, Vaaxjo, Sweden. Many new
results, e.g. Feynman problem of qubit-ization of Fermi fields solved
Factorization, reduction and embedding in integrable cellular automata
Factorized dynamics in soliton cellular automata with quantum group symmetry
is identified with a motion of particles and anti-particles exhibiting pair
creation and annihilation. An embedding scheme is presented showing that the
D^{(1)}_n-automaton contains, as certain subsectors, the box-ball systems and
all the other automata associated with the crystal bases of non-exceptional
affine Lie algebras. The results extend the earlier ones to higher
representations by a certain reduction and to a wider class of boundary
conditions.Comment: LaTeX2e, 20 page
Bethe ansatz at q=0 and periodic box-ball systems
A class of periodic soliton cellular automata is introduced associated with
crystals of non-exceptional quantum affine algebras. Based on the Bethe ansatz
at q=0, we propose explicit formulas for the dynamical period and the size of
certain orbits under the time evolution in A^{(1)}_n case.Comment: 12 pages, Introduction expanded, Summary added and minor
modifications mad
On the structure of Clifford quantum cellular automata
We study reversible quantum cellular automata with the restriction that these
are also Clifford operations. This means that tensor products of Pauli
operators (or discrete Weyl operators) are mapped to tensor products of Pauli
operators. Therefore Clifford quantum cellular automata are induced by
symplectic cellular automata in phase space. We characterize these symplectic
cellular automata and find that all possible local rules must be, up to some
global shift, reflection invariant with respect to the origin. In the one
dimensional case we also find that every uniquely determined and
translationally invariant stabilizer state can be prepared from a product state
by a single Clifford cellular automaton timestep, thereby characterizing these
class of stabilizer states, and we show that all 1D Clifford quantum cellular
automata are generated by a few elementary operations. We also show that the
correspondence between translationally invariant stabilizer states and
translationally invariant Clifford operations holds for periodic boundary
conditions.Comment: 28 pages, 2 figures, LaTe
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