10,614 research outputs found
Universality and programmability of quantum computers
Manin, Feynman, and Deutsch have viewed quantum computing as a kind of
universal physical simulation procedure. Much of the writing about quantum
logic circuits and quantum Turing machines has shown how these machines can
simulate an arbitrary unitary transformation on a finite number of qubits. The
problem of universality has been addressed most famously in a paper by Deutsch,
and later by Bernstein and Vazirani as well as Kitaev and Solovay. The quantum
logic circuit model, developed by Feynman and Deutsch, has been more prominent
in the research literature than Deutsch's quantum Turing machines. Quantum
Turing machines form a class closely related to deterministic and probabilistic
Turing machines and one might hope to find a universal machine in this class. A
universal machine is the basis of a notion of programmability. The extent to
which universality has in fact been established by the pioneers in the field is
examined and this key notion in theoretical computer science is scrutinised in
quantum computing by distinguishing various connotations and concomitant
results and problems.Comment: 17 pages, expands on arXiv:0705.3077v1 [quant-ph
Succinctness of two-way probabilistic and quantum finite automata
We prove that two-way probabilistic and quantum finite automata (2PFA's and
2QFA's) can be considerably more concise than both their one-way versions
(1PFA's and 1QFA's), and two-way nondeterministic finite automata (2NFA's). For
this purpose, we demonstrate several infinite families of regular languages
which can be recognized with some fixed probability greater than by
just tuning the transition amplitudes of a 2QFA (and, in one case, a 2PFA) with
a constant number of states, whereas the sizes of the corresponding 1PFA's,
1QFA's and 2NFA's grow without bound. We also show that 2QFA's with mixed
states can support highly efficient probability amplification. The weakest
known model of computation where quantum computers recognize more languages
with bounded error than their classical counterparts is introduced.Comment: A new version, 21 pages, late
Undecidability of the Spectral Gap (full version)
We show that the spectral gap problem is undecidable. Specifically, we
construct families of translationally-invariant, nearest-neighbour Hamiltonians
on a 2D square lattice of d-level quantum systems (d constant), for which
determining whether the system is gapped or gapless is an undecidable problem.
This is true even with the promise that each Hamiltonian is either gapped or
gapless in the strongest sense: it is promised to either have continuous
spectrum above the ground state in the thermodynamic limit, or its spectral gap
is lower-bounded by a constant in the thermodynamic limit. Moreover, this
constant can be taken equal to the local interaction strength of the
Hamiltonian.Comment: v1: 146 pages, 56 theorems etc., 15 figures. See shorter companion
paper arXiv:1502.04135 (same title and authors) for a short version omitting
technical details. v2: Small but important fix to wording of abstract. v3:
Simplified and shortened some parts of the proof; minor fixes to other parts.
Now only 127 pages, 55 theorems etc., 10 figures. v4: Minor updates to
introductio
The semaphore codes attached to a Turing machine via resets and their various limits
We introduce semaphore codes associated to a Turing machine via resets.
Semaphore codes provide an approximation theory for resets. In this paper we
generalize the set-up of our previous paper "Random walks on semaphore codes
and delay de Bruijn semigroups" to the infinite case by taking the profinite
limit of -resets to obtain -resets. We mention how this opens new
avenues to attack the P versus NP problem.Comment: 28 pages; Sections 3-6 appeared in a previous version of
arXiv:1509.03383 as Sections 9-12 (the split of the previous paper was
suggested by the journal); Sections 1-2 and 7 are ne
Revisiting Reachability in Timed Automata
We revisit a fundamental result in real-time verification, namely that the
binary reachability relation between configurations of a given timed automaton
is definable in linear arithmetic over the integers and reals. In this paper we
give a new and simpler proof of this result, building on the well-known
reachability analysis of timed automata involving difference bound matrices.
Using this new proof, we give an exponential-space procedure for model checking
the reachability fragment of the logic parametric TCTL. Finally we show that
the latter problem is NEXPTIME-hard
Reachability for infinite time Turing machines with long tapes
Infinite time Turing machine models with tape length , denoted
, strengthen the machines of Hamkins and Kidder [HL00] with tape
length . A new phenomenon is that for some countable ordinals ,
some cells cannot be halting positions of given trivial input. The
main open question in [Rin14] asks about the size of the least such ordinal
.
We answer this by providing various characterizations. For instance,
is the least ordinal with any of the following properties: (a) For some
, there is a -writable but not -writable subset of
. (b) There is a gap in the -writable ordinals. (c)
is uncountable in . Here denotes the
supremum of -writable ordinals, i.e. those with a -writable
code of length .
We further use the above characterizations, and an analogue to Welch's
submodel characterization of the ordinals , and , to
show that is large in the sense that it is a closure point of the
function , where denotes the
supremum of the -accidentally writable ordinals
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