17,268 research outputs found
A Survey on Continuous Time Computations
We provide an overview of theories of continuous time computation. These
theories allow us to understand both the hardness of questions related to
continuous time dynamical systems and the computational power of continuous
time analog models. We survey the existing models, summarizing results, and
point to relevant references in the literature
Resource Bounded Immunity and Simplicity
Revisiting the thirty years-old notions of resource-bounded immunity and
simplicity, we investigate the structural characteristics of various immunity
notions: strong immunity, almost immunity, and hyperimmunity as well as their
corresponding simplicity notions. We also study limited immunity and
simplicity, called k-immunity and feasible k-immunity, and their simplicity
notions. Finally, we propose the k-immune hypothesis as a working hypothesis
that guarantees the existence of simple sets in NP.Comment: This is a complete version of the conference paper that appeared in
the Proceedings of the 3rd IFIP International Conference on Theoretical
Computer Science, Kluwer Academic Publishers, pp.81-95, Toulouse, France,
August 23-26, 200
Oracles Are Subtle But Not Malicious
Theoretical computer scientists have been debating the role of oracles since
the 1970's. This paper illustrates both that oracles can give us nontrivial
insights about the barrier problems in circuit complexity, and that they need
not prevent us from trying to solve those problems.
First, we give an oracle relative to which PP has linear-sized circuits, by
proving a new lower bound for perceptrons and low- degree threshold
polynomials. This oracle settles a longstanding open question, and generalizes
earlier results due to Beigel and to Buhrman, Fortnow, and Thierauf. More
importantly, it implies the first nonrelativizing separation of "traditional"
complexity classes, as opposed to interactive proof classes such as MIP and
MA-EXP. For Vinodchandran showed, by a nonrelativizing argument, that PP does
not have circuits of size n^k for any fixed k. We present an alternative proof
of this fact, which shows that PP does not even have quantum circuits of size
n^k with quantum advice. To our knowledge, this is the first nontrivial lower
bound on quantum circuit size.
Second, we study a beautiful algorithm of Bshouty et al. for learning Boolean
circuits in ZPP^NP. We show that the NP queries in this algorithm cannot be
parallelized by any relativizing technique, by giving an oracle relative to
which ZPP^||NP and even BPP^||NP have linear-size circuits. On the other hand,
we also show that the NP queries could be parallelized if P=NP. Thus, classes
such as ZPP^||NP inhabit a "twilight zone," where we need to distinguish
between relativizing and black-box techniques. Our results on this subject have
implications for computational learning theory as well as for the circuit
minimization problem.Comment: 20 pages, 1 figur
Anticoncentration theorems for schemes showing a quantum speedup
One of the main milestones in quantum information science is to realise
quantum devices that exhibit an exponential computational advantage over
classical ones without being universal quantum computers, a state of affairs
dubbed quantum speedup, or sometimes "quantum computational supremacy". The
known schemes heavily rely on mathematical assumptions that are plausible but
unproven, prominently results on anticoncentration of random prescriptions. In
this work, we aim at closing the gap by proving two anticoncentration theorems
and accompanying hardness results, one for circuit-based schemes, the other for
quantum quench-type schemes for quantum simulations. Compared to the few other
known such results, these results give rise to a number of comparably simple,
physically meaningful and resource-economical schemes showing a quantum speedup
in one and two spatial dimensions. At the heart of the analysis are tools of
unitary designs and random circuits that allow us to conclude that universal
random circuits anticoncentrate as well as an embedding of known circuit-based
schemes in a 2D translation-invariant architecture.Comment: 12+2 pages, added applications sectio
Unitary matrix integrals in the framework of Generalized Kontsevich Model. I. Brezin-Gross-Witten Model
We advocate a new approach to the study of unitary matrix models in external
fields which emphasizes their relationship to Generalized Kontsevich Models
(GKM) with non-polynomial potentials. For example, we show that the partition
function of the Brezin-Gross-Witten Model (BGWM), which is defined as an
integral over unitary matrices, , can also be considered as a GKM with potential . Moreover, it can be interpreted as the generating functional for
correlators in the Penner model. The strong and weak coupling phases of the
BGWM are identified with the "character" (weak coupling) and "Kontsevich"
(strong coupling) phases of the GKM, respectively. This sort of GKM deserves
classification as one (i.e. or ) when in the Kontsevich
phase. This approach allows us to further identify the
Harish-Chandra-Itzykson-Zuber (IZ) integral with a peculiar GKM, which arises
in the study of theory and, further, with a conventional 2-matrix model
which is rewritten in Miwa coordinates. Inspired by the considered unitary
matrix models, some further extensions of the GKM treatment which are inspired
by the unitary matrix models which we have considered are also developed. In
particular, as a by-product, a new simple method of fixing the Ward identities
for matrix models in an external field is presented.Comment: FIAN/TD-16/93, ITEP-M6/93, UBC/S-93/93 (39 pages
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