22,563 research outputs found
Average-Case Complexity
We survey the average-case complexity of problems in NP.
We discuss various notions of good-on-average algorithms, and present
completeness results due to Impagliazzo and Levin. Such completeness results
establish the fact that if a certain specific (but somewhat artificial) NP
problem is easy-on-average with respect to the uniform distribution, then all
problems in NP are easy-on-average with respect to all samplable distributions.
Applying the theory to natural distributional problems remain an outstanding
open question. We review some natural distributional problems whose
average-case complexity is of particular interest and that do not yet fit into
this theory.
A major open question whether the existence of hard-on-average problems in NP
can be based on the PNP assumption or on related worst-case assumptions.
We review negative results showing that certain proof techniques cannot prove
such a result. While the relation between worst-case and average-case
complexity for general NP problems remains open, there has been progress in
understanding the relation between different ``degrees'' of average-case
complexity. We discuss some of these ``hardness amplification'' results
Computational Complexity for Physicists
These lecture notes are an informal introduction to the theory of
computational complexity and its links to quantum computing and statistical
mechanics.Comment: references updated, reprint available from
http://itp.nat.uni-magdeburg.de/~mertens/papers/complexity.shtm
Computational complexity of the landscape I
We study the computational complexity of the physical problem of finding
vacua of string theory which agree with data, such as the cosmological
constant, and show that such problems are typically NP hard. In particular, we
prove that in the Bousso-Polchinski model, the problem is NP complete. We
discuss the issues this raises and the possibility that, even if we were to
find compelling evidence that some vacuum of string theory describes our
universe, we might never be able to find that vacuum explicitly.
In a companion paper, we apply this point of view to the question of how
early cosmology might select a vacuum.Comment: JHEP3 Latex, 53 pp, 2 .eps figure
Applying computational complexity to the emergence of classicality
Can the computational complexity theory of computer science and mathematics
say something new about unresolved problems in quantum physics? Particularly,
can the P versus NP question in the computational complexity theory be a factor
in the elucidation of the emergency of classicality in quantum mechanics? The
paper compares two different ways of deriving classicality from the quantum
formalism resulted from two differing hypotheses regarding the P versus NP
question -- the approach of the quantum decoherence theory implying that P = NP
and the computational complexity approach which assumes that P is not equal to
NP.Comment: 14 pages, Latex; typos correcte
Physical portrayal of computational complexity
Computational complexity is examined using the principle of increasing
entropy. To consider computation as a physical process from an initial instance
to the final acceptance is motivated because many natural processes have been
recognized to complete in non-polynomial time (NP). The irreversible process
with three or more degrees of freedom is found intractable because, in terms of
physics, flows of energy are inseparable from their driving forces. In
computational terms, when solving problems in the class NP, decisions will
affect subsequently available sets of decisions. The state space of a
non-deterministic finite automaton is evolving due to the computation itself
hence it cannot be efficiently contracted using a deterministic finite
automaton that will arrive at a solution in super-polynomial time. The solution
of the NP problem itself is verifiable in polynomial time (P) because the
corresponding state is stationary. Likewise the class P set of states does not
depend on computational history hence it can be efficiently contracted to the
accepting state by a deterministic sequence of dissipative transformations.
Thus it is concluded that the class P set of states is inherently smaller than
the set of class NP. Since the computational time to contract a given set is
proportional to dissipation, the computational complexity class P is a subset
of NP.Comment: 16, pages, 7 figure
Direct certification of a class of quantum simulations
One of the main challenges in the field of quantum simulation and computation
is to identify ways to certify the correct functioning of a device when a
classical efficient simulation is not available. Important cases are situations
in which one cannot classically calculate local expectation values of state
preparations efficiently. In this work, we develop weak-membership formulations
of the certification of ground state preparations. We provide a non-interactive
protocol for certifying ground states of frustration-free Hamiltonians based on
simple energy measurements of local Hamiltonian terms. This certification
protocol can be applied to classically intractable analog quantum simulations:
For example, using Feynman-Kitaev Hamiltonians, one can encode universal
quantum computation in such ground states. Moreover, our certification protocol
is applicable to ground states encodings of IQP circuits demonstration of
quantum supremacy. These can be certified efficiently when the error is
polynomially bounded.Comment: 10 pages, corrected a small error in Eqs. (2) and (5
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