14,987 research outputs found
NP-complete Problems and Physical Reality
Can NP-complete problems be solved efficiently in the physical universe? I
survey proposals including soap bubbles, protein folding, quantum computing,
quantum advice, quantum adiabatic algorithms, quantum-mechanical
nonlinearities, hidden variables, relativistic time dilation, analog computing,
Malament-Hogarth spacetimes, quantum gravity, closed timelike curves, and
"anthropic computing." The section on soap bubbles even includes some
"experimental" results. While I do not believe that any of the proposals will
let us solve NP-complete problems efficiently, I argue that by studying them,
we can learn something not only about computation but also about physics.Comment: 23 pages, minor correction
Why Philosophers Should Care About Computational Complexity
One might think that, once we know something is computable, how efficiently
it can be computed is a practical question with little further philosophical
importance. In this essay, I offer a detailed case that one would be wrong. In
particular, I argue that computational complexity theory---the field that
studies the resources (such as time, space, and randomness) needed to solve
computational problems---leads to new perspectives on the nature of
mathematical knowledge, the strong AI debate, computationalism, the problem of
logical omniscience, Hume's problem of induction, Goodman's grue riddle, the
foundations of quantum mechanics, economic rationality, closed timelike curves,
and several other topics of philosophical interest. I end by discussing aspects
of complexity theory itself that could benefit from philosophical analysis.Comment: 58 pages, to appear in "Computability: G\"odel, Turing, Church, and
beyond," MIT Press, 2012. Some minor clarifications and corrections; new
references adde
Solving Set Cover with Pairs Problem using Quantum Annealing
Here we consider using quantum annealing to solve Set Cover with Pairs (SCP), an NP-hard combinatorial optimization problem that plays an important role in networking, computational biology, and biochemistry. We show an explicit construction of Ising Hamiltonians whose ground states encode the solution of SCP instances. We numerically simulate the time-dependent Schrƶdinger equation in order to test the performance of quantum annealing for random instances and compare with that of simulated annealing. We also discuss explicit embedding strategies for realizing our Hamiltonian construction on the D-wave type restricted Ising Hamiltonian based on Chimera graphs. Our embedding on the Chimera graph preserves the structure of the original SCP instance and in particular, the embedding for general complete bipartite graphs and logical disjunctions may be of broader use than that the specific problem we deal with
Statistical mixtures of states can be more quantum than their superpositions: Comparison of nonclassicality measures for single-qubit states
A bosonic state is commonly considered nonclassical (or quantum) if its
Glauber-Sudarshan function is not a classical probability density, which
implies that only coherent states and their statistical mixtures are classical.
We quantify the nonclassicality of a single qubit, defined by the vacuum and
single-photon states, by applying the following four well-known measures of
nonclassicality: (1) the nonclassical depth, , related to the minimal
amount of Gaussian noise which changes a nonpositive function into a
positive one; (2) the nonclassical distance , defined as the Bures distance
of a given state to the closest classical state, which is the vacuum for the
single-qubit Hilbert space; together with (3) the negativity potential (NP) and
(4) concurrence potential, which are the nonclassicality measures corresponding
to the entanglement measures (i.e., the negativity and concurrence,
respectively) for the state generated by mixing a single-qubit state with the
vacuum on a balanced beam splitter. We show that complete statistical mixtures
of the vacuum and single-photon states are the most nonclassical single-qubit
states regarding the distance for a fixed value of both the depth
and NP in the whole range of their values, as well as the NP for a
given value of such that . Conversely, pure states are the
most nonclassical single-qubit states with respect to for a given ,
NP versus , and versus NP. We also show the "relativity" of these
nonclassicality measures by comparing pairs of single-qubit states: if a state
is less nonclassical than another state according to some measure then it might
be more nonclassical according to another measure. Moreover, we find that the
concurrence potential is equal to the nonclassical distance for single-qubit
states.Comment: 12 pages, 3 figures, and 3 table
Fast optimization algorithms and the cosmological constant
Denef and Douglas have observed that in certain landscape models the problem
of finding small values of the cosmological constant is a large instance of an
NP-hard problem. The number of elementary operations (quantum gates) needed to
solve this problem by brute force search exceeds the estimated computational
capacity of the observable universe. Here we describe a way out of this
puzzling circumstance: despite being NP-hard, the problem of finding a small
cosmological constant can be attacked by more sophisticated algorithms whose
performance vastly exceeds brute force search. In fact, in some parameter
regimes the average-case complexity is polynomial. We demonstrate this by
explicitly finding a cosmological constant of order in a randomly
generated -dimensional ADK landscape.Comment: 19 pages, 5 figure
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