30 research outputs found
A Linear-Optical Proof that the Permanent is #P-Hard
One of the crown jewels of complexity theory is Valiant's 1979 theorem that
computing the permanent of an n*n matrix is #P-hard. Here we show that, by
using the model of linear-optical quantum computing---and in particular, a
universality theorem due to Knill, Laflamme, and Milburn---one can give a
different and arguably more intuitive proof of this theorem.Comment: 12 pages, 2 figures, to appear in Proceedings of the Royal Society A.
doi: 10.1098/rspa.2011.023
Simply Exponential Approximation of the Permanent of Positive Semidefinite Matrices
We design a deterministic polynomial time approximation algorithm for
the permanent of positive semidefinite matrices where . We write a natural convex relaxation and show that its optimum solution
gives a approximation of the permanent. We further show that this factor
is asymptotically tight by constructing a family of positive semidefinite
matrices
Monotone Projection Lower Bounds from Extended Formulation Lower Bounds
In this short note, we reduce lower bounds on monotone projections of
polynomials to lower bounds on extended formulations of polytopes. Applying our
reduction to the seminal extended formulation lower bounds of Fiorini, Massar,
Pokutta, Tiwari, & de Wolf (STOC 2012; J. ACM, 2015) and Rothvoss (STOC 2014;
J. ACM, 2017), we obtain the following interesting consequences.
1. The Hamiltonian Cycle polynomial is not a monotone subexponential-size
projection of the permanent; this both rules out a natural attempt at a
monotone lower bound on the Boolean permanent, and shows that the permanent is
not complete for non-negative polynomials in VNP under monotone
p-projections.
2. The cut polynomials and the perfect matching polynomial (or "unsigned
Pfaffian") are not monotone p-projections of the permanent. The latter, over
the Boolean and-or semi-ring, rules out monotone reductions in one of the
natural approaches to reducing perfect matchings in general graphs to perfect
matchings in bipartite graphs.
As the permanent is universal for monotone formulas, these results also imply
exponential lower bounds on the monotone formula size and monotone circuit size
of these polynomials.Comment: Published in Theory of Computing, Volume 13 (2017), Article 18;
Received: November 10, 2015, Revised: July 27, 2016, Published: December 22,
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Polynomial computational complexity of matrix elements of finite-rank-generated single-particle operators in products of finite bosonic states
It is known that computing the permanent of the matrix , where is a
finite-rank matrix, requires a number of operations polynomial in the matrix
size. Motivated by the boson-sampling proposal of restricted quantum
computation, I extend this result to a generalization of the matrix permanent:
an expectation value in a product of a large number of identical bosonic states
with a bounded number of bosons. This result complements earlier studies on the
computational complexity in boson sampling and related setups. The proposed
technique based on the Gaussian averaging is equally applicable to bosonic and
fermionic systems. This also allows us to improve an earlier polynomial
complexity estimate for the fermionic version of the same problem.Comment: 4 pages, introduction and conclusion expanded, minor style
correction
Boson sampling with integrated optical circuits
Simulating the evolution of non-interacting bosons through a linear transformation acting on the system’s Fock state is strongly believed to be hard for a classical computer. This is commonly known as the Boson Sampling problem, and
has recently got attention as the first possble way to demonstrate the superior computational power of quantum devices over classical ones. In this paper we describe the quantum optics approach to this problem, highlighting the role of integrated optical circuits
The Power of Quantum Fourier Sampling
A line of work initiated by Terhal and DiVincenzo and Bremner, Jozsa, and
Shepherd, shows that quantum computers can efficiently sample from probability
distributions that cannot be exactly sampled efficiently on a classical
computer, unless the PH collapses. Aaronson and Arkhipov take this further by
considering a distribution that can be sampled efficiently by linear optical
quantum computation, that under two feasible conjectures, cannot even be
approximately sampled classically within bounded total variation distance,
unless the PH collapses.
In this work we use Quantum Fourier Sampling to construct a class of
distributions that can be sampled by a quantum computer. We then argue that
these distributions cannot be approximately sampled classically, unless the PH
collapses, under variants of the Aaronson and Arkhipov conjectures.
In particular, we show a general class of quantumly sampleable distributions
each of which is based on an "Efficiently Specifiable" polynomial, for which a
classical approximate sampler implies an average-case approximation. This class
of polynomials contains the Permanent but also includes, for example, the
Hamiltonian Cycle polynomial, and many other familiar #P-hard polynomials.
Although our construction, unlike that proposed by Aaronson and Arkhipov,
likely requires a universal quantum computer, we are able to use this
additional power to weaken the conjectures needed to prove approximate sampling
hardness results