10 research outputs found
A full complexity dichotomy for immanant families
Given an integer and an irreducible character of
for some partition of , the immanant
maps matrices
to . Important special cases
include the determinant and permanent, which are the immanants associated with
the sign and trivial character, respectively.
It is known that immanants can be evaluated in polynomial time for characters
that are close to the sign character: Given a partition of with
parts, let count the boxes to the right of the first
column in the Young diagram of . For a family of partitions ,
let and write Imm
for the problem of evaluating on input and
. If , then Imm is known to be
polynomial-time computable. This subsumes the case of the determinant. On the
other hand, if , then previously known hardness results
suggest that Imm cannot be solved in polynomial time. However, these
results only address certain restricted classes of families .
In this paper, we show that the parameterized complexity assumption FPT
#W[1] rules out polynomial-time algorithms for Imm for any
computationally reasonable family of partitions with
. We give an analogous result in algebraic complexity under
the assumption VFPT VW[1]. Furthermore, if even grows
polynomially in , we show that Imm is hard for #P and VNP.
This concludes a series of partial results on the complexity of immanants
obtained over the last 35 years.Comment: 28 pages, to appear at STOC'2
The Computational Complexity of Quantum Determinants
In this work, we study the computational complexity of quantum determinants,
a -deformation of matrix permanents: Given a complex number on the unit
circle in the complex plane and an matrix , the -permanent of
is defined as where
is the inversion number of permutation in the symmetric group on
elements. The function family generalizes determinant and permanent, which
correspond to the cases and respectively.
For worst-case hardness, by Liouville's approximation theorem and facts from
algebraic number theory, we show that for primitive -th root of unity
for odd prime power , exactly computing -permanent is
-hard. This implies that an efficient algorithm for
computing -permanent results in a collapse of the polynomial hierarchy.
Next, we show that computing -permanent can be achieved using an oracle that
approximates to within a polynomial multiplicative error and a membership
oracle for a finite set of algebraic integers. From this, an efficient
approximation algorithm would also imply a collapse of the polynomial
hierarchy. By random self-reducibility, computing -permanent remains to be
hard for a wide range of distributions satisfying a property called the strong
autocorrelation property. Specifically, this is proved via a reduction from
-permanent to -permanent for points on the unit circle.
Since the family of permanent functions shares common algebraic structure,
various techniques developed for the hardness of permanent can be generalized
to -permanents
Immanants and their applications in quantum optics
The regular representation of Sn appears quite naturally in the combinatorial problem
of the redistribution of quantum particles though an n-channel interferometer. By using
tools from representation theory, it has been shown that the coincidence rate can expressed
in terms of linear combinations of permuted immanants of the scattering matrix that
describes the interferometer. This thesis introduces the delay matrix, whose entries are
functions of the relative time delays between particles. The delay matrix is used with
Gamas’ theorem to determine exactly which immanants appear in the coincidence rate
for a given set of time delays, which improves our understanding of the Hong-Ou-Mandel
effect for many-particle systems. Both bosonic and fermionic systems are considered in
this thesis
Quantum simulation of partially distinguishable boson sampling
Boson Sampling is the problem of sampling from the same output probability
distribution as a collection of indistinguishable single photons input into a
linear interferometer. It has been shown that, subject to certain computational
complexity conjectures, in general the problem is difficult to solve
classically, motivating optical experiments aimed at demonstrating quantum
computational "supremacy". There are a number of challenges faced by such
experiments, including the generation of indistinguishable single photons. We
provide a quantum circuit that simulates bosonic sampling with arbitrarily
distinguishable particles. This makes clear how distinguishabililty leads to
decoherence in the standard quantum circuit model, allowing insight to be
gained. At the heart of the circuit is the quantum Schur transform, which
follows from a representation theoretic approach to the physics of
distinguishable particles in first quantisation. The techniques are quite
general and have application beyond boson sampling.Comment: 25 pages, 4 figures, 2 algorithms, comments welcom
Non-perturbative renormalization on the lattice
Strongly-interacting theories lie at the heart of elementary particle physics. Their distinct behaviour shapes our world sui generis. We are interested in lattice simulations of supersymmetric models, but every discretization of space-time inevitably breaks supersymmetry
and allows renormalization of relevant susy-breaking operators. To understand the role of such operators, we study renormalization group trajectories of the nonlinear O(N) Sigma model (NLSM). Similar to quantum gravity, it is believed to adhere to the asymptotic safety scenario. By combining the demon method with blockspin transformations, we compute the global flow diagram. In two dimensions, we reproduce asymptotic freedom and in three dimensions, asymptotic safety is demonstrated. Essential for these results is the application of a novel optimization scheme to treat truncation errors. We proceed with a lattice simulation of the supersymmetric nonlinear O(3) Sigma model. Using an original discretization that requires to fine tune only a single operator, we argue that the continuum limit successfully leads to the correct continuum physics. Unfortunately, for large lattices, a sign problem challenges the applicability of Monte Carlo methods. Consequently, the last chapter of this thesis is spent on an assessment of the fermion-bag method. We find that sign fluctuations are thereby significantly reduced for the susy NLSM. The proposed discretization finally promises a direct confirmation of supersymmetry restoration in the continuum limit. For a complementary analysis, we study the one-flavor Gross-Neveu model which has a complex phase problem. However, phase fluctuations for Wilson fermions are very small and no conclusion can be drawn regarding the potency of the fermion-bag approach for this model