Abstracts - Invited Speakers
Variational Reduced Density Matrix Theory:
Successes and Failures
by
Paul W. Ayers
Department of Chemistry; McMaster Univ.; Hamilton ON L8S 4M1
Coauthors: Dimitri Van Neck, Patrick Bultinck, Helen Van Aggelen,
Brecht Verstichel
Because the molecular Hamiltonian contains only one-body and two-body
operators, the two-electron reduced density matrix contains all
the information needed to evaluate the energy, as well as most of
the other properties of interest to chemists and molecular physicists.
A straightforward minimization of the energy is confounded by the
N-representability problem, which can only be addressed approximately.
The resulting theory has both advantages and disadvantages compared
to more traditional wavefunction-based approaches. The biggest advantage
is that it performs well even when the molecule of interest has
strong multireference character and the “Hartree-Fock plus
correction” wavefunction paradigms fail. Also, as a lower-bound
method, it provides a complementary tool to variational wavefunction
approaches. The biggest disadvantages are the computational cost
(which may yet be surmounted) and problems with dissociation and
degeneracy that seem to afflict all approaches based on a “reduced”
description of the system.
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The reduced density matrix method and
some complexity issues
by
Bastiaan Braams
Emory University
I will review the classical formalism of electronic structure theory
that is based on the two-body reduced density matrix and associated
representability conditions. The N-representability problem is unsolved
and results from computational complexity theory indicate that it
won't be solved in any concise form. I hope on the one hand to make
clear that it doesn't matter, and on the other hand to outline the
challenge of obtaining what could be called anyway a solution to
the N-representability problem. Finally I will discuss the computational
complexity of the Hartree-Fock problem. As an algebraic problem
with general 2-body terms in the hamiltonian (not limited to Coulomb
interaction) it is NP-hard, while in the setting of parameterized
complexity theory, with the number of electrons as the parameter,
it is W[1]-hard.
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The DMRG and Correlator Product States
by
Garnet Kin-Lic Chan
Department of Chemistry and Chemical Biology, Cornell University,
Ithaca NY14853-1301
I will describe applications of the DMRG to quantum chemistry.
In addition, I will also talk about a new class of variational wavefunctions
- the correlator product states - that generalise the DMRG in a
practical way to encode higher dimensional correlations.
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Estimating the spectrum of an operator
- a technique for the analysis of spectral relations via representation
theory
by
Matthias Christandl
Ludwig-Maximilians-Universität München
The optimal way of estimating the spectrum of an operator is by
projection onto irreducible representations of the unitary group
and symmetric group. Formulated in this form by Keyl and Werner,
the mathematical content of spectrum estimation has been discovered
independently and in many variations by mathematicians and physicists
during the last decades.
In this talk, I will show how spectrum estimation can be applied
in order to study relations among operators via representation theory
(and vice versa). Examples are: 1) the study of Horn's problem (addition
of Hermitian operators) via Littlewood-Richardson coefficients and
2) the study of the quantum marginal problem via the Kronecker coefficients
of the symmetric group.
This is a joint work with Graeme Mitchison and Aram Harrow.
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Simulating strongly correlated fermions
by
J. Eisert
University of Potsdam
Coauthors: C. Pineda, T. Barthel
We introduce a scheme for efficiently describing pure states of
strongly correlated fermions in higher dimensions using unitary
circuits. A local way of computing local expectation values is presented.
We formulate a dynamical reordering scheme, corresponding to time-adaptive
Jordan-Wigner transformation that avoids non-local string operators
and only keeps suitably ordered the causal cone. Primitives of such
a reordering scheme are highlighted. Fermionic unitary circuits
can be contracted with the same complexity as in the spin case.
The scheme gives rise to a variational description of fermionic
models that does not suffer from a sign problem. We present a numerical
example on a 27x27 fermionic lattice model to show the functioning
of the approach. We also discuss the contraction complexity of general
fermionic networks.
See http://arxiv.org/abs/0905.0669,
for related work see also http://arxiv.org/abs/0904.4151
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The Pauli exclusion principle and
beyond
by
Alexander Klyachko
Bilkent University
The original Pauli exclusion principle claims that no quantum state
can be occupied by more than one electron. This can be stated as
inequality (y|r|y) = 1 on one-electron density matrix r. Nowadays
the Pauli principle is replaced by skew symmetry of the multilelectron
state. In this talk I review a recent solution of a longstanding
problem about impact of this replacement on the electron density
matrix. It goes far beyond the original Pauli principle and leads
to numerous additional constraints. If time permits I will also
discuss some physical implications.
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Quantum marginal problem
by
Alexnder Klyachko
Bilkent University
Classical marginal problem is about existence of a body with given
projections onto some coordinate subspaces. In the talk I primary
address to a quantum version of this propblem about existence of
a state of a multicomponent quantum system with given reduced density
matrices. I will explain its reduction to Schubert calculus and
its connection with representations of the symmetric group. Some
applications to quantum information will be discussed.
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N-representability is QMA-complete
by
Yi-Kai Liu
Caltech
Coauthors: Matthias Christandl and Frank Verstraete
We study the computational complexity of the N-representability
problem in quantum chemistry. We show that this problem is QMA-complete,
which is the quantum generalization of NP-complete. Our proof uses
a simple mapping from spin systems to fermionic systems, together
with a convex optimization technique that reduces the problem of
finding ground states to N-representability.
(This talk will include a short introduction to complexity theory
and QMA-completeness.)
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QMA-complete problems for stoquastic Hamiltonians
and Markov matrices
by
Peter Love
Haverford College
Coauthors: Stephen Jordan
We show that finding the lowest eigenvalue of a 3-local symmetric
stochastic matrix is QMA-complete. We also show that finding the
highest energy of a stoquastic Hamiltonian is QMA-complete and that
adiabatic quantum computation in the highest energy state and certain
other excited states of a stoquastic Hamiltonian is universal. These
results give a new QMA-complete problem arising in the classical
setting of Markov chains, and new adiabatically universal Hamiltonians
which arise in many physical systems.
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de Finetti theorems for quantum states
by
Robert Koenig (Caltech) and Graeme Mitchison (University of Cambridge)
Institute for Quantum Information, Caltech, MC 305-16, Pasadena
91125, USA// DAMTP, University of Cambridge, Wilberforce Road, Cambridge
CB3 0WA UK
Coauthors: (partly based on joint work with M. Christandl, R. Renner
and M. Wolf)
We give an introduction to de Finetti theorems for quantum states.
Such a theorem asserts that, given a symmetric state on a system
composed of n subsystems, the state obtained by tracing out n-k
of the subsystems can be represented approximately as a convex sum
of product states. Furthermore, the precision of the approximation
increases as k/n decreases, i.e. as a larger fraction of the subsystems
are traced out. This can be proved using simple representation-theoretic
ideas, which can also be extended to prove a number of interesting
variants of the de Finetti theorem. Finally, we consider states
that are both symmetric and unitarily invariant (symmetric Werner
states). These give a rich supply of examples and have a close connection
to the de Finetti theorem for probability distributions. They can
also be used to establish limits on the closeness of approximation
by product states, and therefore provide information about the tightness
of de Finetti theorems.
joint work with Matthias Christandl and Renato Renner
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Non-commutative polynomial optimization
and the varianional RDM method
by
Stefano Pironio
Group of Applied Physics, University of Geneva
Coauthors: Artur Garcia, Miguel Navascues, Antonio Acin
A standard problem in optimization theory is to find the minimum
of a polynomial function subject to polynomial inequality constraints.
We introduce a generalization of this problem where the optimization
variables are not real numbers, but non-commutative variables, i.e.,
operators acting on Hilbert spaces of arbitrary dimension. We show
how semidefinite programming (SDP) can be used to solve this problem.
Specifically, we introduce a sequence of SDP relaxations of the
original problem, whose optima converge monotically to the global
optimum.
Our method can find applications to compute the ground state energy
of quantum many-body systems. In particular, it gives a new interpretation
to and should strengthens the RDM method used in quantum chemistry
to compute electronic energies. Our method provides a computation
technique for many-body systems that is not based on states (and
thus directly linked to entanglement) but that is rather based on
the algebraic structure of quantum operators.
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De Finetti and entropies
by
Renato Renner
ETH Zurich
Coauthors: Matthias Christandl, Graeme Mitchison, Robert Koenig
The estimation of the entropy of permutation invariant systems
plays a crucial role in various information-theoretic applications,
in particular in the context of quantum cryptography.
In this talk, I will review the connections between (generalized)
entropies and de Finetti type theorems (including their quantum
versions). In particular, I will show how de Finetti's theorem can
be used to derive bounds on entropic quantities.
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Density matrices in real-space renormalization
group methods
by
Frank Verstraete
University of Vienna
We will discuss the role of density matrices in real-space renormalization
group methods, and include detours via computational complexity
and density functional theory.
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Bosonic N-representability problem is also
QMA-complete
by
Tzu-Chieh Wei
Institute for Quantum Computing, University of Waterloo
Coauthors: Michele Mosca and Ashwin Nayak
Computing the ground-state energy of interacting electron (fermion)
problems has recently been shown to be hard for QMA, a quantum analogue
of the complexity class NP. Fermionic problems are usually hard,
a phenomenon widely attributed to the so-called sign problem occurring
in Quantum Monte Carlo simulations. The corresponding bosonic problems
are, according to conventional wisdom, tractable. Here, we discuss
the complexity of interacting boson problems and show that they
are also QMA-hard. In addition, we show that the bosonic version
of the so-called N-representability problem is QMA-complete, as
hard as its fermionic version. As a consequence, these problems
are unlikely to have efficient quantum algorithms.
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A new inequality for the von Neumann
entropy
by
Andreas Winter
University of Bristol
Coauthors: Noah Linden, Ben Ibinson
This talk will be mainly based on a paper with N Linden. Pippenger
has initiated the generalization of the programme to find all the
"laws of information theory" to quantum entropy. The standard
inequalities derive from strong subadditivity (SSA). SSA of the
von Neumann entropy, proved in 1973 by Lieb and Ruskai, is a cornerstone
of quantum coding theory. All other known inequalities for entropies
of quantum systems may be derived from it. Here we prove a new inequality
for the von Neumann entropy which we show is independent of strong
subadditivity: it is an inequality which is true for any four party
quantum state, provided that it satisfies three linear relations
(constraints) on the entropies of certain reduced states. In the
talk I will also discuss the possibility of finding an unconstrained
inequality (work with N Linden and B Ibinson).
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Bell inequalities and joint measurability
- relating classical and quantum marginal problems
by
Michael M. Wolf
Niels Bohr Institute
Coauthors: David Perez-Garcia, Carlos Fernandez
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