375 research outputs found
The Complexity of the Consistency and N-representability Problems for Quantum States
QMA (Quantum Merlin-Arthur) is the quantum analogue of the class NP. There
are a few QMA-complete problems, most notably the ``Local Hamiltonian'' problem
introduced by Kitaev. In this dissertation we show some new QMA-complete
problems.
The first one is ``Consistency of Local Density Matrices'': given several
density matrices describing different (constant-size) subsets of an n-qubit
system, decide whether these are consistent with a single global state. This
problem was first suggested by Aharonov. We show that it is QMA-complete, via
an oracle reduction from Local Hamiltonian. This uses algorithms for convex
optimization with a membership oracle, due to Yudin and Nemirovskii.
Next we show that two problems from quantum chemistry, ``Fermionic Local
Hamiltonian'' and ``N-representability,'' are QMA-complete. These problems
arise in calculating the ground state energies of molecular systems.
N-representability is a key component in recently developed numerical methods
using the contracted Schrodinger equation. Although these problems have been
studied since the 1960's, it is only recently that the theory of quantum
computation has allowed us to properly characterize their complexity.
Finally, we study some special cases of the Consistency problem, pertaining
to 1-dimensional and ``stoquastic'' systems. We also give an alternative proof
of a result due to Jaynes: whenever local density matrices are consistent, they
are consistent with a Gibbs state.Comment: PhD thesis. Yay, no more grad school!! (Finished in August, but did
not get around to posting it until now.) 91 pages, a few figures, some boring
sections. Has detailed proofs of results in quant-ph/0604166 and
quant-ph/0609125. Ch.4 is a preliminary sketch of 0712.1388. Ch.5 is
quant-ph/060301
The Quantum PCP Conjecture
The classical PCP theorem is arguably the most important achievement of
classical complexity theory in the past quarter century. In recent years,
researchers in quantum computational complexity have tried to identify
approaches and develop tools that address the question: does a quantum version
of the PCP theorem hold? The story of this study starts with classical
complexity and takes unexpected turns providing fascinating vistas on the
foundations of quantum mechanics, the global nature of entanglement and its
topological properties, quantum error correction, information theory, and much
more; it raises questions that touch upon some of the most fundamental issues
at the heart of our understanding of quantum mechanics. At this point, the jury
is still out as to whether or not such a theorem holds. This survey aims to
provide a snapshot of the status in this ongoing story, tailored to a general
theory-of-CS audience.Comment: 45 pages, 4 figures, an enhanced version of the SIGACT guest column
from Volume 44 Issue 2, June 201
N-representability is QMA-complete
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, as well as a convex optimization technique
that reduces the problem of finding ground states to N-representability
Computational Complexity in Electronic Structure
In quantum chemistry, the price paid by all known efficient model chemistries
is either the truncation of the Hilbert space or uncontrolled approximations.
Theoretical computer science suggests that these restrictions are not mere
shortcomings of the algorithm designers and programmers but could stem from the
inherent difficulty of simulating quantum systems. Extensions of computer
science and information processing exploiting quantum mechanics has led to new
ways of understanding the ultimate limitations of computational power.
Interestingly, this perspective helps us understand widely used model
chemistries in a new light. In this article, the fundamentals of computational
complexity will be reviewed and motivated from the vantage point of chemistry.
Then recent results from the computational complexity literature regarding
common model chemistries including Hartree-Fock and density functional theory
are discussed.Comment: 14 pages, 2 figures, 1 table. Comments welcom
Testing non-isometry is QMA-complete
Determining the worst-case uncertainty added by a quantum circuit is shown to
be computationally intractable. This is the problem of detecting when a quantum
channel implemented as a circuit is close to a linear isometry, and it is shown
to be complete for the complexity class QMA of verifiable quantum computation.
This is done by relating the problem of detecting when a channel is close to an
isometry to the problem of determining how mixed the output of the channel can
be when the input is a pure state. How mixed the output of the channel is can
be detected by a protocol making use of the swap test: this follows from the
fact that an isometry applied twice in parallel does not affect the symmetry of
the input state under the swap operation.Comment: 12 pages, 3 figures. Presentation improved, results unchange
The computational complexity of density functional theory
Density functional theory is a successful branch of numerical simulations of
quantum systems. While the foundations are rigorously defined, the universal
functional must be approximated resulting in a `semi'-ab initio approach. The
search for improved functionals has resulted in hundreds of functionals and
remains an active research area. This chapter is concerned with understanding
fundamental limitations of any algorithmic approach to approximating the
universal functional. The results based on Hamiltonian complexity presented
here are largely based on \cite{Schuch09}. In this chapter, we explain the
computational complexity of DFT and any other approach to solving electronic
structure Hamiltonians. The proof relies on perturbative gadgets widely used in
Hamiltonian complexity and we provide an introduction to these techniques using
the Schrieffer-Wolff method. Since the difficulty of this problem has been well
appreciated before this formalization, practitioners have turned to a host
approximate Hamiltonians. By extending the results of \cite{Schuch09}, we show
in DFT, although the introduction of an approximate potential leads to a
non-interacting Hamiltonian, it remains, in the worst case, an NP-complete
problem.Comment: Contributed chapter to "Many-Electron Approaches in Physics,
Chemistry and Mathematics: A Multidisciplinary View
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