253 research outputs found
Yang-Baxter operators need quantum entanglement to distinguish knots
Any solution to the Yang-Baxter equation yields a family of representations
of braid groups. Under certain conditions, identified by Turaev, the
appropriately normalized trace of these representations yields a link
invariant. Any Yang-Baxter solution can be interpreted as a two-qudit quantum
gate. Here we show that if this gate is non-entangling, then the resulting
invariant of knots is trivial. We thus obtain a general connection between
topological entanglement and quantum entanglement, as suggested by Kauffman et
al.Comment: 12 pages, 2 figure
Partial-indistinguishability obfuscation using braids
An obfuscator is an algorithm that translates circuits into
functionally-equivalent similarly-sized circuits that are hard to understand.
Efficient obfuscators would have many applications in cryptography. Until
recently, theoretical progress has mainly been limited to no-go results. Recent
works have proposed the first efficient obfuscation algorithms for classical
logic circuits, based on a notion of indistinguishability against
polynomial-time adversaries. In this work, we propose a new notion of
obfuscation, which we call partial-indistinguishability. This notion is based
on computationally universal groups with efficiently computable normal forms,
and appears to be incomparable with existing definitions. We describe universal
gate sets for both classical and quantum computation, in which our definition
of obfuscation can be met by polynomial-time algorithms. We also discuss some
potential applications to testing quantum computers. We stress that the
cryptographic security of these obfuscators, especially when composed with
translation from other gate sets, remains an open question.Comment: 21 pages,Proceedings of TQC 201
An efficient high dimensional quantum Schur transform
The Schur transform is a unitary operator that block diagonalizes the action
of the symmetric and unitary groups on an fold tensor product of a vector space of dimension . Bacon, Chuang and Harrow
\cite{BCH07} gave a quantum algorithm for this transform that is polynomial in
, and , where is the precision. In a
footnote in Harrow's thesis \cite{H05}, a brief description of how to make the
algorithm of \cite{BCH07} polynomial in is given using the unitary
group representation theory (however, this has not been explained in detail
anywhere. In this article, we present a quantum algorithm for the Schur
transform that is polynomial in , and using a
different approach. Specifically, we build this transform using the
representation theory of the symmetric group and in this sense our technique
can be considered a "dual" algorithm to \cite{BCH07}. A novel feature of our
algorithm is that we construct the quantum Fourier transform over the so called
\emph{permutation modules}, which could have other applications.Comment: 21 page
Quantum Groups and Noncommutative Geometry
Quantum groups emerged in the latter quarter of the 20th century as, on the
one hand, a deep and natural generalisation of symmetry groups for certain
integrable systems, and on the other as part of a generalisation of geometry
itself powerful enough to make sense in the quantum domain. Just as the last
century saw the birth of classical geometry, so the present century sees at its
end the birth of this quantum or noncommutative geometry, both as an elegant
mathematical reality and in the form of the first theoretical predictions for
Planck-scale physics via ongoing astronomical measurements. Noncommutativity of
spacetime, in particular, amounts to a postulated new force or physical effect
called cogravity.Comment: 72 pages, many figures; intended for wider theoretical physics
community (special millenium volume of JMP
Many-body models for topological quantum information
We develop and investigate several quantum many-body spin models of use for topological quantum information processing and storage. These models fall into two categories: those that are designed to be more realistic than alternative models with similar phenomenology, and those that are designed to have richer phenomenology than related models. In the first category, we present a procedure to obtain the Hamiltonians of the toric code and Kitaev quantum double models as the perturbative low-energy limits of entirely two-body Hamiltonians. This construction reproduces the target models' behavior using only couplings which are natural in terms of the original Hamiltonians. As an extension of this work, we construct parent Hamiltonians involving only local 2-body interactions for a broad class of Projected Entangled Pair States (PEPS). We define a perturbative Hamiltonian with a finite order low energy effective Hamiltonian that is a gapped, frustration-free parent Hamiltonian for an encoded version of a desired PEPS. For topologically ordered PEPS, the ground space of the low energy effective Hamiltonian is shown to be in the same phase as the desired state to all orders of perturbation theory. We then move on to define models that generalize the phenomenology of several well-known systems. We first define generalized cluster states based on finite group algebras, and investigate properties of these states including their PEPS representations, global symmetries, relationship to the Kitaev quantum double models, and possible applications. Finally, we propose a generalization of the color codes based on finite groups. For non-Abelian groups, the resulting model supports non-Abelian anyonic quasiparticles and topological order. We examine the properties of these models such as their relationship to Kitaev quantum double models, quasiparticle spectrum, and boundary structure
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