38 research outputs found
Classical simulation of Yang-Baxter gates
A unitary operator that satisfies the constant Yang-Baxter equation
immediately yields a unitary representation of the braid group B n for every . If we view such an operator as a quantum-computational gate, then
topological braiding corresponds to a quantum circuit. A basic question is when
such a representation affords universal quantum computation. In this work, we
show how to classically simulate these circuits when the gate in question
belongs to certain families of solutions to the Yang-Baxter equation. These
include all of the qubit (i.e., ) solutions, and some simple families
that include solutions for arbitrary . Our main tool is a
probabilistic classical algorithm for efficient simulation of a more general
class of quantum circuits. This algorithm may be of use outside the present
setting.Comment: 17 pages. Corrected error in proof of Theorem
Quantum Invariants of 3-manifolds and NP vs #P
The computational complexity class #P captures the difficulty of counting the
satisfying assignments to a boolean formula. In this work, we use basic tools
from quantum computation to give a proof that the SO(3)
Witten-Reshetikhin-Turaev (WRT) invariant of 3-manifolds is #P-hard to
calculate. We then apply this result to a question about the combinatorics of
Heegaard splittings, motivated by analogous work on link diagrams by M.
Freedman. We show that, if , then there
exist infinitely many Heegaard splittings which cannot be made logarithmically
thin by local WRT-preserving moves, except perhaps via a superpolynomial number
of steps. We also outline two extensions of the above results. First, adapting
a result of Kuperberg, we show that any presentation-independent approximation
of WRT is also #P-hard. Second, we sketch out how all of our results can be
translated to the setting of triangulations and Turaev-Viro invariants.Comment: 22 pages, 5 figure
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
Quantum Algorithms for Invariants of Triangulated Manifolds
One of the apparent advantages of quantum computers over their classical
counterparts is their ability to efficiently contract tensor networks. In this
article, we study some implications of this fact in the case of topological
tensor networks. The graph underlying these networks is given by the
triangulation of a manifold, and the structure of the tensors ensures that the
overall tensor is independent of the choice of internal triangulation. This
leads to quantum algorithms for additively approximating certain invariants of
triangulated manifolds. We discuss the details of this construction in two
specific cases. In the first case, we consider triangulated surfaces, where the
triangle tensor is defined by the multiplication operator of a finite group;
the resulting invariant has a simple closed-form expression involving the
dimensions of the irreducible representations of the group and the Euler
characteristic of the surface. In the second case, we consider triangulated
3-manifolds, where the tetrahedral tensor is defined by the so-called Fibonacci
anyon model; the resulting invariant is the well-known Turaev-Viro invariant of
3-manifolds.Comment: 19 pages, 7 figure
Quantum non-malleability and authentication
Abstract: In encryption, non-malleability is a highly desirable property: it ensures that adversaries cannot manipulate the plaintext by acting on the ciphertext. Ambainis et al. gave a definition of non-malleability for the encryption of quantum data. In this work, we show that this definition is too weak, as it allows adversaries to ``inject'' plaintexts of their choice into the ciphertext. We give a new definition of quantum non-malleability which resolves this problem. Our definition is expressed in terms of entropic quantities, considers stronger adversaries, and does not assume secrecy. Rather, we prove that quantum non-malleability implies secrecy; this is in stark contrast to the classical setting, where the two properties are completely independent. For unitary schemes, our notion of non-malleability is equivalent to encryption with a two-design (and hence also to the definition of Ambainis et al.).
Our techniques also yield new results regarding the closely-related task of quantum authentication. We show that ``total authentication'' (a notion recently proposed by Garg et al.) can be satisfied with two-designs, a significant improvement over their eight-design-based construction. We also show that, under a mild adaptation of the rejection procedure, both total authentication and our notion of non-malleability yield quantum authentication as defined by Dupuis et al