11,898 research outputs found
Universal Test for Quantum One-Way Permutations
The next bit test was introduced by Blum and Micali and proved by Yao to be a
universal test for cryptographic pseudorandom generators. On the other hand, no
universal test for the cryptographic one-wayness of functions (or permutations)
is known, though the existence of cryptographic pseudorandom generators is
equivalent to that of cryptographic one-way functions. In the quantum
computation model, Kashefi, Nishimura and Vedral gave a sufficient condition of
(cryptographic) quantum one-way permutations and conjectured that the condition
would be necessary. In this paper, we affirmatively settle their conjecture and
complete a necessary and sufficient for quantum one-way permutations. The
necessary and sufficient condition can be regarded as a universal test for
quantum one-way permutations, since the condition is described as a collection
of stepwise tests similar to the next bit test for pseudorandom generators.Comment: 12 pages, 3 figures. The previous version included some error. This
is a corrected version. Fortunately, the proof is simplified and results are
improve
Comparing the states of many quantum systems
We investigate how to determine whether the states of a set of quantum
systems are identical or not. This paper treats both error-free comparison, and
comparison where errors in the result are allowed. Error-free comparison means
that we aim to obtain definite answers, which are known to be correct, as often
as possible. In general, we will have to accept also inconclusive results,
giving no information. To obtain a definite answer that the states of the
systems are not identical is always possible, whereas, in the situation
considered here, a definite answer that they are identical will not be
possible. The optimal universal error-free comparison strategy is a projection
onto the totally symmetric and the different non-symmetric subspaces, invariant
under permutations and unitary transformations. We also show how to construct
optimal comparison strategies when allowing for some errors in the result,
minimising either the error probability, or the average cost of making an
error. We point out that it is possible to realise universal error-free
comparison strategies using only linear elements and particle detectors, albeit
with less than ideal efficiency. Also minimum-error and minimum-cost strategies
may sometimes be realised in this way. This is of great significance for
practical applications of quantum comparison.Comment: 13 pages, 2 figures. Corrected a misprint on p. 7 and added a few
references. Accepted for publication in J Mod Op
Deciding universality of quantum gates
We say that collection of -qudit gates is universal if there exists
such that for every every -qudit unitary operation
can be approximated with arbitrary precision by a circuit built from gates of
the collection. Our main result is an upper bound on the smallest with
the above property. The bound is roughly , where is the number of
levels of the base system (the '' in the term quit.) The proof is based
on a recent result on invariants of (finite) linear groups.Comment: 8 pages, minor correction
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
Towards topological quantum computer
One of the principal obstacles on the way to quantum computers is the lack of
distinguished basis in the space of unitary evolutions and thus the lack of the
commonly accepted set of basic operations (universal gates). A natural choice,
however, is at hand: it is provided by the quantum R-matrices, the entangling
deformations of non-entangling (classical) permutations, distinguished from the
points of view of group theory, integrable systems and modern theory of
non-perturbative calculations in quantum field and string theory. Observables
in this case are (square modules of) the knot polynomials, and their pronounced
integrality properties could provide a key to error correction. We suggest to
use R-matrices acting in the space of irreducible representations, which are
unitary for the real-valued couplings in Chern-Simons theory, to build a
topological version of quantum computing.Comment: 14 page
Statistical Zero Knowledge and quantum one-way functions
One-way functions are a very important notion in the field of classical
cryptography. Most examples of such functions, including factoring, discrete
log or the RSA function, can be, however, inverted with the help of a quantum
computer. In this paper, we study one-way functions that are hard to invert
even by a quantum adversary and describe a set of problems which are good such
candidates. These problems include Graph Non-Isomorphism, approximate Closest
Lattice Vector and Group Non-Membership. More generally, we show that any hard
instance of Circuit Quantum Sampling gives rise to a quantum one-way function.
By the work of Aharonov and Ta-Shma, this implies that any language in
Statistical Zero Knowledge which is hard-on-average for quantum computers,
leads to a quantum one-way function. Moreover, extending the result of
Impagliazzo and Luby to the quantum setting, we prove that quantum
distributionally one-way functions are equivalent to quantum one-way functions.
Last, we explore the connections between quantum one-way functions and the
complexity class QMA and show that, similarly to the classical case, if any of
the above candidate problems is QMA-complete then the existence of quantum
one-way functions leads to the separation of QMA and AvgBQP.Comment: 20 pages; Computational Complexity, Cryptography and Quantum Physics;
Published version, main results unchanged, presentation improve
Wrapping interactions and the genus expansion of the 2-point function of composite operators
We perform a systematic analysis of wrapping interactions for a general class
of theories with color degrees of freedom, including N=4 SYM. Wrapping
interactions arise in the genus expansion of the 2-point function of composite
operators as finite size effects that start to appear at a certain order in the
coupling constant at which the range of the interaction is equal to the length
of the operators. We analyze in detail the relevant genus expansions, and
introduce a strategy to single out the wrapping contributions, based on adding
spectator fields. We use a toy model to demonstrate our procedure, performing
all computations explicitly. Although completely general, our treatment should
be particularly useful for applications to the recent problem of wrapping
contributions in some checks of the AdS/CFT correspondence.Comment: 41 pages, LaTeX, 12 figures, minor changes, final version in NP
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