67 research outputs found
Quantum algorithms for highly non-linear Boolean functions
Attempts to separate the power of classical and quantum models of computation
have a long history. The ultimate goal is to find exponential separations for
computational problems. However, such separations do not come a dime a dozen:
while there were some early successes in the form of hidden subgroup problems
for abelian groups--which generalize Shor's factoring algorithm perhaps most
faithfully--only for a handful of non-abelian groups efficient quantum
algorithms were found. Recently, problems have gotten increased attention that
seek to identify hidden sub-structures of other combinatorial and algebraic
objects besides groups. In this paper we provide new examples for exponential
separations by considering hidden shift problems that are defined for several
classes of highly non-linear Boolean functions. These so-called bent functions
arise in cryptography, where their property of having perfectly flat Fourier
spectra on the Boolean hypercube gives them resilience against certain types of
attack. We present new quantum algorithms that solve the hidden shift problems
for several well-known classes of bent functions in polynomial time and with a
constant number of queries, while the classical query complexity is shown to be
exponential. Our approach uses a technique that exploits the duality between
bent functions and their Fourier transforms.Comment: 15 pages, 1 figure, to appear in Proceedings of the 21st Annual
ACM-SIAM Symposium on Discrete Algorithms (SODA'10). This updated version of
the paper contains a new exponential separation between classical and quantum
query complexit
Representing Equivalence Problems for Combinatorial Objects
Methods for representing equivalence problems of various combinatorial objects
as graphs or binary matrices are considered. Such representations can be used
for isomorphism testing in classification or generation algorithms.
Often it is easier to consider a graph or a binary matrix isomorphism problem
than to implement heavy algorithms depending especially on particular combinatorial
objects. Moreover, there already exist well tested algorithms for the graph isomorphism
problem (nauty) and the binary matrix isomorphism problem as well (Q-Extension).
ACM Computing Classification System (1998): F.2.1, G.4
On Self-Dual Quantum Codes, Graphs, and Boolean Functions
A short introduction to quantum error correction is given, and it is shown
that zero-dimensional quantum codes can be represented as self-dual additive
codes over GF(4) and also as graphs. We show that graphs representing several
such codes with high minimum distance can be described as nested regular graphs
having minimum regular vertex degree and containing long cycles. Two graphs
correspond to equivalent quantum codes if they are related by a sequence of
local complementations. We use this operation to generate orbits of graphs, and
thus classify all inequivalent self-dual additive codes over GF(4) of length up
to 12, where previously only all codes of length up to 9 were known. We show
that these codes can be interpreted as quadratic Boolean functions, and we
define non-quadratic quantum codes, corresponding to Boolean functions of
higher degree. We look at various cryptographic properties of Boolean
functions, in particular the propagation criteria. The new aperiodic
propagation criterion (APC) and the APC distance are then defined. We show that
the distance of a zero-dimensional quantum code is equal to the APC distance of
the corresponding Boolean function. Orbits of Boolean functions with respect to
the {I,H,N}^n transform set are generated. We also study the peak-to-average
power ratio with respect to the {I,H,N}^n transform set (PAR_IHN), and prove
that PAR_IHN of a quadratic Boolean function is related to the size of the
maximum independent set over the corresponding orbit of graphs. A construction
technique for non-quadratic Boolean functions with low PAR_IHN is proposed. It
is finally shown that both PAR_IHN and APC distance can be interpreted as
partial entanglement measures.Comment: Master's thesis. 105 pages, 33 figure
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