1,478 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
Quantum Query Complexity of Boolean Functions under Indefinite Causal Order
The standard model of quantum circuits assumes operations are applied in a
fixed sequential "causal" order. In recent years, the possibility of relaxing
this constraint to obtain causally indefinite computations has received
significant attention. The quantum switch, for example, uses a quantum system
to coherently control the order of operations. Several ad hoc computational and
information-theoretical advantages have been demonstrated, raising questions as
to whether advantages can be obtained in a more unified complexity theoretic
framework. In this paper, we approach this problem by studying the query
complexity of Boolean functions under general higher order quantum
computations. To this end, we generalise the framework of query complexity from
quantum circuits to quantum supermaps to compare different models on an equal
footing. We show that the recently introduced class of quantum circuits with
quantum control of causal order cannot lead to any reduction in query
complexity, and that any potential advantage arising from causally indefinite
supermaps can be bounded by the polynomial method, as is the case with quantum
circuits. Nevertheless, we find some functions for which the minimum error with
which they can be computed using two queries is strictly lower when exploiting
causally indefinite supermaps.Comment: 6+11 page
Unconventional machine learning of genome-wide human cancer data
Recent advances in high-throughput genomic technologies coupled with
exponential increases in computer processing and memory have allowed us to
interrogate the complex aberrant molecular underpinnings of human disease from
a genome-wide perspective. While the deluge of genomic information is expected
to increase, a bottleneck in conventional high-performance computing is rapidly
approaching. Inspired in part by recent advances in physical quantum
processors, we evaluated several unconventional machine learning (ML)
strategies on actual human tumor data. Here we show for the first time the
efficacy of multiple annealing-based ML algorithms for classification of
high-dimensional, multi-omics human cancer data from the Cancer Genome Atlas.
To assess algorithm performance, we compared these classifiers to a variety of
standard ML methods. Our results indicate the feasibility of using
annealing-based ML to provide competitive classification of human cancer types
and associated molecular subtypes and superior performance with smaller
training datasets, thus providing compelling empirical evidence for the
potential future application of unconventional computing architectures in the
biomedical sciences
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