Continuous-Variable Quantum Computation of Oracle Decision Problems
Date
2013-01-07
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Abstract
Quantum information processing is appealing due its ability to
solve certain problems quantitatively faster than classical
information processing. Most quantum algorithms have been studied
in discretely parameterized systems, but many quantum systems are
continuously parameterized. The field of quantum optics in
particular has sophisticated techniques for manipulating
continuously parameterized quantum states of light, but the lack
of a code-state formalism has hindered the study of quantum
algorithms in these systems. To address this situation, a
code-state formalism for the solution of oracle decision problems
in continuously-parameterized quantum systems is developed.
In discrete-variable quantum computation, oracle decision problems
exploit quantum parallelism through the use of the Hadamard
transform. The challenge in continuous-variable quantum
computation is to exploit similar quantum parallelism by
generalizing the Hadamard transform to the continuous Fourier
transform while avoiding non-renormalizable states. This
straightforward relationship between the operators in the discrete
and continuous settings make oracle decision problems the ideal
test-bed. However, as the formalism results in a representation of
discrete information strings as proper code states, the approach
also allows for the study of a wider range of quantum algorithms
in continuously-parameterized quantum systems having both finite-
and infinite-dimensional Hilbert spaces.
In the infinite-dimensional case, we study continuous-variable
quantum algorithms for the solution of the Deutsch--Jozsa oracle
decision problem implemented within a single harmonic-oscillator.
Orthogonal states are used as the computational bases, and we show
that, contrary to a previous claim in the literature, this
implementation of quantum information processing has limitations
due to a position-momentum trade-off of the Fourier transform. We
further demonstrate that orthogonal encoding bases are not unique,
and using the coherent states of the harmonic oscillator as the
computational bases, our formalism enables quantifying the
relative performances of different choices of the encoding bases.
We extend our formalism to include quantum algorithms in the
continuously parameterized yet finite-dimensional Hilbert space of
a coherent spin system. We show that the highest-squeezed spin
state possible can be approximated by a superposition of two
states thus transcending the usual model of using a single basis
state as algorithm input. As a particular example, we show that
the close Hadamard oracle-decision problem, which is related to
the Hadamard codewords of digital communications theory, can be
solved quantitatively more efficiently using this computational
model than by any known classical algorithm.
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Physics--Theory
Citation
Adcock, M. R. (2013). Continuous-Variable Quantum Computation of Oracle Decision Problems (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca. doi:10.11575/PRISM/25442