125 research outputs found
Quantum Algorithms for Abelian Difference Sets and Applications to Dihedral Hidden Subgroups
Difference sets are basic combinatorial structures that have applications in signal processing, coding theory, and cryptography. We consider the problem of identifying a shifted version of the characteristic function of a (known) difference set and present a general algorithm that can be used to tackle any hidden shift problem for any difference set in any abelian group. We discuss special cases of this framework which include a) Paley difference sets based on quadratic residues in finite fields which allow to recover the shifted Legendre function quantum algorithm, b) Hadamard difference sets which allow to recover the shifted bent function quantum algorithm, and c) Singer difference sets which allow us to define instances of the dihedral hidden subgroup problem which can be efficiently solved on a quantum computer
Quantum algorithms for algebraic problems
Quantum computers can execute algorithms that dramatically outperform
classical computation. As the best-known example, Shor discovered an efficient
quantum algorithm for factoring integers, whereas factoring appears to be
difficult for classical computers. Understanding what other computational
problems can be solved significantly faster using quantum algorithms is one of
the major challenges in the theory of quantum computation, and such algorithms
motivate the formidable task of building a large-scale quantum computer. This
article reviews the current state of quantum algorithms, focusing on algorithms
with superpolynomial speedup over classical computation, and in particular, on
problems with an algebraic flavor.Comment: 52 pages, 3 figures, to appear in Reviews of Modern Physic
Normalizer Circuits and Quantum Computation
(Abridged abstract.) In this thesis we introduce new models of quantum
computation to study the emergence of quantum speed-up in quantum computer
algorithms.
Our first contribution is a formalism of restricted quantum operations, named
normalizer circuit formalism, based on algebraic extensions of the qubit
Clifford gates (CNOT, Hadamard and -phase gates): a normalizer circuit
consists of quantum Fourier transforms (QFTs), automorphism gates and quadratic
phase gates associated to a set , which is either an abelian group or
abelian hypergroup. Though Clifford circuits are efficiently classically
simulable, we show that normalizer circuit models encompass Shor's celebrated
factoring algorithm and the quantum algorithms for abelian Hidden Subgroup
Problems. We develop classical-simulation techniques to characterize under
which scenarios normalizer circuits provide quantum speed-ups. Finally, we
devise new quantum algorithms for finding hidden hyperstructures. The results
offer new insights into the source of quantum speed-ups for several algebraic
problems.
Our second contribution is an algebraic (group- and hypergroup-theoretic)
framework for describing quantum many-body states and classically simulating
quantum circuits. Our framework extends Gottesman's Pauli Stabilizer Formalism
(PSF), wherein quantum states are written as joint eigenspaces of stabilizer
groups of commuting Pauli operators: while the PSF is valid for qubit/qudit
systems, our formalism can be applied to discrete- and continuous-variable
systems, hybrid settings, and anyonic systems. These results enlarge the known
families of quantum processes that can be efficiently classically simulated.
This thesis also establishes a precise connection between Shor's quantum
algorithm and the stabilizer formalism, revealing a common mathematical
structure in several quantum speed-ups and error-correcting codes.Comment: PhD thesis, Technical University of Munich (2016). Please cite
original papers if possible. Appendix E contains unpublished work on Gaussian
unitaries. If you spot typos/omissions please email me at JLastNames at
posteo dot net. Source: http://bit.ly/2gMdHn3. Related video talk:
https://www.perimeterinstitute.ca/videos/toy-theory-quantum-speed-ups-based-stabilizer-formalism
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