34 research outputs found
Weak Fourier-Schur sampling, the hidden subgroup problem, and the quantum collision problem
Schur duality decomposes many copies of a quantum state into subspaces
labeled by partitions, a decomposition with applications throughout quantum
information theory. Here we consider applying Schur duality to the problem of
distinguishing coset states in the standard approach to the hidden subgroup
problem. We observe that simply measuring the partition (a procedure we call
weak Schur sampling) provides very little information about the hidden
subgroup. Furthermore, we show that under quite general assumptions, even a
combination of weak Fourier sampling and weak Schur sampling fails to identify
the hidden subgroup. We also prove tight bounds on how many coset states are
required to solve the hidden subgroup problem by weak Schur sampling, and we
relate this question to a quantum version of the collision problem.Comment: 21 page
The Symmetric Group Defies Strong Fourier Sampling
The dramatic exponential speedups of quantum algorithms over their best existing classical counterparts were ushered in by the technique of Fourier sampling, introduced by Bernstein and Vazirani and developed by Simon and Shor into an approach to the hidden subgroup problem. This approach has proved successful for abelian groups, leading to efficient algorithms for factoring, extracting discrete logarithms, and other number-theoretic problems. We show, however, that this method cannot resolve the hidden subgroup problem in the symmetric groups, even in the weakest, information-theoretic sense. In particular, we show that the Graph Isomorphism problem cannot be solved by this approach. Our work implies that any quantum approach based upon the measurement of coset states must depart from the original framework by using entangled measurements on multiple coset states
Quantum algorithm for the Boolean hidden shift problem
The hidden shift problem is a natural place to look for new separations
between classical and quantum models of computation. One advantage of this
problem is its flexibility, since it can be defined for a whole range of
functions and a whole range of underlying groups. In a way, this distinguishes
it from the hidden subgroup problem where more stringent requirements about the
existence of a periodic subgroup have to be made. And yet, the hidden shift
problem proves to be rich enough to capture interesting features of problems of
algebraic, geometric, and combinatorial flavor. We present a quantum algorithm
to identify the hidden shift for any Boolean function. Using Fourier analysis
for Boolean functions we relate the time and query complexity of the algorithm
to an intrinsic property of the function, namely its minimum influence. We show
that for randomly chosen functions the time complexity of the algorithm is
polynomial. Based on this we show an average case exponential separation
between classical and quantum time complexity. A perhaps interesting aspect of
this work is that, while the extremal case of the Boolean hidden shift problem
over so-called bent functions can be reduced to a hidden subgroup problem over
an abelian group, the more general case studied here does not seem to allow
such a reduction.Comment: 10 pages, 1 figur
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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