CO 781
Topics in Quantum Information
Spring 2009
Lectures
Complete references to the works mentioned below are available here.
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May 5, 7, 12: [Kerenidis and de Wolf, 2004]
Locally decodable codes (LDCs),
Hadamard code,
exponential lower bound for 2-query codes (statement),
outline of proof,
XOR of two bits with one quantum query,
learning two bits with one quantum query,
construction of 1-quantum query LDC,
random access codes (RAC) and a lower bound on their size,
construction of a quantum RAC from a 1-quantum query LDC, implication
for the size of 2-query LDCs, simpler proof using
the Katz-Trevisan reduction to an XOR decoder.
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June 2: [Beals, Buhrman, Cleve, Mosca, de Wolf, 2001]
Representation of Boolean functions by multi-linear polynomials;
ANDn, ORn, PARITYn;
degree of a function; approximation of a Boolean function by
polynomials; approximation degree; Example of two-bit OR; polynomials
from quantum black-box algorithms; approximation degree for
ORn (statement); applications of polynomial
representations: black-box or decision tree complexity, learning, circuit
lower bounds (sketch); lower bound for approximation degree of
PARITYn: symmetrization, degree lower bound,
consequence for black-box complexity.
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June 4: [Ambainis, Childs, Reichardt, Spalek, Zhang, 2007]
Guest lecture by Andrew Childs.
Formulae over AND, OR, NOT; NAND formulae; approximation degree of a
function defined by a read-once formula; consequence for PAC learning;
certificates for value of formula; randomized algorithm for evaluation;
correspondence of value of a formula with eigenvectors of the associated
tree; detecting eigenvectors with non-zero amplitude on the root
(statement); description of the quantum algorithm.
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June 16, 18: [Aaronson, 2005]
The complexity class PP, complete problem for PP, power of the class
(w.r.t. NP, BPP, PH, PSPACE), comparison of exponential sums is in PP,
closure under complementation and intersection (also discussed NP, BPP),
the class PostBQP, probability amplification, NP in PostPP, closure of
PostBQP under complementation and intersection, PostBQP in PP
(express success probability as an exponential sum), PP in
PostBQP (map inputs in and not in the language to different halves of
the Bloch sphere, and detect which half).
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June 23, 24: [Hoyer, Lee, Spalek, 2007]
The adversary bound for quantum query complexity, sqrt(n) lower bound
for ORn, consequences for formula size, and classical
decision tree complexity, preliminaries---Hilbert-Schmidt inner product,
matrix norms and their properties, partial trace, proof of the lower
bound.
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June 25 [Laplante, Lee, Szegedy, 2005]
Formulae over AND, OR, NOT, size and depth of formulae,
the Karchmer-Wigderson relation, two-party communication protocols,
protocol partition number bound for formula size, monochromatic rectangle
partition size bound, subadditivity and monotonicity of spectral norm,
the adversary bound for formula size.
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June 30, July 2 [Aaronson, 2006]
Decision trees, examples: ORn and MAJORITYn,
radomized decision trees, local search, example: search on n-dimensional
hypercube (deterministic versus randomized), sketch of randomized
algorithm for general graphs, relational adversary bound
(statement), example: PARITYn, proof of the lower bound,
potential function, initial and final values, and bound on the
increment in the potential function.
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July 15, 16, 23 [Regev, 2005]
Solving CVPd with a sampler for dual lattice vectors, and
solver for LWE, the reduction and proofs using the smoothing parameter,
properties of Fourier transforms, and Gaussians.
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July 28, 29 [Hastings, 2007]
The 1-d, nearest-neighbour Local Hamiltonian problem, its complexity,
and heuristics to solve it, Matrix product states, an example, its low Schmidt
rank and small description, Hastings' area law for gapped systems,
implication for existence of MPS approximation to the ground state, and
to the complexity of gapped local Hamiltonian problem.