Lectures

Complete references to the works mentioned below are available here.

• 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.

• May 14,19,21,28: [Aharonov and Regev, 2005]
  Lattices, Shortest and Closest Vector Problems, and their gapped versions, Turing reduction from SVP to CVP, Containment in NP, Gap-CVP(sqrt(n)) in co-NP (statement), Sketch of [Goldreich and Goldwasser, 2000] and [Aharonov and Regev, 2003], outline of proof of [Aharonov and Regev, 2005], Fourier transform, the witness function and its properties, approximation of the witness function, the verifier, soundness and completeness of the proof system.

  Note: There is no lecture on May 26. Make-up lectures will be announced.

• June 2: [Beals, Buhrman, Cleve, Mosca, de Wolf, 2001]
  Representation of Boolean functions by multi-linear polynomials; AND_n, OR_n, PARITY_n; 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 OR_n (statement); applications of polynomial representations: black-box or decision tree complexity, learning, circuit lower bounds (sketch); lower bound for approximation degree of PARITY_n: symmetrization, degree lower bound, consequence for black-box complexity.

• 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.

• Note: There are no lectures on June 9,11. Make-up lectures on June 24, and July 15.

• 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).

• June 23, 24: [Hoyer, Lee, Spalek, 2007]
  The adversary bound for quantum query complexity, sqrt(n) lower bound for OR_n, 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.

• 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.

• June 30, July 2 [Aaronson, 2006]
  Decision trees, examples: OR_n and MAJORITY_n, 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: PARITY_n, proof of the lower bound, potential function, initial and final values, and bound on the increment in the potential function.

• July 2, 7, 8, 14 [Peikert, 2009]
  Learning With Errors (LWE) problem, equivalence of decision and search versions, equivalence of worst-case and average-case for the decision version, a public key encryption scheme with passive security based on LWE, proofs of correctness and security (including leftover hash lemma), reduction from GapSVP to LWE: the components from previous works, the algorithm, correctness.

  Note: There is no lecture on July 21. Make up lecture on July 29.

• July 15, 16, 23 [Regev, 2005]
  Solving CVP_d 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.

• 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.