6.896 Quantum Complexity Theory
October 9th, 2008
Lecture 11
Lecturer: Scott Aaronson
In this class, well nish the proof for D(f ) = O(Q(f )6 ) for all total Boolean functions f .
1
Block Sensitivity
We already saw the rst step in the proof: D(f ) C (f )2
6.896 Quantum Complexity Theory
October 7th, 2008
Lecture 10
Lecturer: Scott Aaronson
Science: We seek out ignorance and try to demolish it.
Last Time: Grovers algorithm and its optimality
D(f ) = deterministic query complexity of Boolean function f : cf
6.896 Quantum Complexity Theory
October 2, 2008
Lecture 9
Lecturer: Scott Aaronson
In this class we discuss Grovers search algorithm as well as the BBBV proof that it is optimal.
1
1.1
Grovers Algorithm
Setup
Given N items cfw_x1 , x2 , ., xN we wish to
6.896 Quantum Complexity Theory
Sep. 30, 2008
Lecture 8
Lecturer: Scott Aaronson
1
Hidden Subgroup Problem
Last time we talked about Shors factoring algorithm without going through all the details. Before
we continue, rst let us say something about the qu
6.896 Quantum Complexity Theory
September 25, 2008
Lecture 7
Lecturer: Scott Aaronson
1
Short review and plan for this lecture
In previous lectures we started building up some intuition into the way quantum algorithms work,
and we have seen examples (Bern
6.896 Quantum Complexity Theory
Sept 23, 2008
Lecture 6
Lecturer: Scott Aaronson
Last Time:
Quantum Error-Correction
Quantum Query Model
Deutsch-Jozsa Algorithm (Computes x y in one query.)
To day:
Bernstein-Vazirini Algorithm
Simons Algorithm
Shors
6.896 Quantum Complexity Theory
September 18, 2008
Lecture 5
Lecturer: Scott Aaronson
Last time we looked at whats known about quantum computation as it relates to classical
complexity classes. Today we talk somewhat more concretely about the prospects fo
6.896 Quantum Complexity Theory
September 16, 2008
Lecture 4
Lecturer: Scott Aaronson
1
1.1
Review of the last lecture
B QP
B QP is a class of languages L (0, 1) , decidable with bounded error probability ( say 1/3 ) by a
uniform family of polynomial-size
6.896 Quantum Complexity Theory
September 9, 2008
Lecture 2
Lecturer: Scott Aaronson
Quick Recap
The central object of study in our class is BQP, which stands for Bounded error, Quantum,
Polynomial time. Informally this complexity class represents problem
Sept 4, 2008
6.896 Quantum Complexity Theory
Lecture 1
Lecturer: Scott Aaronson
1
1.1
Introduction
Course structure
Prof. Aaronson wants people to participate and show up. There will also be projects which
could contain some original research, not necessa
6.845 Problem Set 4: Quantum Lower Bounds and More
1. Let f be the log2 N -level AND/OR tree. Show that Q (f ) =
reduce a PARITY problem of size N to f .]
N . [Hint: Show that you can
2. Suppose f : cfw_0, 1N cfw_0, 1 is a symmetric Boolean function: tha
6.845 Problem Set 3: Quantum Algorithms and Lower Bounds
n
1. In the Bernstein-Vazirani problem, we are given oracle access to a Boolean function f : cfw_0, 1 cfw_0, 1,
n
and promised that there exists a string s cfw_0, 1 such that f (x) = s x (mod 2) for
6.845 Problem Set 2: Basic Training for the BQP Army
Do any
7 of the 10 problemsthe remaining 3 are extra credit.
1. Distinguishing two quantum states.
(a) Show that there exists a measurement that, given as input either | = a|0 + b|1 or | =
a|0 b|1, for
6.845 Problem Set 1: Quantum Basics
1. Stochastic and unitary matrices.
(a) Show that a matrix A Rnn maps every nonnegative vector v Rn 0 to a nonnegative vector
Av with the same L1 -norm, if and only if A is stochastic (that is, A is a nonnegative matrix