Cs395t Notes: Quantum Complexity Theory

“The big secret of quantum mechanics is how simple it is once you take the physics out of it.” The course website is http://www.scottaaronson.com/qct2016/, and the syllabus is at http://www. scottaaronson.com/qct2016/syllabus-qct2016.pdf. We’ll be mostly following lecture notes found at http://www.scottaaronson.com/barbados-2016.pdf. This lecture’s goal is to acquaint the listener with the basic concepts and notation that we’ll use in the rest of the course; it’s not presented as review, but everything else in the course depends on it. For this material, there are many excellent references, some of which are listed in the syllabus. Quantum mechanics has a very underserved reputation for being very complicated. Mysterious, yes; counterintuitive, yes; but complicated is a bit much. All sorts of interesting consequences follow from a single change to the laws of probability, crucial to physics at the subatomic level, but thought to apply to everything in the universe. A probability of something happening is a real number p ∈ [0, 1]: it makes no sense to ask what a probability of −1/3 is, much less i/3. But quantum mechanics assigns a more general number, an amplitude α ∈ C, to an event. The thesis of quantum mechanics is that any isolated physical system’s state can be described by a vector of its amplitudes. In particular, systems in quantum mechanics have a dimension; intuitively, if there are N different things you can observe, the system is N-dimensional. The simplest quantum systems are two-dimensional, where there are two possinilities ) and 1. These systems have a special name: qubits. In general, we think of the state of a quantum system as a unit vector ψ ∈ CN of length 1. These vectors are denoted using a notation that Paul Dirac invented in the 1930s, the Dirac ket notation. The syntax looks a little jarring at first, but is convenient in a lot of ways. A ket is a vector |v〉: a qubit has two basis vectors |0〉, representing an outcome of 0 and |1〉, similarly an outcome of 1, so a general state is |v〉 = α|0〉+ β|1〉, representing a linear combination, or superposition, of the two options: α, β ∈ C are complex numbers, and