Skip to content

Update lec_19_quantum.md #184

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Open
BenjiKan wants to merge 1 commit into
base: master
Choose a base branch
from
Open

Update lec_19_quantum.md #184

Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
6 changes: 3 additions & 3 deletions lec_19_quantum.md
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -187,9 +187,9 @@ relatively short resource that contains essentially everything we discuss here.
See also this [blog post of Aaronson](http://www.scottaaronson.com/blog/?p=208) for a high level explanation of Shor's algorithm which ends with links to several more detailed expositions.
See also [this lecture](http://www.scottaaronson.com/democritus/lec14.html) of Aaronson for a great discussion of the feasibility of quantum computing (Aaronson's [course lecture notes](http://www.scottaaronson.com/democritus/default.html) and the [book](http://www.amazon.com/Quantum-Computing-since-Democritus-Aaronson/dp/0521199565) that they spawned are fantastic reads as well).

__States:__ We will consider a simple quantum system that includes $n$ objects (e.g., electrons/photons/transistors/etc..) each of which can be in either an "on" or "off" state - i.e., each of them can encode a single _bit_ of information, but to emphasize the "quantumness" we will call it a _qubit_.
A _probability distribution_ over such a system can be described as a $2^n$ dimensional vector $v$ with non-negative entries summing up to $1$, where for every $x\in\{0,1\}^n$, $v_x$ denotes the probability that the system is in state $x$.
As we mentioned, quantum mechanics allows negative (in fact even complex) probabilities and so a _quantum state_ of the system can be described as a $2^n$ dimensional vector $v$ such that $\|v\|^2 = \sum_x |v_x|^2 = 1$.
__States:__ We will consider a simple quantum system that includes $n$ objects (e.g., electrons/photons/transistors/etc.) each of which can be in either an "on" or "off" state - i.e., each of them can encode a single _bit_ of information, but to emphasize the "quantumness" we will call it a _qubit_.
A _probability distribution_ over a classical system can be described as a $2^n$ dimensional vector $v$ with non-negative entries summing up to $1$, where for every $x\in\{0,1\}^n$, $v_x$ denotes the probability that the system is in state $x$.
As we mentioned, quantum mechanics allows negative (in fact even complex) probability amplitudes and so a _quantum state_ of the system can be described as a $2^n$ dimensional unit norm vector $v$ of amplitudes such that $\|v\|^2 = \sum_x |v_x|^2 = 1$.

__Measurement:__ Suppose that we were in the classical probabilistic setting, and that the $n$ bits are simply random coins.
Thus we can describe the _state_ of the system by the $2^n$-dimensional vector $v$ such that $v_x=2^{-n}$ for all $x$.
Expand Down