Contents:
In this minicourse, we will introduce the essentials of the fascinating field of quantum computation and its classical simulability. The scope of the course will be self-contained.
We will start from a general overview of the basics of quantum information, but will progress towards more advanced topics requiring a higher level of mathematical abstraction. In particular, we intend to address current research topics, including: approaches to the demonstration of quantum computational supremacy; useful and reliable quantum simulators; verification of quantum computations; connections with machine learning, etc.
The course is aimed at graduate students or advanced undergrads of all areas of physics, computer science, and mathematics. Basic knowledge of quantum mechanics is required and will be assumed. David Mermin Cambridge University Press , as a preparation for the course.
Minicourse program: PDF updated on October 9, The registration will be on October 15 at the institute at am. List of Participants: Updated on October The easiest way to reach us is by subway or bus, please find instructions here. Ground transportation from Congonhas Airport to the Universe Flat. Ground transportation from The Universe Flat to the institute. Home About us. To put it another way, the strings of ones and zeros form the basis for a vector space where our machine state lives. We can write down the state of a QC by writing out a sum like so:.
Where X , Y , etc are strings of ones and zeros, and a , b , etc are the amplitudes for the respective components X, Y, etc. So if we apply Mix twice to any qubit in any state we get back to where we started. In other words, Mix is it's own inverse.
This book aims to provide a pedagogical introduction to the subjects of quantum information and quantum computation. Topics include non-locality of quantum. quantum information theory and quantum cryptography. Our intent is not to quantum computing course over the past few years. Phillip Kaye.
If we have two qubits a and b initialized to zero and we perform a sequence of quantum operations on a and then do the same to b , we would expect to wind up with a and b having the same value, and we do. In this example, a and b are completely independent. If we measure one it should have no effect on the other. If the operations were more complicated than a simple Not ;Mix , we might be tempted to perform them only once on a and then copy the value from a to b.
OK, we can't really copy because it's not a reversible operation , but we can initialize b to zero and CNot b,a which accomplishes the same goal. The spectrum of a and b look correct. And indeed if we were to measure just a or b we would get the same result as above. The difference lies in what happens when we measure both a and b. Keep in mind that the outcome of a measurement is random, so if you're repeating this experiment, your mileage may vary.
By measuring a , we collapsed the superposition of b. This is because a and b were entangled in what physicists call an EPR pair, after Einstein, Podolsky, and Rosen, all of whom used this in an attempt to show that quantum mechanics was an incomplete theory.
John Bell, however, later demonstrated entanglement in real particles by experimental refutation of the Bell Inequality which formalized the EPR thought experiment. What happens when you try to copy one quantum variable onto another? You wind up with entanglement rather than a real copy. Suppose we are given a function that takes a one bit argument and returns one bit. And to keep things on the up and up, let's require that this be a pseudo-classical function.
If we hand it a classical bit 0 or 1 as an argument, it will return a classical bit. The first two are constant, meaning they output the same value regardless of input. The second two are balanced meaning the output is 0 half the time and 1 half the time. Classically there's no way to determine if f is constant or balanced without evaluating the function twice. Deutches problem asks us to determine whether f is constant or balanced by evaluating f only once.
Here's how it works. First, we have to construct a pseudo-classic operator in qcl that evaluates f x. To do this we'll define a qufunct with arguments for input and output. For example:. You can comment out either the CNot or Not lines to get one of the other three possible functions. Now let's reset the quantum memory, and run Deutches algorithm.
It works by first putting the in and out bits into a superposition of four basis states. At line 9 we run the quantum function F which XORs the value of f in into the out qubit. The function F is pseudo-classical, meaning it swaps basis vectors around without changing any amplitudes. By applying the F function to a superposition state, we have effectively applied F to all four basis states in one fell swoop.
This is what's called "quantum parallelism" and it's a key element of QC. Our simulator will, of course, have to apply F to each of the basis states in turn, but a real QC would apply F to the combined state as a single operation. If we had not run F at line 9, this would have brought us back to the state we had at line 4 this is because Mix is its own inverse. But because we swapped amplitudes with F , undoing the superposition puts us into a different state than where we were at line 9. Specifically the in qubit is now set to "1" rather than "0".
It's also instructive to note that the amplitude of -1 in line 15 is unimportant. A quantum state is a vector whose overall length is of no interest to us as long as it's not zero. Only the direction of the vector, the ratios between the component amplitudes, is important. By keeping quantum states as unit vectors, the transformations are all unitary. The book is certainly not only one for beginners, as the contributors have done a very good job at getting to the heart of their topics. See the review.
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