Quantum Class 2, Thu 2020-09-03
Table of contents
1 Handwritten notes, video transcripts, and other files
They are accessible from the Files tab on the top menu bar.
So far, they include scans of my handwritten notes during the lectures, and webex's attempt at a written transcript of the lecture videos. It's actually useful.
2 Lecture videos
Let's try RPI's mediasite: https://mediasite.mms.rpi.edu/Mediasite5/Channel/quantum
3 Asking questions in class
Please ask questions. I try to have the chat window open most of the time. If I ignore you, ok to unmute your mike to ask a question.
Constructive opinions and comments are also welcome.
4 Quantum computing in the news
https://www.telegraph.co.uk/technology/2020/09/02/britain-must-act-fast-prevent-brain-drain-quantum-computing/ - good summary of programs around the world.
https://venturebeat.com/2020/09/02/xanadu-photonics-quantum-cloud-platform/
5 Class 1 extras
5.1 Youtube videos to watch
These are the ones I tried unsuccessfully to show in class 1.
A beginner's guide to quantum computing | Shohini Ghose https://www.youtube.com/watch?v=QuR969uMICM 10:04
- Quantum Computing in Under 11 Minutes daytonellwanger
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https://www.youtube.com/watch?v=TAzZKAdX2Tw 10:56 more technical
5.3 Review
5.3.1 Classical computation
Bit.
Its 2 possible values are 0 or 1.
Byte.
Has 8 bits.
It has 256 possible values.
Bits are transformed with gates, like nand, nor, and, or, xor, not, ...
They generally destroy info, and are not invertible.
More complex circuits, like adders, are formed from a combo of these gates.
5.3.2 Quantum computation
Qbit, $q$.
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Its state is a linear combo of two basis states, $|0>$ and $|1>$:
$q = a|0> + b|1>$ ,
where $a$ and $b$ are complex numbers, and $ | a | ^2 + | b | ^2 = 1$.
IOW, its state is a superposition of those two basis states, with those weights.
It is wrong to think that $q$ is really in one of the two states, but you don't know which one. This is the hidden variable theory. It has been proved experimentally to be false.
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$q$ is really in both states simultaneously.
Alice laughed. "There's no use trying," she said: "one can't believe impossible things." "I daresay you haven't had much practice," said the Queen. "When I was your age, I always did it for half-an-hour a day. Why, sometimes I've believed as many as six impossible things before breakfast." - Through the Looking-Glass, and What Alice Found There (1871), by Lewis Carroll (Charles Lutwidge Dodgson).
You cannot observe its state, unless it is $|0>$ and $|1>$, in which case you observe $0$ or $1$. This is the classical case.
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Otherwise you observe it with a measurement operator that transforms it to either $|0>$ and $|1>$, with probabilities
$| a | ^2$ and $| b | ^2 = 1$, respectively.
$a$ and $b$ are complex.
That measurement changes $q$; it no longer has its old value.
$q$, that is, $q$ 's value, can be considered to be a vector of length one: $$\begin{pmatrix} a | 0> \\ b | 1> \end{pmatrix} $$ or simply $$\begin{pmatrix}a\\b\end{pmatrix}$$.
You operate on $q$ with a matrix multiplication: $q_2 = M q$.
This changes $q$; the old value is no longer available.
No cloning: You cannot copy a qbit, but can move it.
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The life cycle of a qbit:
Create a qbit with a classical value, 0 or 1.
Operate on it with matrices, which rotate it to have complex weights.
Transform it back to 0 or 1 and read it.
So far, not very powerful.
Now, let $q$ be a system with two qbits, i.e., a 2-vector of qbits.
6 Next class
Class 3 is on Tues Sept 8 because of Labor Day
7 Today
7.2 Entanglement
Crazy counterintuitive idea that's the basis for quantum speedup.
Classical metaphor for entanglement:
Start with a piece of paper.
Tear it into two halves.
Put each half into an envelope, seal them, and mix them up, so that you can't tell which half is in which envelope.
Address and mail one envelope to a friend in Australia, and the other to a friend in Greenland.
When the Australian opens his envelope, he knows what the Greenlander will find in his.
However that doesn't let the Australian send any info to the Greenlander, or vv.
This has been demonstrated with real qbits transported 1000 miles apart.
Entanglement means that if you measure one qbit then what you observe restricts what would be observed when you measure the other qbit.
However that does not let you communicate.
7.3 Two qbit operators
Now, let $q$ be a system with two qbits, i.e., a 2-vector of qbits.
$q$ is now a linear combo of 4 basis values, $ | 00>$, $ | 01>$, $ | 10>$, $ | 11>$.
$q = a_0 | 00> + a_1 | 01> + a_2 | 10> + a_3 | 11> $
where $a_i$ are complex and $ \sum | a_i | ^2 = 1$.
$q$ exists in all 4 states simultaneously.
If $q$ is a vector with n component qbits, then it exists in $2^n$ states simultaneously.
This is part of the reason that quantum computation is powerful.
A measurement operator applied to $q$ will rotate it to a basis {00, 01, 10, 11}, so that it will be observed in one of those four cases, with probabilities $ | a_i | ^2$.
You operate on $q$ by multiplying it by a 4x4 matrix operator.
The matrices are all invertible, and all leave $ | q | = 1$.
You set the initial value of $q$ by setting its two qbits each to 0 or 1.
How this is done depends on the particular hw.
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I.e., initially, $q_1 = \begin{pmatrix}a_1 | 0> \\b_1 | 1> \end{pmatrix}$ and $q_2 = \begin{pmatrix}a_2 | 0> \\b_2 | 1> \end{pmatrix}$, and so
$$q = \begin{pmatrix} q_1 \\ q_2 \end{pmatrix} = \begin{pmatrix} a_1 a_2 | 00 > \\ a_1 b_2 | 01 > \\ a_2 b_1 | 10 > \\ b_1 b_2 | 11 > \end{pmatrix}$$.
Here, the combined state is the tensor product of the individual qbits.
For $n$ qbits, the tensor product is a vector with $2^n$ elements, one element for each possible value of each qbit.
Each element of the tensor product has a complex weight.
You transform a state by multiplying it by a matrix.
The matrix is invertible.
The transformation doesn't destroy information.
When you measure a state, it collapses into one of the component states. (This may be inaccurate.)
You don't need to bring in consciousness etc. The collapse happens because the measurement causes the state to interact with the outside world.
The probability of collapsing into a particular state is the squared magnitude of its complex weight.
For some sets of weights, particularly after a transformation, the combined state cannot be separated into a tensor product of individual qbits. In this case, the individual qbits are entangled.
That is the next part of why quantum computation is powerful.
Entanglement means that if you measure one qbit then what you observe restricts what would be observed when you measure the other qbit.
However that does not let you communicate.
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From page 171 of
Quantum Computing for Computer Scientists 1st Edition
All quantum algorithms work with the following basic framework:
The system will start with the qubits in a particular classical state.
From there the system is put into a superposition of many states.
This is followed by acting on this superposition with several unitary operations.
And finally, a measurement of the qubits.
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Ways to look at measurement:
Converts qbit to classical bit.
Is an interaction with the external world.
Information is not lost, but leaks into the external world.
Is an operator represented by a matrix.
that is Hermitian, i.e., it's equal to its complex transpose.
For physical systems, some operators compute the system's momentum, position, or energy.
The matrix's eigenvalues are real.
The result of the operation is one of the eigenvalues.
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More from
Quantum Computing for Computer Scientists 1st Edition
Compare byte with qbyte.
State of byte is 8 bits.
State of qbyte is 256 complex weights.
They all get modified by each operation (aka matrix multiply).
That is the power of quantum computing.
The current limitations are that IBM does only a few qbits and that the operation is noisy.