Quantum Class 5, Mon 2020-09-14
Table of contents
1 Today
We're doing chapters 4 and 5 of Quantum Computing for Computer Scientists.
I'm typing the important points into this blog.
2 Chap 4, pp 103ff
2.1 4.1 Quantum states, p 103
This restates more solidly what they stated more informally earlier.
2.1.1 Particle on line: 1st motivating example
Consider a particle on a line, with possible positions $x_0, x_1, ... x_{n-1}$.
We don't do the continuous limit here.
$|x_i>$ means the particle is at position $x_i$.
We describe the state of the system by using a complex vector $[c_0, c_1, ... c_{n-1}]^T$.
$|x_0> \longmapsto [1, 0, ...]^T$. $\longmapsto$ means maps to.
$|x_1> \longmapsto [0, 1, 0, ...]^T$.
$|x_{n-1}> \longmapsto [0, 0, ..., 1]^T$.
Posit All vectors in $C^n$ are legal states for the particle.
$|\psi>$ is a general state
$|\psi> = c_0 | x_0> + c_1 | x_1 + ...$
$|\psi> \longmapsto [c_0, c_1, ...]^T$.
This is a linear combo of the $|x_i>$.
That's superposition.
Observing it causes transition to one state, with prob eqn 4.5.
$p(x_i) = \frac{ | c_i | ^2}{\sum_j | c_j | ^2 } $
$| \psi> \leadsto | x_i >$ .
$\leadsto$ means the first quantum state transitions to the second.
2.1.2 Spin: 2nd motivating example
Spin is a property of most quantum particles.
You can measure its spin wrt the axis of your choice (subject to your equipment).
Wrt any given axis, the spin is up $|\uparrow>$ or down $\downarrow>$.
Particle is in a superposition of both spins.
$|\psi = c_0 | \uparrow> + c_1 | \downarrow>$.
Probability of observing in $|\uparrow>$
2.1.3 Measurement operator and Transition amplitude
Measurement operator will cause state to transition to one of a set of specific states (depending on the operator).
$|\psi> \leadsto | \psi' >$.
Application of inner product.
Probability of transitioning from $\psi$ to $\psi'$ when measured = $<\psi' | \psi>$.
2.2 4.2 Observables, p 115
Each observable thing, e.g, position, has a Hermitian operator.
The possible observable values are its eigenvalues.
In the particle example, P = diag[$x_0, x_1, ...$].
If we do two observations, the second is affected by what we see in the first.
Skip pages 118-125.
2.3 4.3 Measuring, p 126
Determine the value of an observable. If the observable is position, then a possible value might be 3.
Measuring causes the system to transition randomly to one eigenvector of the observable.
The probability is the transition amplitude.
(Not in book) One way to look at measuring is that the system is interacting with the outside world. Both are being affected, and the combo is reversible.
2.4 4.4 Dynamics, p 129
Evolution of a system is a unitary operator.
Reversible.
Schrodingers equation.
2.5 4.5 Assembling quantum systems, p 132
Tensoring the state spaces of the parts.
Entanglement again.
Example: create two paired spin particles from something with no spin. If you observe one particle to be $|\uparrow>$, then when you observe the other, it must be $|\downarrow>$. This is because spin is conserved.
Example 4.5.1, p 133
Exercise 4.5.3, p 135.
3 Chap 5, Architecture, pp 138ff
We've seen much of this earlier.
This puts classical gates and bits into quantum notation. This may help you understand quantum gates better.
3.1 5.1 Bits and qubits
Spelling: some say qbit others qubit.
3.2 5.2 Classical gates. p 144
Sample notation: AND $\!|01\!\!>\,\,=\,\,|0\!\!>$
There is a gate that will clone its input, i.e., make it fan out to two outputs.
3.3 5.3 Reversible gates, p 151
Erasing info dissipates energy.
3.4 5.4 Quantum gates, p 158
Block sphere. I don't personally find it that interesting. You may differ.
Can add one more input to make any gate into a controlled gate. E.g. ${}^c NOT$.
The classical cloning gate doesn't clone quantum states.
No-cloning.
Transporting or swapping is fine. (Star Trek got it right.)