Reading for this week:

Wednesday, 4/17/24 (week 3, lesson 2):

1. Read 3.8 (all subsections)
2. Read lecture slides on the Qubit, pp. 15-23
3. HW 1 due F! Upload pdf to Canvas.
4. WA 1: Addition of binary numbers, F, beginning of class. Upload pdf to Canvas when work is complete. Closed book.

Monday, 4/15/24 (week 3, lesson 1):

1. Read 1.6, 2.8 (all subsections), 3.5, 3.7 (but not 3.7.1)
2. Read lecture slides on Quantum superposition, pp. 28-32
3. Read lecture slides on the Qubit, pp. 1-14

Review

Last time we talked about the wave function for a hydrogen atom in the ground state:

note that there are other states, but for our example we just say the others are negligible. The states $|100⟩$ and $|200⟩$ are the basis states of the electron, and the state of the electron is ${\mathrm{\Psi }}_{electron}$.

The Qubit

The Single-Qubit State

Classical computers store information in bits. Each bit can be either in state 0 or state 1. Quantum computers store information in "quantum bits" (for short qubits). The general single-qubit state:

where $a,b\in \mathbb{C}$ and $|a{|}^2+|b{|}^2=1$ (because the probability has to be a total of 1). We can call $|0⟩$ and $|1⟩$ really anything, maybe $|\updownarrow ⟩$ or something similar. We can always relabel things via Linear Algebra.

Here, notice that anything can theoretically be a qubit, as long as it has just two distinct states (on observation).

Like any wave function, the probability to be in any of the possible states needs to be 1, so then we get the $|a{|}^2+|b{|}^2=1$ as shown below:

Notice the distributive manner of the braket notation here. Also really $⟨X|Y⟩=\chi (X=Y)$, because it's essentially doing an inner product of orthogonal basis vectors. See Chapter 6 - Inner Product Spaces#6.B Orthonormal Bases for more information on this.

Notice if we have $n$ states then:

a qubit with $n$ states is a qutrit. These come into play when extra states just happen to come up (which definitely can happen). An electron can have 2 states, but bigger particles like electrons in Silicon could have way more.

Information is stored in the coefficients of each bit/quantum bit, namely $a,b$. Notice for classical bits, we'd have:

so using $|a{|}^2+|b{|}^2=1$ then if we know one, we know the other as a result. So we have:

• one piece of information (choose $a$, then $b$ is determined from it)
• 2 options/states ($a,b$)
• Here $a$ is 0 or 1, and that's it. Same with $b$.

But in the quantum version:

So then:

• $|a|,|b|,{\varphi }_a,{\varphi }_b$ are all values, where $|a|,|b|\in \mathbb{R}$ is from 0 to 1, and ${\varphi }_a,{\varphi }_b\in [0,2\pi ]$. As such, then
• As such, then we have two pieces of information (the complex information from our $a$ and $b$)
• There's a continuum of possible values for our information (anything from $\mathbb{R}$ rather than from $\{0,1\}$)
• Here, the $|a{|}^2$ is the probability of finding our qubit in state $|0⟩$, and for $|1⟩$ it's $|b{|}^2$.

Writing Vectors in ${\mathbb{C}}^n$

We have other ways of writing a single-qubit state. As a vector in $\mathbb{H}$ or our "Hilbert Space" (the space of wavefunctions), the basis axes in Hilbert space (similar to $\hat{i}$ and $\hat{j}$ from ${\mathbb{R}}^2$) the complex numbers $a,b$ are the components of each basis state:

We can, since $a,b$ are complex numbers, use the angular versions of them:

But notice the geometric considerations! If $|a{|}^2+|b{|}^2=1$ then this represents a circle of radius 1 in ${\mathbb{C}}^2$. Since $|a|,|b|\ge 0$ then:

We just rename the respective angle $\varphi$ to $\theta /2$. We'll see why later on.

Visualization of a Single-Qubit State

Every measurable quantity in quantum mechanics has the form:

We can prove that a difference in the phase of $\mathrm{\Psi }$ will not change the outcome:

So, WLOG, we can rewrite our single-qubit state as:

We'll get to the bloc sphere next time!