### 2020-07-06 › math

Last Friday I finished by first research experience for undergraduates (REU)! I learned about quantum transfer methods on graphs and quantum computation. The final draft for the project can be found here (if that link is gone, please email me for a copy). I had the great pleasure to work with Whitney Drazen (our advisor) and Noble Mushtak on this project! Read more for a bit of a rushed summary.

I'm a little burned out right now, so I might write about this more later. I'll give a brief summary here, so please forgive any errors.

In classical computer/information science, data can be represented using *bits*,
which are either $0$ or $1$.
We can consider a quantum analogue to this binary system, called a *quantum bit*, or *qubit*.
Like a classical bit, a qubit can have the states $0$ or $1$, usually denoted by the ket notations
$| 0 \rangle$ and $| 1 \rangle$.
Unlike a classical bit, a qubit $q$ can exist in a *superposition* of $| 0 \rangle$ and $| 1 \rangle$,
that is, a general state $| \psi \rangle$ of $q$ is linear combination
$\alpha | 0 \rangle + \beta | 1 \rangle$, where are $\alpha$ and $\beta$
are complex numbers such that $|\alpha|^2 + |\beta|^2 = 1$.
This gives that $| \psi \rangle$ is a unit vector is a two-dimensional $\mathbb{C}$-vector space $V$,
called the state space of $q$.
Another caveat: while the state of a qubit is
a superposition of $| 0 \rangle$ and $| 1 \rangle$, when we "observe" it, the state
collapses to either $| 0 \rangle$ and $| 1 \rangle$, and the state is lost
(the probability of collapsing to $| 0 \rangle$ is $|\alpha|^2$ and the probability
of collapsing to $| 1 \rangle$ is $|\beta|^2$).

Now suppose we have $n$ qubits $q_1, \dots, q_n$, with state spaces $V_1, \dots, V_n$. A naive approach to an $n$-qubit system might be to take the product of the $V_i$. But due to quantum entanglement, this is not the right notion. Instead, we take the tensor product $V = V_1 \otimes \cdots \otimes V_n$. Then entanglement is expressed as the fact that an arbitrary element of $V$ is not an elementary tensor $u \otimes v$, but instead a linear combination of the basis vectors. These basis vectors ($2^n$ of them) are precisely

$$ | b_1 \cdots b_n \rangle := | b_1 \rangle \otimes \cdots \otimes | b_n \rangle, $$

where $b_i \in \{0, 1\}$ (this is a bit of an abuse of notation).

Now one might like to ask the question: *can we copy a qubit's state?*
In the classical setting, this is as easy as inspecting the state of the bit and copying the value.
Well, not so in the quantum case.
This is due to the *no-cloning theorem*, which gives that there does not exist
an operator which can clone an *arbitrary* quantum state.

What's the next thing we can try?
There is a concept called *perfect state transfer*.
We might want to consider the problem of moving a qubit's state in some kind of network
(usually something which is called a *quantum spin network*).
Suppose this network has $n$ qubits.
We can model the network as an undirected graph $G$ with $n$ vertices.
The state of $G$ at a given time $t$ can be expressed as a vector $| \Psi(t) \rangle$ with $n$ components,
one for each vertex in $G$.
The axioms of quantum mechanics give that the state $| \Psi(t) \rangle$ of $G$
evolves over time according to Schrödinger's equation,

$$i\hbar\frac{d}{dt} | \Psi(t) \rangle = \mathcal{H} | \Psi(t) \rangle,$$

where $\hbar$ is the reduced Planck constant (in mathematical fashion, we can take $\hbar = 1$), and $\mathcal{H}$ is the Hamiltonian of the system. The Hamiltonian can be expressed in different ways depending on the context, but it involves the adjacency matrix $A$ of $G$. Solving Schrödinger's equation gives

$$ | \Psi(t) \rangle = e^{-it\mathcal{H}/\hbar} | \Psi(0) \rangle.$$

Now we are interested in the case where one qubit is "excited" and the others are $| 0 \rangle$. Through some math involving the Hamiltonian, we get that the evolution of this system is given by

$$ U(t) := e^{-itA}.$$
This operator is called the *quantum walk operator*.

Now say we have two qubits $q_i$ and $q_j$ in $G$. We say that perfect state transfer occurs at time $t$ from $q_i$ to $q_j$ if $| U(t)_{ij} | = 1$, where $U(t)_{ij}$ is the $q_i,q_j$-entry of $U(t)$ (we're indexing the vertices by the qubits). This represents the idea that at time $t$, the quantum state at $q_i$ was transfered to $q_j$ with probability $1$.

We can use various methods from spectral graph theory and algebraic graph theory to find out properties of the graph $G$ via the eigenvalues and eigenvectors of its adjacency matrix $A$, and more.

Our project examined conditions for a more general version of perfect state transfer, where instead of
transfering between two qubits, we consider a transfer to a superposition over a subset of ther vertex set.
This is called *fractional revival*, and comes up in entanglement generation (although I know little about this).

We found examples of graphs which exhibit a certain property called *fractional cospectrality*, which
can be used sometimes to rule out fractional revival.

We might also try to figure out which graphs have *pretty good fractional revival*, which is an $\epsilon$-close
version of fractional revival which is a bit less strict.

Okay I think that's enough for right now, I might talk about this later.

Thanks for reading! :)