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My question is why do we need entanglement at all? If we can hold $n$ bits in superposition, wouldn't we still be able to surpass a supercomputer once we had say over 100-bits?

Is it because we need to represent the two basis vectors concurrently and need to have them influence each other on the $0$, $1$ and every state in between and entanglement provides the means?

All the stuff I find on the web references amounts to something like

Quantum entanglement is a strange phenomenon of the universe in which two subatomic particles are linked together such that when one particle is changed, the other particle is also changed, even if the two particles are separated by large distances. This inexplicable phenomena came to be known as “Quantum entanglement”, a term originally coined by Erwin Schrödinger, another early adopter and theorist of the quantum world.

That still does not explain why we need entanglement at all (I am not a Physicist). My intuition is telling me that there is still a lot here we do not understand and maybe that makes an intuitive explanation difficult, but I could be wrong. Maybe the answer is just that that is how nature is doing it and you just accept it.

I have researched the following articles, but none of them answers my question

Is there any reference that better explains this?

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Almost every quantum state is entangled$^1$. It is therefore unsurprising that quantum algorithms bump into them every so often. In fact, an attempt to navigate the space of all quantum states without going through entangled states is so constraining as to rule out nearly all interesting quantum algorithms.

One way to understand this is to realize that entanglement is the quantum analogue of classical correlations. The latter are so fundamental to classical computing that it is hard to even imagine what computing would look like without them. For example, consider what a Turing machine could do if it were not allowed to produce correlations between its internal state and the tape?

Thus, perhaps a better way to think about entanglement is not as something we need, but rather as something nearly unavoidable when quantum phenomena are at play. In particular, in order to exploit the exponential dimension of the Hilbert space of $n$ qubits we have to use entangled states.

We can make this quantitative. A single qubit has two independent real degrees of freedom that correspond to the two angles on the Bloch sphere. We can describe all unentangled states of $n$ qubits by specifying these two parameters per qubit for a total of $2n$ real parameters. On the other hand, an arbitrary pure possibly entangled state of $n$ qubits is described by $2^n-2$ real parameters.

Imagine you wish to store a state $|\psi\rangle$ of 72-qubit Bristlecone processor in the memory of a classical computer using single-precision floating point numbers$^2$. If $|\psi\rangle$ is not entangled, this can be accomplished with 576 bytes of memory. We can even write it down on a piece of paper. Moreover, every time we add a qubit, we need to add just 8 more bytes.

Now, if $|\psi\rangle$ is arbitrary and possibly entangled then we need about 16 billion terabytes of memory to store $|\psi\rangle$ in single-precision. That's more than 2 terabytes of RAM for every person alive today. Moreover, every time we add a qubit this number doubles.


$^1$ Technically: unentangled states form a zero measure subset of all states.
$^2$ One single-precision floating point number requires four bytes of storage.

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    $\begingroup$ So many intro materials on quantum computing spend so much time on the Bloch sphere and how a single qubit represents arbitrary superposition of states and maybe mention entanglement at the end, but entanglement is the only property that makes quantum computing worth pursuing $\endgroup$ May 30, 2023 at 17:21
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Quantum entanglement is used in many algorithms such as HHL, which solves a linear system of equations. HHL uses entanglement by entangling our data to an ancilla qubit so we measure that qubit repeatedly to get a certain result that depends on our data. When that ancilla gives the result we want, then, after unentangling the data to avoid errors, our circuit returns the solution we want. If you would like to look more into HHL, here is the original paper, and here is a paper that goes into more depth. Hope that helps!

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    $\begingroup$ Sorry, here they are in order: original paper Step-by-step paper $\endgroup$
    – Saksham
    May 30, 2023 at 16:03
  • $\begingroup$ What do you mean by "unentangling the data to avoid errors"? Even after the flag register indicates a success, there is still entanglement in the output register $|x\rangle$. $\endgroup$ May 30, 2023 at 17:19
  • $\begingroup$ @MarkSpinelli The original paper disentangles the phase qubits from the ancilla to avoid entanglement errors, but the other paper I attached says it is not significant. The output of the algorithm is meant to be a one state in the ancilla followed by n 0s followed by qubits storing the data for the solution x to the problem. Regardless, of whether there is entanglement or not though, my point was that we use entanglement with the ancilla. Sorry for any confusion! $\endgroup$
    – Saksham
    May 31, 2023 at 16:27

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