Are quantum states like the W, Bell, GHZ, and Dicke state actually used in quantum computing research?

Question

I recently started studying quantum computing and learned about several well-known quantum states such as the W state, GHZ state, and Dicke state. I noticed that there are also some questions here on Stack Exchange regarding quantum state preparation for these states. However, when I looked further, I realized that there are fewer types of well-known quantum states than I expected. Why might that be the case? My questions are as followings:

1. Are quantum states like the W, Bell, GHZ, and Dicke state actually used in quantum computing research?
2. Are there any other 'useful' quantum states?
3. Is it uncommon to prepare an arbitrary quantum state? For instance, consider the case of preparing the state $(\alpha \vert 00 \rangle + \beta \vert 11\rangle)$ for arbitrary $\alpha$ and $\beta$. For example, would preparing a specific state, such as $\sqrt{\frac{3}{10}} \vert 00 \rangle + \sqrt{\frac{7}{10}} \vert 11 \rangle$, be considered too contrived?
4. I found that in Hamiltonian simulation, the input consists of $\hat{H}$, $\epsilon$, and the initial state $\vert \psi_0 \rangle$. How is $\vert \psi_0 \rangle$ determined?

Answer 1

TL;DR: Yes, these states have wide-ranging applications in quantum computing research. See below for a non-exhaustive list of examples.

Bell, GHZ and cat states

Bell state $(|00\rangle+|11\rangle)/\sqrt{2}$, aka EPR pair, is the primary "currency" used for quantifying the amount of entanglement, for example in entanglement distillation schemes such as this one. GHZ state is used in quantum game theory where it is an entangled resource state that enables a quanutm strategy that wins with probability $100\%$ in a game where the best classical strategy wins only $75\%$ of the time.

Both EPR pair and the GHZ state $(|000\rangle+|111\rangle)/\sqrt{2}$ are special cases of the so-called "cat states" $(|00\ldots 0\rangle+|11\ldots 1\rangle)/\sqrt{2}$ which are for example useful for fault-tolerant syndrome measurement.

W and Dicke states

The W state $(|001\rangle+|010\rangle+|100\rangle)/\sqrt{3}$ is a special case of Dicke states \begin{align} |D_k^n\rangle=\frac{1}{\sqrt{{n \choose k}}}\sum_{\substack{x\in\{0,1\}^n\\|x|=k}}|x\rangle\tag1 \end{align} where $|x|$ denotes the Hamming weight of bit string $x$. Dicke states have applications in quantum game theory, quantum error correction and quantum optimization (see below).

Amplitude embedding

Preparation of a state $\sum_{k} w_k|k\rangle$ with given amplitudes $w_k$ arises in quantum machine learning where it is called amplitude embedding. Both amplitude embedding and Dicke states are used in Decoded Quantum Interferometry (disclosure: I'm a co-author).

Other states

A large class of interesting quantum states often encountered in quantum computing research is the set of Absolutely Maximally Entangled (AME) states which have connections to quantum games, quantum error correction, and AdS/CFT correspondence. See also Table of AME states.

Answer 2

There are indeed different paradigms when studying quantum states and although the states you mention have probably not been studied with a certain application in mind, there are various occasions where those very structured states come in handy. I cannot give you an exhaustive list, because this is a very broad question. I hope these references help you dig deeper into the topics you are interested in.

Some examples in answer to point 1:

• $W$ and, more generally, Dicke states are the ground state of a Hamiltonian that only consists of hopping terms of the form $\sigma^+_i \sigma^-_j +H.c.$. Although there are some more steps in the way, this can be interesting to study weakly interacting fermions or bosons.
• Let me refer you to the Introduction section of this paper for more applications of Dicke states.
• GHZ and Bell states are often used as a resource of entanglement, their applications are very versatile ranging from process tomography over error mitigation to quantum communication and cryptography. A pedagogic example is probably the Quantum teleportation protocol, where Alice can teleport a qubit to Bob only using classical communication if they share a Bell pair in the first place.

To point 2., I am also disclaiming completeness. Certainly a large class of "useful" or rather interesting states are matrix product states (MPS). These are states of bounded entanglement and can therefore be efficiently handled on a classical computer, or sometimes even with pen and paper. Especially in the discipline of quantum phases of matter, MPS play a prominent role and there is some effort to prepare those states as initial states for more complicated quantum algorithms.

Your third question sounds a lot like amplitude encoding, a method that is used to encode classical data as amplitudes of computational basis states. These kinds of states can be useful for problems with large data sets as in quantum machine learning. There is some effort to implement simple function values, but in general and particularly if the amplitudes are sufficiently random, even the approximate preparation of such a state could be hard.

Point 4. is finally easy. The initial state is arbitrary, but fixed. If you want to carry out a Hamiltonian simulation experiment, you typically have an initial state in mind and you have to find a way to prepare the desired initial state before applying an algorithm that evolves the state in time. Depending on the complexity of said initial state, this can be very easy or very hard, on near-term hardware sometimes even prohibitive.

Answer 3

Regarding the specific question of whether states of the form $\alpha \vert 00 \rangle + \beta \vert 11 \rangle$ for arbitrary $\alpha,\beta$ are useful: they are for instance used to maximize Bell inequality violation under imperfect-detector conditions, as discussed in Eq. (32) of this paper; see, e.g., Eq. (2) in this preprint for a non-paywalled description.