Today we will look at Dicke state generation. The purpose is to create eigenstates corresponding to the Hamiltonian as quantum state generation for combinatorial optimization problems such as QAOA, where an XY mixer is used to search for a solution. The search using quantum entanglement can considerably increase the rate of correct solutions since only a subspace search is required.
There are many proposals for Dicke state generation, but in this article, we will review a technique for creating a Dicke state from classical bits.
Deterministic Preparation of Dicke States
Andreas Bärtschi, Stephan Eidenbenz
https://arxiv.org/abs/1904.07358
Best title.
Dicke States
This one can be expressed by the following equation.
It is an entangled state where there are n qubits and k of them are 1 and the remaining n-k are 0.
In the paper, it is made step by step using an operation called Split & Cyclic Shift unitary.
The method we will use here is to convert a classical bitstring to a Dicke state, but by using SCS, the computation can be done successfully and recursively.
SCS_{n,k} corresponds to the operation of shifting the entire sequence by moving the first 0 of the qubits to the last.
Since the number of bits is 3 and the number of ones is 1, SCS_{3,1} multiplies the original sequence by a factor of \sqrt{k/n}, taking the first zero to the end and shifting the whole sequence forward to the quantum state \sqrt{(n-k)/n} coefficients. By doing this recursively, we end up with a Dicke state!
image:https://arxiv.org/pdf/1904.07358.pdf
I'm going to make a little test.
Next, let's shift it by one: if there is no 1, it will still the same.
The result is that the Dicke state can be created.
The actual circuit is composed of CX and CCRY, and the angles are also specified.
image:https://arxiv.org/pdf/1904.07358.pdf
It specifies how to generate a Dicke state which is quite useful. I'd love to use this next time we do a QAOA! That's all!