Efficient preparation of Dicke states

We present an algorithm utilizing mid-circuit measurement and feedback that prepares Dicke states with polylogarithmically many ancillas and polylogarithmic depth. Our algorithm uses only global mid-circuit projective measurements and adaptively-chosen global rotations. This improves over prior work that was only efficient for Dicke states of low weight, or was not efficient in both depth and width. Our algorithm can also naturally be implemented in a cavity QED context using polylogarithmic time, zero ancillas, and atom-photon coupling scaling with the square root of the system size.

💡 Research Summary

The paper introduces a remarkably simple yet powerful algorithm for preparing Dicke states—symmetric superpositions of n spin‑½ particles with a fixed Hamming weight—using only global operations and a collective Jz measurement. The authors assume three primitive capabilities: (1) the ability to initialize any qubit in the |0⟩ state on demand, (2) the ability to apply a uniform rotation about the y‑axis, U(θ)=e^{-iθJ_y}, to all qubits simultaneously, and (3) the ability to perform a mid‑circuit measurement of the total z‑magnetization Jz, which is equivalent to measuring the Hamming weight of the computational basis string.

Starting from the all‑up state |j,j⟩ (=|0⟩^{⊗n}), the algorithm proceeds iteratively. In each iteration the current value of m (the eigenvalue of Jz) determines a rotation angle θ_m. After applying U(θ_m) the collective Jz measurement is performed. If the outcome equals the desired target m_t (the Dicke state |j,m_t⟩ is then prepared), the loop terminates. Otherwise the new measurement outcome m′ is fed back to select the next angle θ_{m′}. To keep the process efficient, the authors add a simple reset rule: whenever |m| exceeds √j the whole register is re‑initialized to |0⟩^{⊗n} (i.e., m=j).

The crucial technical contribution is the choice of the rotation angles. By representing Dicke states through their Husimi‑Q distribution on a collective Bloch sphere, each state appears as a narrow ring at latitude r_m=√{j(j+1)−m²}. A global rotation tilts the ring by an angle θ, and the measurement projects it onto a Dicke ring. The authors argue that the optimal θ is the one that makes the two rings tangent, maximizing their overlap. Analytically this yields
θ_{m_t,m}=arcsin