Depth-Efficient Quantum Circuit Synthesis for Deterministic Dicke State Preparation

TL;DR

This paper introduces depth-efficient, deterministic quantum circuits for preparing Dicke states, optimizing for different qubit connectivity constraints and establishing near-optimal depth bounds.

Contribution

It presents new quantum circuit designs for Dicke state preparation that are optimized for all-to-all and grid connectivity, improving upon previous bounds and proving near-optimality.

Findings

Circuit depth improved to $O( ext{log}(k) ext{log}(n/k)+k)$ for all-to-all connectivity.

New depth bounds for grid connectivity surpass previous results, with optimal-depth circuits for certain parameters.

Established lower bounds of $ ext{Omega}( ext{log}(n))$ and $ ext{Omega}(n_2)$, confirming near-optimality of the proposed circuits.

Abstract

The $n$-qubit $k$-weight Dicke states $|D^n_k\rangle$, defined as the uniform superposition of all computational basis states with exactly $k$ qubits in state $|1\rangle$, form a basis of the symmetric subspace and represent an important class of entangled quantum states with broad applications in quantum computing. We propose deterministic quantum circuits for Dicke state preparation under two commonly seen qubit connectivity constraints: 1. All-to-all qubit connectivity: our circuit has depth $O(\log(k)\log(n/k)+k)$, which improves the previous best bound of $O(k\log(n/k))$. 2. Grid qubit connectivity ($(n_1\times n_2)$-grid, $n_1\le n_2$): (a) For $k\ge n_2/n_1$, we design a circuit with depth $O(k\log(n/k)+n_2)$, surpassing the prior $O(\sqrt{nk})$ bound. (b) For $k< n_2/n_1$, we design an optimal-depth circuit with depth $O(n_2)$. Furthermore, we establish the depth lower bounds of $\Omega(\log(n))$ for all-to-all qubit connectivity and $\Omega(n_2)$ for $(n_1\times n_2)$-grid connectivity constraints, demonstrating the near-optimality of our constructions.