Polylogarithmic-Weight Dicke States in QAC$^0$ and Arbitrary Symmetric States in QAC$^0_f$

Abstract

An $n$-qubit Dicke state of weight $k$, is the uniform superposition over all $n$-bit strings of Hamming weight $k$. Dicke states are central to quantum algorithms exhibiting speedups, such as Decoded Quantum Interferometry (Jordan et al., \emph{Nature}, 2025). In the NISQ era, quantum hardware is constrained by both depth and locality, motivating the question of which global operations suffice to prepare such states. QAC$^0$, the quantum analogue of AC$^0$, minimally extends local $O(1)$-depth quantum circuits by allowing arbitrary-width Toffoli (reversible AND) gates. We show that Dicke states of $\mathrm{polylog}(n)$ weight can be prepared in QAC$^0$. This gives the first QAC$^0$ construction of any super-constant-weight $n$-qubit Dicke state, since previous constructions relied on the much more powerful FANOUT$_n$ gate. In general, we show that any weight-$k$ Dicke state can be constructed using FANOUT$_{\min(k,n-k)}$ gates. Combined with recent hardness results, this yields a tight characterization: for $k \leq n/2$, a $n$-qubit weight-$k$ Dicke state can be prepared in QAC$^0$ if and only if FANOUT$_k$ $\in$ QAC$^0$. We develop a limited-fanout state-synthesis toolkit for QAC$^0$ that yields further constant-depth, poly$(n)$-ancilla constructions: 1. Every $n$-qubit symmetric state supported on Hamming weight $\leq k$ can be prepared using FANOUT$_k$ gates. 2. Every $O(\log n)$-qubit state can be prepared using quantum random-access memory (QRAM$_n$), which refers to a coherent indexing gate. QRAM$_n$ is a potentially weaker resource than FANOUT$_n$ and can be implemented in QAC$^0_f$.