Super-Constant Weight Dicke States in Constant Depth Without Fanout

• URL: http://arxiv.org/abs/2604.15298v1
• Date: Thu, 16 Apr 2026 17:57:08 GMT
• Title: Super-Constant Weight Dicke States in Constant Depth Without Fanout
• Abstract summary: An $n$-qubit Dicke state of weight $k$ is the uniform superposition over all $n$-bit strings of Hamming weight $k$.<n>We give explicit constant-depth circuits that prepare $n$-qubit Dicke states for all $k textpolylog(n)$.<n>We show that any weight-$k$ Dicke state can be constructed with access to FANOUT$_min(k,n-k)$, rather than FANOUT$_n$.
• Score: 1.4018975578160688
• License: http://creativecommons.org/licenses/by/4.0/
• 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 an entanglement resource with important practical applications in the NISQ era and, for instance, play a central role in Decoded Quantum Interferometry (DQI). Furthermore, any symmetric state can be expressed as a superposition of Dicke states. First, we give explicit constant-depth circuits that prepare $n$-qubit Dicke states for all $k \leq \text{polylog}(n)$, using only multi-qubit Toffoli gates and single-qubit unitaries. This gives the first $\text{QAC}^0$ construction of super-constant weight Dicke states. Previous constant-depth constructions for any super-constant $k$ required the FANOUT$_n$ gate, while $\text{QAC}^0$ is only known to implement FANOUT$_k$ for $k$ up to $\text{polylog}(n)$. Moreover, we show that any weight-$k$ Dicke state can be constructed with access to FANOUT$_{\min(k,n-k)}$, rather than FANOUT$_n$. Combined with recent hardness results, this yields a tight characterization: for $k \leq n/2$, weight-$k$ Dicke states can be prepared in $\text{QAC}^0$ if and only if FANOUT$_k \in \text{QAC}^0$. We further extend our techniques to show that, in fact, \emph{any} superposition of $n$-qubit Dicke states of weight at most $k$ can be prepared in $\text{QAC}^0$ with access to FANOUT$_k$. Taking $k = n$, we obtain the first $O(1)$-depth unitary construction for arbitrary symmetric states. In particular, any symmetric state can be prepared in constant depth on quantum hardware architectures that support FANOUT$_n$, such as trapped ions with native global entangling operations.

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