Super-Constant Weight Dicke States in Constant Depth Without Fanout

Lucas Gretta, Meghal Gupta, Malvika Raj Joshi

Apr 16, 2026

arXiv:2604.15298v1 PDF

quant-ph(primary)cs.DS

#434of 2593·Quantum Physics

#434 of 2593 · Quantum Physics

Tournament Score

1483±31

10501750

61%

Win Rate

Wins

Losses

Matches

Rating

8/ 10

Significance8.5

Rigor9

Novelty8

Clarity8.5

Tournament Score

1483±31

10501750

61%

Win Rate

Wins

Losses

Matches

Rating

8/ 10

Significance

Rigor

Novelty

Clarity

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.

AI Impact Assessments

(3 models)

Scientific Impact Assessment

Core Contribution

This paper resolves an open question in quantum circuit complexity by establishing a tight characterization of the resources needed to prepare Dicke states in constant depth. The main result shows that for $k \leq n/2$ , the weight- $k$ Dicke state $|D_k^n\rangle$ can be prepared in QAC⁰ if and only if FANOUT $_k \in$ QAC⁰. This is achieved through two key contributions:

1. QAC⁰[FANOUT $_{k}$ ] sufficiency: Weight- $k$ Dicke states on $n$ qubits can be prepared using only FANOUT $_{k}$ gates (rather than FANOUT $_{n}$ ), closing the gap between the FANOUT $_{k}$ lower bound of [GGJ26] and the FANOUT $_{n}$ upper bound of [JV26].

2. First super-constant weight Dicke states in QAC⁰: For $k = \text{polylog}(n)$ , since FANOUT $_{k}$ is implementable in QAC⁰, this yields the first construction of $|D_k^n\rangle$ for $k = \omega(1)$ without any FANOUT gates—only multi-qubit Toffoli gates and single-qubit unitaries.

3. Arbitrary symmetric states: Any symmetric state supported on weights $\leq k$ can be prepared in QAC⁰[FANOUT $_{k}$ ], giving the first $O (1)$ -depth unitary construction for arbitrary symmetric states (using FANOUT $_{n}$ ).

Methodological Rigor

The technical approach is sophisticated and well-executed. The central challenge—that amplitude amplification fails when $k = \omega(1)$ because the probability of hitting exactly $k$ ones in Bernoulli sampling becomes subconstant—is addressed through a creative "bucketing" strategy:

Bucket expansion: Starting from

|D_{k^3}^k\rangle

(preparable with FANOUT

_{k}

), each qubit is expanded into

n / k^{3}

qubits via controlled-W maps, capturing most weight-

k

strings by a birthday paradox argument.

Damped binomial distribution: The "missing" strings (those with bucket collisions) are recovered through a carefully designed intermediate distribution

S_{k}^{m}

with the crucial property that all weight-

k

strings with the same occupancy pattern receive equal amplitude (Claim 5.4).

Deferred post-selection: Rather than amplifying each occupancy class separately (which fails for small

j

), the authors combine all contributions and show the total marked weight is

\Theta(1)

, enabling a single Grover amplification step.

The proofs are complete and rigorous, with careful tracking of normalization constants and probability bounds. The construction is exact and clean (ancillae returned to $|0\rangle$ ), which is stronger than approximate constructions. The intermediate primitives (Lemma 4.1 for amplitude adjustment, Lemma 4.8 for controlled state constructions, Corollary 4.4 for arbitrary $O(\log n)$ -qubit states) are clearly presented and likely useful beyond this specific application.

Potential Impact

Theoretical significance: The tight equivalence between Dicke state preparation and FANOUT is a clean structural result in quantum circuit complexity. It adds to the growing understanding of the QAC⁰ vs. QAC⁰ $_{f}$ separation question, which is a quantum analog of the classical AC⁰ vs. TC⁰ question. The result elegantly delineates which Dicke states are "easy" (polylog weight) and which require additional resources.

Practical relevance: Dicke states are central to several applications:

Decoded Quantum Interferometry (DQI), a promising near-term quantum optimization approach

Quantum metrology and sensing

As a basis for the symmetric subspace

The polylog-weight Dicke state construction in QAC⁰ is particularly relevant for NISQ implementations, as it requires only Toffoli gates and single-qubit unitaries—no fanout. The arbitrary symmetric state result is relevant for trapped-ion architectures with native global entangling operations.

Toolkit contributions: The primitives developed (controlled circuits with limited fanout, amplitude adjustment, controlled state preparation via reflection-based techniques) constitute a useful toolkit for constant-depth quantum circuit design that will likely find applications in other state preparation problems.

Timeliness & Relevance

This paper is highly timely. It directly follows and completes the picture initiated by [JV26] (constant-depth Dicke states with FANOUT $_{n}$ ) and [GGJ26] (FANOUT $_{k}$ lower bound), both from 2026. The question of whether QAC⁰ = QAC⁰ $_{f}$ remains one of the central open problems in quantum circuit complexity, and understanding which tasks require which level of fanout is a current frontier. The connection to DQI [JSW+25] adds practical motivation from a very recent algorithmic development.

Strengths

Tight characterization: The matching upper and lower bounds yield a clean equivalence, which is rare and satisfying in complexity theory.

Novel techniques: The bucketing strategy, damped binomial distribution, and deferred post-selection represent genuinely new ideas for constant-depth circuit design.

Generality: Extension to arbitrary symmetric states is a substantial bonus, not merely incremental.

Cleanness and exactness: The constructions are exact (not approximate) and clean (ancillae uncomputed), the strongest possible form.

Clear exposition: The overview section effectively communicates the high-level strategy before diving into technical details.

Limitations

Polynomial ancilla overhead: The constructions require poly(

n

) ancillae, which may be large in practice. The constant in the depth bound and the polynomial degree are not optimized.

Gap for intermediate $k$ : For

k

between polylog(

n

) and

n / 2

, the result is conditional on FANOUT

_k \in

QAC⁰, which remains open.

Practical circuit compilation: The paper does not discuss concrete gate counts or compilation to native gate sets beyond the abstract QAC⁰ model.

No experimental validation or numerical benchmarking of the circuits' practical performance.

Rating:8/ 10

Significance 8.5Rigor 9Novelty 8Clarity 8.5

Generated Apr 17, 2026

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