Super-Constant Weight Dicke States in Constant Depth Without Fanout

Lucas Gretta; Malvika Raj Joshi; Meghal Gupta

arxiv: 2604.15298 · v1 · submitted 2026-04-16 · 🪐 quant-ph · cs.DS

Super-Constant Weight Dicke States in Constant Depth Without Fanout

Lucas Gretta , Meghal Gupta , Malvika Raj Joshi This is my paper

Pith reviewed 2026-05-10 11:13 UTC · model grok-4.3

classification 🪐 quant-ph cs.DS

keywords Dicke statesQAC0constant depthToffoli gatesfanoutsymmetric statesquantum circuits

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The pith

Explicit constant-depth circuits prepare polylog-weight Dicke states using only Toffoli gates.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper provides explicit constructions of constant-depth quantum circuits that prepare n-qubit Dicke states of weight k up to polylogarithmic in n. These use only multi-qubit Toffoli gates and single-qubit unitaries, without requiring the FANOUT gate for copying to all qubits. A sympathetic reader would care because Dicke states serve as key entanglement resources for applications in the NISQ era and in algorithms like Decoded Quantum Interferometry, and they help build any symmetric quantum state. The result establishes the first such constructions in the QAC^0 model, which previously could only handle constant-weight cases without fanout. It also gives a tight characterization showing that preparing weight-k Dicke states in QAC^0 is equivalent to having FANOUT_k in QAC^0 for k up to n/2.

Core claim

We give explicit constant-depth circuits that prepare n-qubit Dicke states for all k ≤ polylog(n), using only multi-qubit Toffoli gates and single-qubit unitaries. This gives the first QAC^0 construction of super-constant weight Dicke states. 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 ≤ n/2, weight-k Dicke states can be prepared in QAC^0 if and only if FANOUT_k ∈ QAC^0. We further extend to show that any superposition of n-qubit Dicke states of weight at most k can be prepared in QAC^0 with access to FANOUT_k. Taking k = n, we obtain the f

What carries the argument

Constant-depth circuits built from unbounded-fan-in multi-qubit Toffoli gates and single-qubit unitaries, with fanout limited to min(k, n-k).

Load-bearing premise

The quantum circuit model allows unbounded fan-in Toffoli gates in constant depth while QAC^0 permits only polylogarithmic fanout.

What would settle it

Finding a weight-k Dicke state for k = polylog(n) that cannot be prepared by any constant-depth circuit using only Toffoli gates and polylog fanout would show the construction fails.

Figures

Figures reproduced from arXiv: 2604.15298 by Lucas Gretta, Malvika Raj Joshi, Meghal Gupta.

**Figure 1.** Figure 1: Visualization of the bucket occupancy of ℓ = k 3 buckets (left) and the corresponding n = mk3 -qubit state on each bucket obtained via a controlled-W preparation on each group of m qubits (right). This gives us an equal superposition over all strings of weight k where each bucket has at most one 1. In order to show that this state has constant fidelity with |Dn k ⟩, it suffices to show that a uniformly ran… view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Explicit constant-depth QAC0 circuits for polylog-weight Dicke states, with a tight FANOUT characterization.

read the letter

The paper's main contribution is explicit constant-depth circuits that prepare Dicke states of weight up to polylog(n) without needing full fanout. It also gives a tight if-and-only-if characterization with FANOUT_k in QAC^0 for k up to n/2. They construct these using multi-qubit Toffoli gates and single-qubit unitaries in the standard quantum circuit model. Previous work needed FANOUT_n for any super-constant k, but here they reduce the requirement to FANOUT_min(k, n-k). This, combined with existing hardness results, pins down exactly when these states are in QAC^0. They further show how to prepare any superposition of such Dicke states up to weight k, which covers all symmetric states in constant depth given FANOUT_n. The work is done well in that the constructions are claimed to be explicit and the argument avoids circularity by importing hardness from prior papers. The model assumptions are standard for this area of quantum circuit complexity, and the stress-test confirms there are no obvious inconsistencies like hidden fanout or depth blowups. Soft spots are limited. The constant depth comes with Toffoli gates whose fan-in grows with k, so up to polylog size, which is allowed but means the gates are not small. Readers will want to see the full circuit diagrams and gate counts in the paper to assess practicality. The extension to symmetric states is nice but relies on having global operations, which is hardware-specific. This paper is aimed at quantum complexity theorists and experimentalists working on NISQ devices that can do global entangling gates, such as trapped ions. Anyone building algorithms around Dicke states or symmetric states, like in Decoded Quantum Interferometry, will find the constructions directly useful. It should go to peer review. The result advances the boundary of what constant-depth quantum circuits can achieve and provides a clean characterization, so referees can verify the details and check for any overlooked implementation issues.

Referee Report

2 major / 2 minor

Summary. The manuscript claims explicit constant-depth circuits (using only multi-qubit Toffoli gates and single-qubit unitaries) that prepare n-qubit Dicke states of weight k for all k ≤ polylog(n), yielding the first QAC^0 construction for super-constant-weight Dicke states. It further shows that any weight-k Dicke state can be prepared using FANOUT_min(k,n-k) rather than FANOUT_n, and combines this with prior hardness results to obtain a tight characterization: for k ≤ n/2, weight-k Dicke states lie in QAC^0 if and only if FANOUT_k does. The techniques extend to preparing arbitrary superpositions of Dicke states of weight at most k using FANOUT_k, and in particular yield the first O(1)-depth unitary construction for arbitrary symmetric states when k = n.

Significance. If the constructions hold, the result substantially clarifies the constant-depth quantum circuit complexity of symmetric states and Dicke states, removing the previous necessity of full FANOUT_n for super-constant k while staying inside the polylog-fanout regime permitted by QAC^0. The characterization directly links Dicke-state preparation to the open FANOUT problem, and the constant-depth symmetric-state construction has direct implications for hardware platforms (e.g., trapped ions) that natively support global entangling operations. The explicit, gate-level constructions constitute a concrete advance over prior non-constructive or fanout-heavy approaches.

major comments (2)

[§3] §3 (main construction): the explicit circuit for weight-k Dicke states with k = polylog(n) must be verified to use only polylogarithmic fanout; the abstract asserts this bound, but the gate-by-gate description should explicitly bound the fanout of every Toffoli and confirm that no implicit copying or broadcasting exceeds the QAC^0 allowance.
[Characterization paragraph] Characterization paragraph (likely §4): the hardness direction imports a prior result on FANOUT_k; the manuscript should state the precise theorem citation and confirm that the reduction from Dicke_k to FANOUT_k preserves constant depth and the exact fanout bound min(k,n-k).

minor comments (2)

[Abstract] The abstract states that 'any symmetric state can be prepared in constant depth on quantum hardware architectures that support FANOUT_n'; a brief remark on the precise depth constant (independent of n) would strengthen the claim.
[Preliminaries] Notation for the superposition of Dicke states (the 'any superposition of n-qubit Dicke states of weight at most k') should be formalized with an explicit state vector or coefficient vector in the preliminaries section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for recommending minor revision. We address the major comments point by point below.

read point-by-point responses

Referee: [§3] §3 (main construction): the explicit circuit for weight-k Dicke states with k = polylog(n) must be verified to use only polylogarithmic fanout; the abstract asserts this bound, but the gate-by-gate description should explicitly bound the fanout of every Toffoli and confirm that no implicit copying or broadcasting exceeds the QAC^0 allowance.

Authors: We appreciate the referee's suggestion for additional clarity on this point. In the revised manuscript, we will add an explicit paragraph in §3 that bounds the fanout of every multi-qubit Toffoli gate appearing in the construction. Each such gate acts on a number of controls and targets bounded by O(k) with k ≤ polylog(n); the circuit description contains no implicit copying, broadcasting, or fanout operations beyond this size. This addition will confirm that the construction remains strictly inside the polylog-fanout regime permitted by QAC^0. revision: yes
Referee: [Characterization paragraph] Characterization paragraph (likely §4): the hardness direction imports a prior result on FANOUT_k; the manuscript should state the precise theorem citation and confirm that the reduction from Dicke_k to FANOUT_k preserves constant depth and the exact fanout bound min(k,n-k).

Authors: We agree that greater precision is warranted here. In the revision we will insert the exact theorem citation for the hardness of FANOUT_k (including the full reference) into the characterization paragraph. We will also add a short sentence confirming that the reduction from weight-k Dicke-state preparation to FANOUT_min(k,n-k) preserves constant depth and does not increase the fanout bound, as the construction invokes the fanout gate directly within a constant-depth block. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper provides an explicit constructive proof of constant-depth QAC^0 circuits for exact weight-k Dicke states (k ≤ polylog(n)) using only unbounded-fan-in Toffolis and single-qubit gates, plus a reduction showing that such states are equivalent to FANOUT_k membership in QAC^0. No parameters are fitted to data, no outputs are defined in terms of themselves, and no load-bearing step reduces by construction or self-citation to the authors' own prior unverified results. The hardness direction for the tight characterization is imported from external prior work rather than derived internally, and the central existence claim remains independently verifiable via the stated circuit constructions. The derivation is self-contained against standard circuit-complexity benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on the standard quantum circuit model and prior definitions of QAC^0 and FANOUT; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)

domain assumption Quantum circuits may use unbounded fan-in multi-qubit Toffoli gates in constant depth
Invoked when stating the gate set for the constructions.
domain assumption QAC^0 is defined as constant-depth circuits with only polylogarithmic fanout
Used to position the result relative to prior circuit classes.

pith-pipeline@v0.9.0 · 5683 in / 1295 out tokens · 32866 ms · 2026-05-10T11:13:19.906873+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

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Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages · cited by 1 Pith paper

[1]

With a new register,B={b 1, b2

LetC 1 be the circuit from Corollary 4.7. With a new register,B={b 1, b2 . . . bℓ}, obtain, |ψ1⟩:=C 1 |ψ0⟩ |0ℓ⟩B (A.19) = X (j,k)∈I δk j · |ek⟩Q |ej⟩A |Dℓ j⟩B .(A.20)

work page
[2]

|ψ2⟩A,B,T :=  O i∈[ℓ] c-C2(bi, Ti)   · |ψ1⟩ |0n⟩T (A.21) = X (j,k)∈I δk j · |ek⟩Q |ej⟩A X y∈{0,1}ℓ:|y|=j |y⟩B  O i:yi=1 |S⟩ Ti    O i:yi=0 |0m⟩Ti   .(A.22)

Letm=n/ℓand letC 2 be the map from Corollary 5.6, then, for eachiprepare|S⟩onT i controlled on 27 bi as below. |ψ2⟩A,B,T :=  O i∈[ℓ] c-C2(bi, Ti)   · |ψ1⟩ |0n⟩T (A.21) = X (j,k)∈I δk j · |ek⟩Q |ej⟩A X y∈{0,1}ℓ:|y|=j |y⟩B  O i:yi=1 |S⟩ Ti    O i:yi=0 |0m⟩Ti   .(A.22)

work page
[3]

Note⟨S|0 m⟩= 0. Then, using an OR gate onT i registers withb i as the target, uncomputeb i to obtain, |ψ3⟩= X (j,k)∈I δk j |ek⟩Q |ej⟩A |0ℓ⟩B X S⊆[ℓ] |S|=j O i∈S |S⟩ Ti !  O i∈[ℓ]\S |0m⟩Ti   .(A.23)

work page
[4]

Due to Claim 5.4, all stringsx∈ {0,1} m with Hamming weightk ′ andocc(x) =jappear with the same amplitude. Therefore the state can be re-written as|ψ⟩=|ψ ′ 3⟩ |0ℓ⟩B, |ψ′ 3⟩= X (j,k)∈I δk j |ek⟩ |ej⟩A kX k′=j γj,k′ |Dm,ℓ j,k′ ⟩T ,(A.24) where|D m,ℓ j,k′ ⟩is the state conditioned on measuring occupancyjand hamming weightk ′ as defined in Definition 5.8 andγ...

work page
[5]

Recall thatq k(j) := Pr |S ⊗j|=k , and is precisely the amplitude on|e k⟩Q |ej⟩A |Dm,ℓ j,k ⟩T . Marking this on ancillaafor allj, kusing Lemma A.2 produces, |ψ4⟩=   X (j,k)∈I δk j p qk(j)|e k⟩Q |ej⟩A |Dm,ℓ j,k ⟩T |1⟩a   +γ bad |bad⟩ |0⟩a (A.25) =   1√ R X (j,k)∈I ηk p pk(j)|e k⟩Q |ej⟩A |Dm,ℓ j,k ⟩T |1⟩a   +γ bad |bad⟩ |0⟩a (A.26) = 1√ R k∗X k=0 ηk...

work page

[1] [1]

With a new register,B={b 1, b2

LetC 1 be the circuit from Corollary 4.7. With a new register,B={b 1, b2 . . . bℓ}, obtain, |ψ1⟩:=C 1 |ψ0⟩ |0ℓ⟩B (A.19) = X (j,k)∈I δk j · |ek⟩Q |ej⟩A |Dℓ j⟩B .(A.20)

work page

[2] [2]

|ψ2⟩A,B,T :=  O i∈[ℓ] c-C2(bi, Ti)   · |ψ1⟩ |0n⟩T (A.21) = X (j,k)∈I δk j · |ek⟩Q |ej⟩A X y∈{0,1}ℓ:|y|=j |y⟩B  O i:yi=1 |S⟩ Ti    O i:yi=0 |0m⟩Ti   .(A.22)

Letm=n/ℓand letC 2 be the map from Corollary 5.6, then, for eachiprepare|S⟩onT i controlled on 27 bi as below. |ψ2⟩A,B,T :=  O i∈[ℓ] c-C2(bi, Ti)   · |ψ1⟩ |0n⟩T (A.21) = X (j,k)∈I δk j · |ek⟩Q |ej⟩A X y∈{0,1}ℓ:|y|=j |y⟩B  O i:yi=1 |S⟩ Ti    O i:yi=0 |0m⟩Ti   .(A.22)

work page

[3] [3]

Note⟨S|0 m⟩= 0. Then, using an OR gate onT i registers withb i as the target, uncomputeb i to obtain, |ψ3⟩= X (j,k)∈I δk j |ek⟩Q |ej⟩A |0ℓ⟩B X S⊆[ℓ] |S|=j O i∈S |S⟩ Ti !  O i∈[ℓ]\S |0m⟩Ti   .(A.23)

work page

[4] [4]

Due to Claim 5.4, all stringsx∈ {0,1} m with Hamming weightk ′ andocc(x) =jappear with the same amplitude. Therefore the state can be re-written as|ψ⟩=|ψ ′ 3⟩ |0ℓ⟩B, |ψ′ 3⟩= X (j,k)∈I δk j |ek⟩ |ej⟩A kX k′=j γj,k′ |Dm,ℓ j,k′ ⟩T ,(A.24) where|D m,ℓ j,k′ ⟩is the state conditioned on measuring occupancyjand hamming weightk ′ as defined in Definition 5.8 andγ...

work page

[5] [5]

Recall thatq k(j) := Pr |S ⊗j|=k , and is precisely the amplitude on|e k⟩Q |ej⟩A |Dm,ℓ j,k ⟩T . Marking this on ancillaafor allj, kusing Lemma A.2 produces, |ψ4⟩=   X (j,k)∈I δk j p qk(j)|e k⟩Q |ej⟩A |Dm,ℓ j,k ⟩T |1⟩a   +γ bad |bad⟩ |0⟩a (A.25) =   1√ R X (j,k)∈I ηk p pk(j)|e k⟩Q |ej⟩A |Dm,ℓ j,k ⟩T |1⟩a   +γ bad |bad⟩ |0⟩a (A.26) = 1√ R k∗X k=0 ηk...

work page