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 $Ω(\log(n))$ for all-to-all qubit connectivity and $Ω(n_2)$ for $(n_1\times n_2)$ -grid connectivity constraints, demonstrating the near-optimality of our constructions.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 $Ω(\log(n))$ for all-to-all qubit connectivity and $Ω(n_2)$ for $(n_1\times n_2)$-grid connectivity constraints, demonstrating the near-optimality of our constructions.

arXiv:2505.15413May 21, 20250 citations

Short-Depth Circuits for Dicke State Preparation

Andreas Bärtschi, Stephan Eidenbenz

We present short-depth quantum circuits to deterministically prepare any Dicke state $\left| {D_k^n} \right\rangle $, which is the equal-amplitude superposition of all n-qubit computational basis states with Hamming Weight k. Dicke states are an important class of entangled quantum states with a large variety of applications, and a long history of experimental creation in physical systems. On the other hand, not much is known regarding efficient scalable quantum circuits for Dicke state preparation on realistic quantum computing hardware connectivities.Concretely, we present preparation circuits for Dicke states $\left| {D_k^n} \right\rangle $ with (i) depth of $\mathcal{O}\left( {k\log \frac{n}{k}} \right)$ for all-to-all connectivity (such as on current ion trap devices); (ii) depth of $\mathcal{O}\left( {k\surd \frac{n}{k}} \right) = \mathcal{O}(\sqrt {nk} )$ for Grid connectivity on grids of size $\Omega \left( {\surd \frac{n}{s}} \right) \times \mathcal{O}(\sqrt {ns} )$ with s ≤ k (such as on most current superconducting qubit devices).Both approaches have a total gate count of $\mathcal{O}(kn)$ , need no ancilla qubits, and generalize to preparation and compression of those symmetric pure states, in which all non-zero amplitudes correspond to states with Hamming weight at most k. Our work significantly improves and expands previous state-of-the art circuits which had depth $\mathcal{O}(n)$ on a Linear Nearest Neighbor connectivity for arbitrary k (FCT 2019 [4]) and depth $\mathcal{O}(\log n)$ on all-to-all connectivity for k = 1 (AQT 2019 [9]).We present short-depth quantum circuits to deterministically prepare any Dicke state $\left| {D_k^n} \right\rangle $, which is the equal-amplitude superposition of all n-qubit computational basis states with Hamming Weight k. Dicke states are an important class of entangled quantum states with a large variety of applications, and a long history of experimental creation in physical systems. On the other hand, not much is known regarding efficient scalable quantum circuits for Dicke state preparation on realistic quantum computing hardware connectivities.Concretely, we present preparation circuits for Dicke states $\left| {D_k^n} \right\rangle $ with (i) depth of $\mathcal{O}\left( {k\log \frac{n}{k}} \right)$ for all-to-all connectivity (such as on current ion trap devices); (ii) depth of $\mathcal{O}\left( {k\surd \frac{n}{k}} \right) = \mathcal{O}(\sqrt {nk} )$ for Grid connectivity on grids of size $\Omega \left( {\surd \frac{n}{s}} \right) \times \mathcal{O}(\sqrt {ns} )$ with s ≤ k (such as on most current superconducting qubit devices).Both approaches have a total gate count of $\mathcal{O}(kn)$, need no ancilla qubits, and generalize to preparation and compression of those symmetric pure states, in which all non-zero amplitudes correspond to states with Hamming weight at most k. Our work significantly improves and expands previous state-of-the art circuits which had depth $\mathcal{O}(n)$ on a Linear Nearest Neighbor connectivity for arbitrary k (FCT 2019 [4]) and depth $\mathcal{O}(\log n)$ on all-to-all connectivity for k = 1 (AQT 2019 [9]).

arXiv:2207.09998IEEE International Conference on Quantum Computing and Engineering (QCE)Sep 1, 202243 citations

Generation of symmetric Dicke states

C. Thiel, Joachim von Zanthier, Thierry Bastin +2 more

Jan 1, 20080 citations

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

Ω(\log(n))

for all-to-all qubit connectivity and

Ω(n_2)

for

(n_1\times n_2)

-grid connectivity constraints, demonstrating the near-optimality of our constructions.

\mathcal{O}\left( {k\log \frac{n}{k}} \right)

for all-to-all connectivity (such as on current ion trap devices); (ii) depth of

\mathcal{O}\left( {k\surd \frac{n}{k}} \right) = \mathcal{O}(\sqrt {nk} )

for Grid connectivity on grids of size

\Omega \left( {\surd \frac{n}{s}} \right) \times \mathcal{O}(\sqrt {ns} )

with s ≤ k (such as on most current superconducting qubit devices).Both approaches have a total gate count of

\mathcal{O}(kn)

, need no ancilla qubits, and generalize to preparation and compression of those symmetric pure states, in which all non-zero amplitudes correspond to states with Hamming weight at most k. Our work significantly improves and expands previous state-of-the art circuits which had depth

\mathcal{O}(n)

on a Linear Nearest Neighbor connectivity for arbitrary k (FCT 2019 [4]) and depth

\mathcal{O}(\log n)

on all-to-all connectivity for k = 1 (AQT 2019 [9]).