In this paper, we study efficient algorithms towards the construction of any arbitrary Dicke state. Our contribution is to use proper symmetric Boolean functions that involve manipulations with Krawtchouk polynomials. Deutsch-Jozsa algorithm, Grover algorithm and the parity measurement technique are stitched together to devise the complete algorithm. Further, motivated by the work of Childs et al (2002), we explore how one can plug the biased Hadamard transformation in our strategy. Our work compares fairly with the results of Childs et al (2002).In this paper, we study efficient algorithms towards the construction of any arbitrary Dicke state. Our contribution is to use proper symmetric Boolean functions that involve manipulations with Krawtchouk polynomials. Deutsch-Jozsa algorithm, Grover algorithm and the parity measurement technique are stitched together to devise the complete algorithm. Further, motivated by the work of Childs et al (2002), we explore how one can plug the biased Hadamard transformation in our strategy. Our work compares fairly with the results of Childs et al (2002).

arXiv:1209.5932Sep 26, 20121 citation

Deterministic Preparation of Arbitrary Spin Eigenfunctions

Wenxuan Tao, Jianan Wang, Fen Zuo

Quantum states with conserved total spins, or spin eigenfunctions, are important for studying quantum chemistry and quantum manybody physics problems. A typical class of spin eigenfunctions are Dicke states, which attain maximal spins. While we already have many efficient quantum algorithms to prepare Dicke states, it is not yet clear if we could do so for arbitrary spin eigenfunctions deterministically. Generalizing Bärtschi and Eidenbenz's elegant algorithms for Dicke state preparation, we successfully prepare arbitrary spin eigenfunctions characterized by branching paths and binary spin trees. As a byproduct, we also develop the corresponding classical algorithms to reconstruct all these spin states.Quantum states with conserved total spins, or spin eigenfunctions, are important for studying quantum chemistry and quantum manybody physics problems. A typical class of spin eigenfunctions are Dicke states, which attain maximal spins. While we already have many efficient quantum algorithms to prepare Dicke states, it is not yet clear if we could do so for arbitrary spin eigenfunctions deterministically. Generalizing Bärtschi and Eidenbenz's elegant algorithms for Dicke state preparation, we successfully prepare arbitrary spin eigenfunctions characterized by branching paths and binary spin trees. As a byproduct, we also develop the corresponding classical algorithms to reconstruct all these spin states.

quant-pharXiv:2608.22892Aug 25, 20260 citations

Constant-depth adaptive preparation of Dicke and symmetric states

Rui Luo, Junjie Chen, Xiongfeng Ma

Efficient preparation of Dicke states and, more generally, permutation-symmetric states is important for quantum metrology, quantum networking, and collective quantum information processing. Measurements and classical feedforward enable low-depth preparations of these states, with a cost of ancillary qubits. In this work, we introduce an exact constant-depth adaptive preparation protocol for arbitrary Dicke- $(n,k)$ states and further symmetric states. We first provide a protocol preparing the uniform subset superposition state, as a primitive, using constant-depth adaptive circuit with $O(k^2\log^2 n)$ ancillary qubits and success probability at least $1/k$ . This yields an exact, probabilistic, constant-depth Dicke-state preparation protocol using $O\left(n^2+k^2\log^2 n+kn\log n\log\log n\right)$ ancillary qubits. Parallel repetition suppresses the failure probability exponentially without increasing the quantum depth. Moreover, the uniform subset superposition state is also of independent interest as the uniform vertex state of the Johnson graph and as the compact uniform subset state appearing in quantum-walk and topological-data-analysis algorithms. We further establish a general lifting framework that coherently combines clean unitary Dicke-state preparation circuits to prepare arbitrary symmetric states with only polynomial ancillary overhead. Combined with recent constant-depth unitary Dicke-state constructions, this gives an exact constant-depth preparation protocol for arbitrary $n$ -qubit symmetric states using $O(n^3\sqrt{\log n})$ ancillary qubits.Efficient preparation of Dicke states and, more generally, permutation-symmetric states is important for quantum metrology, quantum networking, and collective quantum information processing. Measurements and classical feedforward enable low-depth preparations of these states, with a cost of ancillary qubits. In this work, we introduce an exact constant-depth adaptive preparation protocol for arbitrary Dicke-$(n,k)$ states and further symmetric states. We first provide a protocol preparing the uniform subset superposition state, as a primitive, using constant-depth adaptive circuit with $O(k^2\log^2 n)$ ancillary qubits and success probability at least $1/k$. This yields an exact, probabilistic, constant-depth Dicke-state preparation protocol using $O\left(n^2+k^2\log^2 n+kn\log n\log\log n\right)$ ancillary qubits. Parallel repetition suppresses the failure probability exponentially without increasing the quantum depth. Moreover, the uniform subset superposition state is also of independent interest as the uniform vertex state of the Johnson graph and as the compact uniform subset state appearing in quantum-walk and topological-data-analysis algorithms. We further establish a general lifting framework that coherently combines clean unitary Dicke-state preparation circuits to prepare arbitrary symmetric states with only polynomial ancillary overhead. Combined with recent constant-depth unitary Dicke-state constructions, this gives an exact constant-depth preparation protocol for arbitrary $n$-qubit symmetric states using $O(n^3\sqrt{\log n})$ ancillary qubits.

quant-pharXiv:2608.01144Aug 4, 20260 citations

Generation of N-particle Dicke-Class States and their Application to Quantum Information Processing

Daniel DeYoung, Kishore T. Kapale

Bulletin of the American Physical SocietyNov 8, 20130 citations

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

Lucas Gretta, Meghal Gupta, Malvika Raj Joshi

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 $^0_f$ .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$^0_f$.

quant-pharXiv:2604.15298Apr 17, 20260 citations

(n,k)

states and further symmetric states. We first provide a protocol preparing the uniform subset superposition state, as a primitive, using constant-depth adaptive circuit with

O(k^2\log^2 n)

ancillary qubits and success probability at least

1/k

. This yields an exact, probabilistic, constant-depth Dicke-state preparation protocol using

O\left(n^2+k^2\log^2 n+kn\log n\log\log n\right)

ancillary qubits. Parallel repetition suppresses the failure probability exponentially without increasing the quantum depth. Moreover, the uniform subset superposition state is also of independent interest as the uniform vertex state of the Johnson graph and as the compact uniform subset state appearing in quantum-walk and topological-data-analysis algorithms. We further establish a general lifting framework that coherently combines clean unitary Dicke-state preparation circuits to prepare arbitrary symmetric states with only polynomial ancillary overhead. Combined with recent constant-depth unitary Dicke-state constructions, this gives an exact constant-depth preparation protocol for arbitrary

n

-qubit symmetric states using

O(n^3\sqrt{\log n})

ancillary qubits.

Polylogarithmic-Weight Dicke States in QAC

^0

and Arbitrary Symmetric States in QAC

^0_f

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