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 ∣Dkn⟩, defined as the uniform superposition of all computational basis states with exactly k qubits in state ∣1⟩, 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(klog(n/k)). 2. Grid qubit connectivity ((n1×n2)-grid, n1≤n2): (a) For k≥n2/n1, we design a circuit with depth O(klog(n/k)+n2), surpassing the prior O(nk) bound. (b) For k<n2/n1, we design an optimal-depth circuit with depth O(n2). Furthermore, we establish the depth lower bounds of Ω(log(n)) for all-to-all qubit connectivity and Ω(n2) for (n1×n2)-grid connectivity constraints, demonstrating the near-optimality of our constructions.
Hamming-weight computation and symmetric Boolean functions are basic primitives for quantum arithmetic and state preparation, where circuit depth and qubit overhead matter. We construct an exact circuit computing the Hamming weight of an n-qubit register with depth O(logn) and size O(nlogn), using only the n input qubits and m=⌈log2(n+1)⌉ binary output qubits. The circuit uses arbitrary one-qubit gates and CNOTs with all-to-all connectivity. Both depth and total width are optimal. Our constructions combine tree structures and pipelining, temporarily using input qubits as workspace. We also give exact circuits for every symmetric Boolean function with the same depth and size bounds, using only m ancillary qubits returned to zero. Finally, we apply the counter to exact preparation of balanced Dicke states for even n. With only O(logn) ancillary qubits, we obtain expected depth O(log2n) using measurements and classical feedforward, and depth O(n1/4logn) using unitary gates alone.
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 O(klogkn) for all-to-all connectivity (such as on current ion trap devices); (ii) depth of O(k√kn)=O(nk) for Grid connectivity on grids of size Ω(√sn)×O(ns) with s ≤ k (such as on most current superconducting qubit devices).Both approaches have a total gate count of 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 O(n) on a Linear Nearest Neighbor connectivity for arbitrary k (FCT 2019 [4]) and depth O(logn) on all-to-all connectivity for k = 1 (AQT 2019 [9]).
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(k2log2n) ancillary qubits and success probability at least 1/k. This yields an exact, probabilistic, constant-depth Dicke-state preparation protocol using O(n2+k2log2n+knlognloglogn) 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(n3logn) ancillary qubits.
Preparing large-qubit Dicke states is of broad interest in quantum computing and quantum metrology. However, the number of qubits available on a single quantum processing unit (QPU) is limited -- motivating the distributed preparation of such states across multiple QPUs as a practical approach to scalability. In this article, we investigate the distributed preparation of n-qubit k-excitation Dicke states D(n,k) across a general number p of QPUs, presenting a distributed quantum circuit (each QPU hosting approximately ⌈n/p⌉ qubits) that prepares the state with communication complexity O(plogk), circuit size O(nk), and circuit depth O(p2k+logklog(n/k)). To the best of our knowledge, this is the first construction to simultaneously achieve logarithmic communication complexity and polynomial circuit size and depth. We also establish a lower bound on the communication complexity of p-QPU distributed state preparation for a general target state. This lower bound is formulated in terms of the canonical polyadic rank (CP-rank) of a tensor associated with the target state. For the special case p=2, we explicitly compute the CP-rank corresponding to the Dicke state D(n,k) and derive a lower bound of ⌈log(k+1)⌉, which shows that the communication complexity of our construction matches this fundamental limit.
Polylogarithmic-Weight Dicke States in QAC0 and Arbitrary Symmetric States in QACf0
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. QAC0, the quantum analogue of AC0, minimally extends local O(1)-depth quantum circuits by allowing arbitrary-width Toffoli (reversible AND) gates.
We show that Dicke states of polylog(n) weight can be prepared in QAC0. This gives the first QAC0 construction of any super-constant-weight n-qubit Dicke state, since previous constructions relied on the much more powerful FANOUTn gate. In general, we show that any weight-k Dicke state can be constructed using FANOUTmin(k,n−k) gates. Combined with recent hardness results, this yields a tight characterization: for k≤n/2, a n-qubit weight-k Dicke state can be prepared in QAC0 if and only if FANOUTk∈ QAC0. We develop a limited-fanout state-synthesis toolkit for QAC0 that yields further constant-depth, poly(n)-ancilla constructions:
1. Every n-qubit symmetric state supported on Hamming weight ≤k can be prepared using FANOUTk gates.
2. Every O(logn)-qubit state can be prepared using quantum random-access memory (QRAMn), which refers to a coherent indexing gate. QRAMn is a potentially weaker resource than FANOUTn and can be implemented in QACf0.
The exact number of CNOT and single qubit gates needed to implement a Quantum Algorithm in a given architecture is one of the central problems of Quantum Computation. In this work we study the importance of concise realizations of Partially defined Unitary Transformations for better circuit construction using the case study of Dicke State Preparation. The Dicke States (∣Dkn⟩) are an important class of entangled states with uses in many branches of Quantum Information. In this regard we provide the most efficient Deterministic Dicke State Preparation Circuit in terms of CNOT and single qubit gate counts in comparison to existing literature. We further observe that our improvements also reduce architectural constraints of the circuits. We implement the circuit for preparing D24⟩ on the "ibmqx2" machine of the IBM QX service and observe that the error induced due to noise in the system is lesser in comparison to the existing circuit descriptions. We conclude by describing the CNOT map of the generic ∣Dkn⟩ preparation circuit and analyze different ways of distributing the CNOT gates in the circuit and its affect on the induced error.