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 ∣Dkn⟩, 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 ∣Dkn⟩ 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]).
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 a divide-and-conquer approach to deterministically prepare Dicke states |Dkn> (i.e. equal-weight superpositions of all n-qubit states with Hamming Weight k) on quantum computers. In an experimental evaluation for up to n=6 qubits on IBM Quantum Sydney and Montreal devices, we achieve significantly higher state fidelity compared to previous results [Mukherjee et.al. TQE'2020, Cruz et.al. QuTe'2019]. The fidelity gains are achieved through several techniques: Our circuits first divide the Hamming weight between blocks of n/2 qubits, and then conquer those blocks with improved versions of Dicke state unitaries [Brtschi et.al. FCT'2019]. Due to the sparse connectivity on IBM's heavy-hex-architectures, these circuits are implemented for linear nearest neighbor topologies. Further gains in (estimating) the state fidelity are due to our use of measurement error mitigation and hardware progress.
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.
Dicke states serve as a critical resource in quantum metrology, communication, and computation. However, unitary preparation of Dicke states is limited to logarithmic depth in standard circuit models and existing constant-depth protocols require measurement and feed-forward. In this work, we present the first unitary, constant-depth protocols for exact Dicke state preparation. We overcome the logarithmic-depth barrier by moving beyond the standard circuit model and leveraging global interactions (native to architectures such as neutral atoms and trapped ions). Specifically, utilizing unbounded CZ gates (i.e. within the QAC0 circuit class), we offer circuits for exact computation of constant-weight Dicke states, using polynomial ancillae, and approximation of weight-1 Dicke states (i.e. W states), using only constant ancillae. Granted additional access to the quantum FAN-OUT operation (i.e. upgrading to the QACf0 circuit class), we also achieve exact and clean preparation of arbitrary-weight Dicke states, with polynomial ancillae. These protocols distinguish the constant-depth capabilities of quantum architectures based on connectivity and offer a novel path toward resolving a long-standing quantum complexity conjecture.