Dicke states are fundamental to advances in quantum communication and computation, yet their efficient preparation has long presented a significant challenge. Vasconcelos, Joshi, and their colleagues have now demonstrated the first unitary, constant‑depth protocols for the exact preparation of these crucial quantum states. By exploiting global interactions—common in architectures such as neutral‑atom arrays and trapped‑ion systems—and employing unbounded CZ gates, the researchers bypass the logarithmic‑depth barrier that has limited standard circuit models. This breakthrough not only highlights the potential of specific quantum architectures but also offers a new approach to a longstanding complexity problem in the field. Preparing Dicke states with constant‑depth circuits is a substantial step toward realizing more complex quantum algorithms and technologies.

Scientists have long faced a challenge in efficiently preparing these highly entangled states. Standard quantum‑circuit methods are limited to logarithmic depth, a barrier that grows with the number of qubits. Existing constant‑depth protocols require measurement and classical feed‑forward, introducing control complexity and potential noise. The new work presents the first unitary, constant‑depth protocols for the exact preparation of Dicke states, overcoming this limitation by moving beyond conventional circuit models.

The team achieved the breakthrough by leveraging architectures with global interactions, such as neutral‑atom and trapped‑ion systems. Using unbounded CZ gates within the QAC⁰ circuit class, they constructed circuits that exactly compute constant‑weight Dicke states while employing only a polynomial number of ancilla qubits. They also demonstrated a constant‑ancilla approximation of weight‑1 Dicke states (the W states) within the same framework, eliminating the need for the depth‑scaling circuits previously required.

Granting access to the FAN‑OUT operation—upgrading to the QAC⁰_f circuit class—extends the method to the exact preparation of arbitrary‑weight Dicke states, again with polynomial ancilla overhead. These protocols showcase the constant‑depth capabilities of quantum architectures based on their connectivity, offering a clear path toward resolving a long‑standing conjecture in quantum‑complexity theory. Generating Dicke states in constant time, without adaptive measurements, represents a significant advancement in quantum information processing. Experiments show that these protocols distinguish the capabilities of different quantum hardware architectures, positioning systems capable of global FAN‑OUT operations (e.g., trapped ions) above those limited to global CZ operations (e.g., neutral atoms). Both architectures, however, outperform those restricted to local connectivity.

From a theoretical perspective, the difficulty of preparing super‑constant‑weight Dicke states without FAN‑OUT suggests that these states can serve as natural benchmarks for separating the QAC⁰ and QAC⁰_f circuit classes, resolving a long‑standing question in quantum complexity. The research establishes that constant‑depth generation of long‑range entanglement is possible in systems with global interactions, consistent with the physics of these systems and bypassing the limitations imposed by local‑interaction models.

---

Constant‑Depth Dicke‑State Preparation via Global Interactions

Dicke states are crucial for communication, computation, and quantum physics, but their preparation has historically been hampered by circuit‑depth limitations. Existing methods typically require logarithmic depth or rely on measurement and classical feed‑forward, adding control complexity and noise. This study pioneers unitary, constant‑depth protocols for exact Dicke‑state preparation, circumventing the logarithmic‑depth barrier by exploiting global interactions native to neutral‑atom and trapped‑ion architectures. The researchers specifically use unbounded CZ gates within the QAC circuit class to construct circuits that exactly compute constant‑weight Dicke states with polynomial ancillae.

They also developed circuits that approximate weight‑1 Dicke states (W states) using only constant ancillae, dramatically reducing resource requirements. By granting access to the quantum FAN‑OUT operation, the system is elevated to the QAC⁰_f circuit class, enabling exact preparation of arbitrary‑weight Dicke states with polynomial ancilla overhead. This advancement distinguishes the constant‑depth capabilities of various quantum architectures, highlighting those based on connectivity and offering a novel solution to a long‑standing complexity conjecture. The protocols demonstrate a computational hierarchy among quantum hardware, placing systems with global FAN‑OUT (trapped ions) above those limited to global CZ (neutral atoms), both of which surpass architectures constrained by local geometry.

The study details a QAC⁰ reduction from constant‑weight Dicke states to the EXACTₖ Boolean function, followed by an implementation of constant‑weight EXACT within the QAC⁰ framework. Researchers then leveraged FAN‑OUT to achieve a constant‑depth construction for arbitrary‑weight Dicke states. The inherent difficulty of preparing super‑constant‑weight Dicke states without FAN‑OUT suggests their utility as witnesses for a state‑synthesis separation between QAC⁰ and QAC⁰_f, resolving a key question in quantum complexity. Precise mathematical definitions of Dicke states—as uniform superpositions over computational‑basis states with a specific Hamming weight—enable rigorous analysis and validation of the protocols.

---

Constant‑Depth Dicke‑State Preparation via QAC Circuits

The breakthrough demonstrates the first unitary, constant‑depth protocols for exact Dicke‑state preparation. By utilizing unbounded CZ gates within the QAC circuit class, the researchers overcame the prior logarithmic‑depth requirement. Experiments revealed circuits that compute constant‑weight Dicke states with polynomial ancillae and approximate weight‑1 Dicke states using only constant ancillae. A direct connection was established between computing the EXACTₖ Boolean function and preparing weight‑k Dicke states in QAC⁰ for constant k.

An explicit polynomial‑size QAC⁰ circuit for exact computation of EXACTₖ was designed, yielding a corresponding circuit for exact preparation of the |Dₙᵏ⟩ Dicke state. A constant‑size QAC⁰ circuit for approximating EXACT₁ achieved a constant‑error approximation of the W state with minimal ancillae. Granting access to FAN‑OUT—upgrading to QAC⁰_f—enabled exact preparation of arbitrary‑weight Dicke states, again with polynomial ancilla overhead. These results bypass the Lieb‑Robinson bounds that typically limit entanglement‑generation time in locally interacting systems.

The protocols deliver constant‑time generation of complex, permutation‑invariant entanglement, consistent with the physics of long‑range interacting systems. They leverage architectures with global interactions, such as trapped‑ion systems employing Mølmer‑Sørensen interactions and neutral‑atom arrays using Rydberg blockade mechanisms. Recent demonstrations of transversal logical multi‑qubit CZ gates and constant‑depth multi‑body logic across 48 logical qubits simultaneously validate the feasibility of these approaches.

Beyond experimental impact, the work has profound implications for quantum‑complexity theory, offering a novel route toward resolving the open question of separating the QAC⁰ and QAC⁰_f circuit classes: protocols achieve preparation of arbitrary‑weight Dicke states in QAC⁰_f, but only constant‑weight states in QAC⁰.