Constant-depth Unitary Preparation Achieves Exact Dicke States with Polynomial Ancillae

Dicke states are fundamental to advances in quantum communication and computation, yet their efficient preparation has long presented a significant challenge. Francisca Vasconcelos and Malvika Raj Joshi, both of UC Berkeley, alongside their colleagues, have now demonstrated the first unitary, constant-depth protocols for the exact preparation of these crucial quantum states. The researchers circumvent the limitations of standard circuit models by utilising global interactions, common in architectures like neutral atoms and trapped ions, and employing unbounded CZ gates. This breakthrough not only overcomes a previously established logarithmic-depth barrier, but also highlights the potential of specific quantum architectures and offers a new approach to a longstanding complexity problem within the field. The ability to prepare Dicke states with constant-depth circuits represents a substantial step towards realising more complex quantum algorithms and technologies.

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

The team achieved this breakthrough by leveraging architectures with global interactions, such as those found in neutral atom and trapped ion systems. Utilizing unbounded CZ gates, within a specific circuit class known as QAC0, they developed circuits for the exact computation of constant-weight Dicke states, employing a polynomial number of ancilla qubits. Furthermore, they demonstrated the constant-ancilla approximation of weight-1 Dicke states, also known as W states, within the same circuit framework. This approach circumvents the need for complex, depth-scaling circuits previously required for generating these entangled states.

Granting access to the FAN-OUT operation, upgrading to the QAC0 f circuit class, the research extends to the exact preparation of arbitrary-weight Dicke states, again with polynomial ancilla overhead. These protocols highlight 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. The ability to generate Dicke states in constant time, without relying on 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, like trapped ions, above those limited to global CZ operations, such as neutral atoms.

Both architectures, however, demonstrably outperform those restricted by local connectivity. This work not only provides a practical pathway for generating complex entanglement but also establishes a computational hierarchy amongst quantum hardware, suggesting that the architecture itself plays a critical role in achieving quantum advantage. From a theoretical perspective, the challenges in preparing super-constant-weight Dicke states without FAN-OUT suggest these states can serve as a natural benchmark 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 limitations imposed by local interaction models.

Constant-Depth Dicke State Preparation via Global Interactions

Dicke states, crucial for communication, computation, and quantum physics, have historically presented a challenge in their preparation due to limitations in circuit depth. Existing methods typically require logarithmic depth or rely on measurement and classical feed-forward, introducing control complexity and potential noise. This study pioneers unitary, constant-depth protocols for exact Dicke state preparation, circumventing the logarithmic-depth barrier by moving beyond standard circuit models and harnessing the power of global interactions native to architectures like neutral atoms and trapped ions. The research team specifically utilizes unbounded CZ gates, operating within the QAC circuit class, to construct circuits for exact computation of constant-weight Dicke states with polynomial ancillae.

Scientists developed circuits capable of approximating weight-1 Dicke states, specifically W states, using only constant ancillae, significantly reducing resource requirements. Further expanding the capabilities of this approach, granting access to the quantum FAN-OUT operation elevates the system to the QAC0f circuit class, enabling exact preparation of arbitrary-weight Dicke states with polynomial ancillae. 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 within quantum complexity theory. The protocols developed demonstrate a computational hierarchy amongst quantum hardware, positioning systems with global FAN-OUT operations, such as trapped ions, above those limited to global CZ operations, like neutral atoms, and both surpassing architectures constrained by local geometry.

The study details a QAC0 reduction from constant-weight Dicke states to EXACTk, followed by implementation of constant-weight EXACT within the QAC0 framework. Researchers addressed the challenges of extending beyond constant-weight states by leveraging the FAN-OUT operation, achieving a constant-depth construction for arbitrary-weight Dicke states. The inherent difficulty in preparing super-constant-weight Dicke states without FAN-OUT suggests their utility as a witness for a state-synthesis separation between QAC0 and QAC0f, a resolution to a key question in quantum complexity. Experiments employ precise mathematical definitions of Dicke states, defined as uniform superpositions over computational basis states with a specific Hamming weight, enabling rigorous analysis and validation of the developed protocols.

Constant-Depth Dicke State Preparation via QAC Circuits

Scientists have achieved a breakthrough in quantum state preparation, demonstrating the first unitary, constant-depth protocols for the exact preparation of Dicke states. The research overcomes a long-standing limitation in standard circuit models, which previously required logarithmic depth for Dicke state preparation, by utilizing unbounded CZ gates within the QAC circuit class. Experiments revealed circuits capable of computing constant-weight Dicke states with polynomial ancillae and approximating weight-1 Dicke states using only constant ancillae. The team measured the performance of these new protocols, establishing a direct connection between computation of the EXACTk Boolean function and preparation of weight-k Dicke states in QAC0, for constant k.

They then designed an explicit polynomial-sized QAC0 circuit for exact computation of EXACTk, resulting in a corresponding circuit for exact preparation of the |Dn k⟩ Dicke state. Further tests proved the creation of a constant-sized QAC0 circuit for approximating EXACT1, achieving constant-error approximation of the W state with minimal ancillae. Granting access to the FAN-OUT operation, upgrading to the QAC0 f circuit class, scientists achieved exact preparation of arbitrary-weight Dicke states, again with polynomial ancillae. Measurements confirm that this bypasses the limitations imposed by Lieb-Robinson bounds, which typically restrict entanglement generation time in systems with local interactions.

The breakthrough delivers constant quantum time generation of complex, permutation-invariant entanglement, consistent with the physics of long-range interacting systems. These protocols leverage architectures with global interactions, such as those found in trapped ion systems utilizing Mølmer, Sørensen interactions and neutral atom arrays employing Rydberg blockade mechanisms. A recent demonstration showcased transversal logical multi-qubit CZ gates and constant-depth multi-body logic across 48 logical qubits simultaneously, validating the feasibility of these approaches. The work’s implications extend to quantum complexity theory, offering a novel path toward resolving a major open question regarding the separation between the QAC0 and QAC0 f circuit classes, as protocols achieve preparation of arbitrary-weight Dicke states in QAC0 f, but only constant-weight states in QAC0.