[Paper Review] Efficiently preparing Schrödinger's cat, fractons and non-Abelian topological order in quantum devices

Ruben Verresen, Nathanan Tantivasadakarn+1 more|arXiv (Cornell University)|Dec 2, 2021

Cold Atom Physics and Bose-Einstein Condensates32 citations

TL;DR

The paper shows measurement-assisted protocols to scalably prepare long-range entangled states—1D GHZ/cat states, 2D toric and color codes, 3D fracton states, and non-Abelian S3 and D4 orders—in Rydberg atom arrays using time evolution under intrinsic Ising interactions followed by selective sublattice measurements, achieving ultra-high fidelities.

ABSTRACT

Long-range entangled quantum states -- like cat states and topological order -- are key for quantum metrology and information purposes, but they cannot be prepared by any scalable unitary process. Intriguingly, using measurements as an additional ingredient could circumvent such no-go theorems. However, efficient schemes are known for only a limited class of long-range entangled states, and their implementation on existing quantum devices via a sequence of gates and measurements is hampered by high overheads. Here we resolve these problems, proposing how to scalably prepare a broad range of long-range entangled states with the use of existing experimental platforms. Our two-step process finds an ideal implementation in Rydberg atom arrays, only requiring time-evolution under the intrinsic atomic interactions, followed by measuring a single sublattice (by using, e.g., two atom species). Remarkably, this protocol can prepare the 1D Greenberger-Horne-Zeilinger (GHZ) 'cat' state and 2D toric code with fidelity per site exceeding $0.9999$, and a 3D fracton state with fidelity $\gtrapprox 0.998$. In light of recent experiments showcasing 3D Rydberg atom arrays, this paves the way to the first experimental realization of fracton order. While the above examples are based on efficiently preparing and measuring cluster states, we also propose a multi-step procedure to create $S_3$ and $D_4$ non-Abelian topological order in Rydberg atom arrays and other quantum devices -- offering a route towards universal topological quantum computation.

Motivation & Objective

• Motivate the need for preparing long-range entangled states for quantum metrology and information tasks.
• Propose a finite-time, measurement-assisted framework to overcome unitary-no-go theorems for state preparation.
• Demonstrate high-fidelity pathways to 1D GHZ/cat states, 2D toric and color codes, and 3D fracton states.
• Extend the approach to non-Abelian topological orders to enable richer quantum information processing.

Proposed method

• Use time-evolution under intrinsic Ising-like interactions in Rydberg atom arrays to generate cluster states.
• Measure a sublattice (potentially dual-species) to project onto long-range entangled states.
• Exploit symmetry-protected topological (SPT) phase structure to ensure robustness against certain imperfections.
• Quantify fidelity via cluster stabilizers and GHZ-fidelity bounds derived from stabilizer expectations.
• Adapt to 2D and 3D lattice geometries (honeycomb, Lieb/square, hexagonal prism) to realize toric, color, and fracton orders.
• Extend to multi-step sequences to realize non-Abelian topological orders (D4, S3) through gauging-like procedures.

Experimental results

Research questions

• RQ1 Can measurement-based protocols, with minimal gate overhead, prepare 1D GHZ/cat states efficiently in Rydberg arrays?
• RQ2 Can similar protocols yield 2D toric and color codes, and 3D fracton states with high fidelity in realistic platforms?
• RQ3 Is it feasible to engineer non-Abelian topological orders (D4, S3) via finite-depth sequences of time evolution and measurements in Rydberg systems?
• RQ4 What are the robustness and fidelity limits imposed by longer-range interactions and imperfect timing in these schemes?

Key findings

• 1D GHZ/cat state prepared with fidelity per site > 0.9999 on idealized Ising dynamics.
• 2D toric code state prepared with fidelity per site > 0.9999 on honeycomb geometry after measuring the A sublattice in Y-basis.
• 3D fracton state prepared with fidelity per site ≈ 0.998 under the fracton-prone 3D hexagonal prism lattice.
• Non-Abelian D4 order realizable via a two-stage, measurement-guided protocol combining toric and color-code resources.
• Possible realization of S3 non-Abelian order through a finite-depth sequence on qutrits leveraging Z3 toric code and gauging concepts.
• Stability analyses indicate robustness against longer-range Ising couplings, with dominant corrections arising at expected finite distances (e.g., r = √7 a for color code; r = 3a for 1D case).