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

TL;DR

The paper tackles the challenge of preparing long-range entangled states that resist scalable unitary synthesis by introducing measurement-assisted protocols aligned with existing quantum hardware, notably Rydberg atom arrays. It demonstrates a two-step approach—dynamic Ising evolution to form cluster states, followed by selective sublattice measurements—to realize 1D GHZ cat states, 2D toric/color codes, and 3D fracton states with fidelities per site approaching 0.9999–0.9998. Beyond Abelian orders, it presents finite-depth schemes for non-Abelian topological orders, namely D4 and S3, using multi-sublattice gauging and measurement sequences, offering a concrete pathway toward universal topological quantum computation. The results, supported by robust analyses against long-range interactions and timing imperfections, position fracton order and non-Abelian anyons within experimental reach and applicable to multiple platforms, with potential impacts on metrology, quantum error correction, and fault-tolerant quantum computing.

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.

Figures (10)

• Figure 1: Fast preparation of GHZ state by measuring 1D cluster state. (a) The protocol starts with a product state in the $X$-basis which we let time-evolve under an Ising interaction. After a time $t=t_\textrm{SPT}$, this produces the cluster state. Measuring the red sites in the $X$-basis leads to a cat state. Using the information of the measurement outcomes, one can flip the appropriate spins to obtain the GHZ state on the blue sites Briegel01. (b) If we do not time-evolve with exactly $t = t_\textrm{SPT}$, the post-measurement state is only an approximate cat state. Nevertheless, the resulting correlation length (within which there is long-range order) is very large even for moderate time deviations. (c) A tensor-network simulation of the protocol for Rydberg atoms interacting with a $1/r^6$ van der Waals interaction. We simulate this for Rb $70S_{1/2}$ with two different lattice spacings, which give different results due to incorporating the fact that the initial product state preparation pulses are not instantaneous (see \ref{['methods']}). (d) Contributions from longer-range van der Waals interactions can be systematically suppressed by interspersing time-evolution with $X$-pulses (orange) on particular sublattices (see \ref{['methods']}). This particular example cancels out couplings at distance $r=2a$, although this was not used to achieve the results in (c).
• Figure 2: From toric and color code to non-Abelian topological order. (a) We place Rydberg atoms on the vertices (red) and bonds (blue) of the honeycomb lattice. Initializing into the product state $|-\rangle^{\otimes N}$ and time-evolving for $t=\pi/V(a)$, we obtain the cluster state whose two types of stabilizers are depicted. Measuring the Rydberg atoms on the red sites in the $Y$-basis produces the toric code state on the blue sites up to known single-site spin flips. Corrections to topological order are given by longer-range van der Waals interactions connecting the red and blue sublattices; the leading correction has a pre-factor $1/\sqrt{7}^6 \approx 0.003$. (b) If one instead puts atoms on the centers (red) and vertices (blue) of the hexagons, then time-evolution and measurement of the red sites produces the color code. Subsequently loading atoms on the links (purple) can produce $D_4$ non-Abelian topological order after another round of time-evolution and measuring the blue sites (see Section \ref{['sec:nonabelian']}).
• Figure 3: Realizing a fractal spin liquid with fracton order. (a) Rydberg atoms are placed at the red and blue sites on the hexagonal prism lattice. After time-evolving under the Rydberg interaction and measuring the red sites, we obtain the ground state of the so-called Sierpinski prism model Yoshida13 (see \ref{['methods']}). (b) Quasiparticles have restricted mobility in the hexagonal layers due to being created at the corners of fractal (Sierpinski) operators.
• Figure 4: Towards $\mathbb Z_3$ and $S_3$ non-Abelian topological order. (a) Time-evolving qutrits (formed by pairs of Rydberg-blockaded atoms) on the vertices and bonds of this 3D configuration and measuring the $A$ sublattice creates the $\mathbb Z_3$ toric code with a non-Abelian defect in the corner. This effectively realizes a square lattice with a disclination; an $e$-anyon traveling around it transforms into its conjugate $\bar{e}$. (b) Starting with the $\mathbb Z_3$ toric code on the $B$ sublattice, we load Rydberg atoms on the $C$ and $D$ sublattices. An appropriate sequence of time-evolution and pulses (see main text), followed by measuring the $C$ sublattice, produces non-Abelian $S_3$ topological order.
• Figure A.1: The distribution of the correlation length $\xi$ over a distance $n = \bar{\xi}$ for $t=0.99 \times t_\textrm{SPT}$ (see Eq. \ref{['eq:postmeasurement_xi']}. The black line is the predicted normal distribution (see Eq. \ref{['eq:normal']}).

Theorems & Definitions (1)

• proof