[Paper Review] Shortest Route to Non-Abelian Topological Order on a Quantum Processor

Nathanan Tantivasadakarn, Ruben Verresen+1 more|arXiv (Cornell University)|Sep 8, 2022

Quantum Computing Algorithms and Architecture44 references4 citations

TL;DR

This paper proposes a single-layer measurement protocol using only finite-depth quantum circuits and single-shot measurements to prepare non-Abelian topological order—specifically $D_4$ anyons—on near-term quantum processors, bypassing the need for feed-forward or adiabatic state preparation. The key result is that certain non-Abelian states with a Lagrangian subgroup can be created efficiently, even on hardware with limited connectivity, as demonstrated via a depth-11 circuit on Google’s quantum processors.

ABSTRACT

A highly coveted goal is to realize emergent non-Abelian gauge theories and their anyonic excitations, which encode decoherence-free quantum information. While measurements in quantum devices provide new hope for scalably preparing such long-range entangled states, existing protocols using the experimentally established ingredients of a finite-depth circuit and a single round of measurement produce only Abelian states. Surprisingly, we show there exists a broad family of non-Abelian states -- namely those with a Lagrangian subgroup -- which can be created using these same minimal ingredients, bypassing the need for new resources such as feed-forward. To illustrate that this provides realistic protocols, we show how $D_4$ non-Abelian topological order can be realized, e.g., on Google's quantum processors using a depth-11 circuit and a single layer of measurements. Our work opens the way towards the realization and manipulation of non-Abelian topological orders, and highlights counter-intuitive features of the complexity of non-Abelian phases.

Motivation & Objective

• To develop a scalable, non-adiabatic method for preparing non-Abelian topological order on near-term quantum processors.
• To overcome the limitation of existing measurement-based protocols that only produce Abelian anyons using finite-depth circuits and single measurement layers.
• To eliminate the need for feed-forward control in preparing non-Abelian anyons, which is a major experimental bottleneck.
• To demonstrate that non-Abelian topological order can be created with a time-independent protocol, independent of system size.
• To identify and realize a minimal quantum circuit architecture for $D_4$ non-Abelian topological order on existing hardware such as Google’s quantum processors.

Proposed method

• The protocol uses a finite-depth quantum circuit (depth-11 for Google’s hardware) to entangle qubits into a state with $D_4$ topological order prior to measurement.
• Single-site projective measurements are applied to gauge-invariant operators associated with a Lagrangian subgroup of anyons, including $e_1e_2$, $m_1m_2$, and $s$ in the $D_4$ theory.
• The measurement process projects the system into a non-Abelian topological order by condensing anyons in the Lagrangian subgroup, which maps to a trivial state under condensation.
• The protocol avoids feed-forward by ensuring the measurement basis is invariant under the relevant symmetry (e.g., swap symmetry between layers), enabling single-shot preparation.
• For hardware with limited connectivity, SWAP gates are used to reconfigure qubit couplings, enabling implementation on square-lattice architectures like Google’s processors.
• The protocol is generalizable to other non-Abelian topological orders with a Lagrangian subgroup, such as $Q_8$ quantum double or doubled Ising, via analogous measurement of Gauss law operators.

Experimental results

Research questions

• RQ1 Can non-Abelian topological order be prepared in a single measurement layer without feed-forward, using only finite-depth unitary circuits?
• RQ2 Which non-Abelian topological orders admit a minimal measurement-based preparation protocol using only standard quantum processor operations?
• RQ3 Can the $D_4$ anyonic model be realized on current superconducting quantum processors with realistic connectivity and gate depth?
• RQ4 What is the minimal number of measurement layers required to prepare non-Abelian topological order, and how does this compare to adiabatic or multi-step protocols?
• RQ5 How can non-Abelian topological order be diagnosed experimentally using entanglement signatures?

Key findings

• A single-layer measurement protocol can prepare $D_4$ non-Abelian topological order on a quantum processor without requiring feed-forward or adiabatic evolution.
• The protocol achieves a depth-11 circuit on Google’s quantum processors (13 layers when decomposed into native CZ gates), independent of system size.
• The method works by measuring Gauss law operators associated with a Lagrangian subgroup of anyons, which ensures the resulting state is non-Abelian topological order.
• The protocol is generalizable to other non-Abelian topological orders such as $Q_8$ quantum double and doubled Ising, provided they admit a Lagrangian subgroup.
• Non-Abelian anyon entanglement serves as a smoking gun signature: inserting non-Abelian anyons changes the entanglement entropy according to their quantum dimension.
• The protocol avoids the need for multi-step or feed-forward protocols, offering a scalable and experimentally feasible route to non-Abelian anyons on near-term devices.