[Paper Review] Non-Abelian Topological Order and Anyons on a Trapped-Ion Processor

Mohsin Iqbal, Nathanan Tantivasadakarn+16 more|arXiv (Cornell University)|May 5, 2023

Quantum optics and atomic interactions74 references18 citations

TL;DR

The paper reports the first unambiguous realization of non-Abelian topological order (D4) on a 27-qubit kagome lattice using adaptive circuits on a trapped-ion processor, demonstrating ground-state degeneracy, non-Abelian anyon braiding, and Borromean-ring braiding.

ABSTRACT

Non-Abelian topological order (TO) is a coveted state of matter with remarkable properties, including quasiparticles that can remember the sequence in which they are exchanged. These anyonic excitations are promising building blocks of fault-tolerant quantum computers. However, despite extensive efforts, non-Abelian TO and its excitations have remained elusive, unlike the simpler quasiparticles or defects in Abelian TO. In this work, we present the first unambiguous realization of non-Abelian TO and demonstrate control of its anyons. Using an adaptive circuit on Quantinuum's H2 trapped-ion quantum processor, we create the ground state wavefunction of $D_4$ TO on a kagome lattice of 27 qubits, with fidelity per site exceeding $98.4\%$. By creating and moving anyons along Borromean rings in spacetime, anyon interferometry detects an intrinsically non-Abelian braiding process. Furthermore, tunneling non-Abelions around a torus creates all 22 ground states, as well as an excited state with a single anyon -- a peculiar feature of non-Abelian TO. This work illustrates the counterintuitive nature of non-Abelions and enables their study in quantum devices.

Motivation & Objective

• Motivate and realize non-Abelian topological order as a platform for fault-tolerant quantum information processing.
• Prepare and control the ground state subspace of a D4 topological order model on a kagome lattice.
• Demonstrate non-Abelian anyon creation, braiding, and fusion within a trapped-ion quantum processor.
• Explore logical sectors and how braiding affects fusion channels in a non-Abelian setting.

Proposed method

• Use an adaptive, finite-depth circuit with measurement and feed-forward to prepare the D4 topological order ground state on 27 qubits arranged on a kagome lattice with periodic boundary conditions.
• Implement the 12-body star operator A_s and three-body triangle operator B_t as per the Hamiltonian H = - sum_s A_s - sum_t B_t (Eq. 1).
• Prepare the ground state by entangling ancilla qubits with non-Clifford exp(iπ/8 ZZZ) gates, then use CNOTs, and perform X-basis ancilla measurements with feed-forward to achieve deterministic state preparation.
• Define and utilize color-decorated logical Z-operators and X-operators to access and toggle between the 22 ground-state sectors.
• Demonstrate non-Abelian anyon braiding, fusion, and Borromean ring braiding through interferometry and Hadamard tests.
• Characterize ground-state degeneracy and fusion outcomes to illustrate non-Abelian fusion channels and the non-trivial action of braiding on the ground-state manifold.

Experimental results

Research questions

• RQ1 Can non-Abelian topological order be prepared and manipulated on a quantum processor with high fidelity?
• RQ2 How do non-Abelian anyons in the D4 model behave under braiding, fusion, and logical sector manipulation on a trapped-ion platform?
• RQ3 What is the ground-state degeneracy structure and its relation to non-Abelian fusion channels in this system?
• RQ4 Can Borromean braiding be observed experimentally and distinguished from Abelian braiding?
• RQ5 What are the practical requirements (circuit depth, feed-forward) to realize such states on near-term quantum devices?

Key findings

• Ground-state fidelity per site exceeds 98.4% before readout error correction, rising to 99.0% after correction.
• The system on 27 qubits realizes D4 non-Abelian topological order with a 22-fold ground-state degeneracy (not a perfect square).
• Non-Abelian anyons m_R, m_G, m_B can be created, moved, and fused, revealing nontrivial fusion channels including excited states with single anyons.
• Braiding a G around a B non-Abelian anyon toggles fusion channels to e_R, demonstrated via Hadamard tests.
• Borromean-ring braiding of three non-Abelian anyons yields a nontrivial phase (approximately 1.02π) detectable via interferometry, distinct from Abelian cases.
• Tunneling non-Abelians around a torus produces all 22 ground states plus a state with a single anyon, consistent with theoretical fusion rules.