[Paper Review] Adaptive constant-depth circuits for manipulating non-abelian anyons

Sergey Bravyi, Isaac H. Kim+2 more|arXiv (Cornell University)|May 4, 2022

Quantum and electron transport phenomena22 citations

TL;DR

The paper shows that for solvable groups G, ground state preparation, creation of anyon pairs at arbitrary distances, and non-destructive topological charge measurements in Kitaev’s quantum double model can be achieved with constant-depth adaptive circuits using local gates and mid-circuit measurements; it also proves that non-adaptive constant-depth circuits cannot implement distant ribbon operators for non-abelian G.

ABSTRACT

We consider Kitaev's quantum double model based on a finite group $G$ and describe quantum circuits for (a) preparation of the ground state, (b) creation of anyon pairs separated by an arbitrary distance, and (c) non-destructive topological charge measurement. We show that for any solvable group $G$ all above tasks can be realized by constant-depth adaptive circuits with geometrically local unitary gates and mid-circuit measurements. Each gate may be chosen adaptively depending on previous measurement outcomes. Constant-depth circuits are well suited for implementation on a noisy hardware since it may be possible to execute the entire circuit within the qubit coherence time. Thus our results could facilitate an experimental study of exotic phases of matter with a non-abelian particle statistics. We also show that adaptiveness is essential for our circuit construction. Namely, task (b) cannot be realized by non-adaptive constant-depth local circuits for any non-abelian group $G$. This is in a sharp contrast with abelian anyons which can be created and moved over an arbitrary distance by a depth-$1$ circuit composed of generalized Pauli gates.

Motivation & Objective

• Motivate and formalize why non-abelian anyons demand deeper circuits than abelian ones.
• Show constant-depth adaptive circuit constructions for ground state preparation, creating anyon pairs, and measuring topological charge for solvable groups G.
• Demonstrate the necessity of adaptivity and establish a depth lower bound for non-adaptive implementations.
• Provide explicit protocols for S3 as an illustrative example and generalize to arbitrary solvable groups.

Proposed method

• Define the quantum double model D(G) and its ribbon operators and topological charge projections.
• Prove a circuit depth lower bound showing non-adaptive constant-depth circuits cannot implement certain anyonic ribbon operators for non-abelian G.
• Construct adaptive constant-depth local circuits that prepare the ground state, implement ribbon operators, and perform topological charge measurements for solvable G.
• Use a combination of constant-depth quantum layers and efficient classical processing informed by mid-circuit measurement outcomes.
• Provide explicit implementation details for G = S3 as a concrete example.

Experimental results

Research questions

• RQ1 Can constant-depth adaptive circuits efficiently realize ground state preparation, anyon creation, and charge measurements in non-abelian quantum double models for solvable G?
• RQ2 Is adaptivity essential for achieving constant-depth implementations of ribbon operators and related topological operations in non-abelian cases?
• RQ3 What are the circuit-depth limits for non-adaptive approaches to creating distant anyon pairs in non-abelian models?
• RQ4 How do the procedures specialize to concrete groups like S3 and extend to general solvable groups?
• RQ5 What are the implications for experimental realizations of non-abelian topological order under noise and hardware constraints?

Key findings

• For non-abelian G, certain ribbon operators require extensive (linear in system size) circuit depth when implemented non-adaptively.
• There exist constant-depth adaptive circuits that realize ground state preparation, creation of anyon pairs at arbitrary distances, and topological charge measurements for any solvable G.
• Adaptivity (mid-circuit measurements with classical processing) is essential for the constant-depth constructions to work.
• The results include explicit constructions for G = S3 and a general framework to extend to arbitrary solvable groups.
• Ribbon operators for abelian groups can be implemented at depth 1, highlighting a sharp contrast with non-abelian cases.