Asymptotically Optimal Topological Quantum Compiling

In a topological quantum computer, braids of non-Abelian anyons in a (2+1)-dimensional space-time form quantum gates, whose fault tolerance relies on the topological, rather than geometric, properties of the braids. Here we propose to create and exploit redundant geometric degrees of freedom to improve the theoretical accuracy of topological single- and two-qubit quantum gates. We demonstrate the power of the idea using explicit constructions in the Fibonacci model. We compar...

Non-Abelian topological orders offer an intriguing path towards fault-tolerant quantum computation, where information can be encoded and manipulated in a topologically protected manner immune to arbitrary local noises and perturbations. However, realizing non-Abelian topologically ordered states is notoriously challenging in both condensed matter and programmable quantum systems, and it was not until recently that signatures of non-Abelian statistics were observed through dig...

A fundamental question in the theory of quantum computation is to understand the ultimate space-time resource costs for performing a universal set of logical quantum gates to arbitrary precision. Here we demonstrate that non-Abelian anyons in Turaev-Viro quantum error correcting codes can be moved over a distance of order the code distance, and thus braided, by a constant depth local unitary quantum circuit followed by a permutation of qubits. Our gates are protected in the s...

Fibonacci string-net condensate, a complex topological state that supports non-Abelian anyon excitations, holds promise for fault-tolerant universal quantum computation. However, its realization by a static-lattice Hamiltonian has remained elusive due to the inherent high-order interactions demanded. Here, we introduce a scalable dynamical string-net preparation (DSNP) approach, suitable even for near-term quantum processors, that can dynamically prepare the state through rec...

We investigate the topological quantum compilation of two-qubit operations within a system of Fibonacci anyons. Our primary goal is to generate gates that are approximately leakage-free and equivalent to the controlled-NOT (CNOT) gate up to single-qubit operations. These gates belong to the local equivalence class [CNOT]. Additionally, we explore which local equivalence classes of two-qubit operations can be naturally generated by braiding Fibonacci anyons. We discovered that...

The Fibonacci topological order is the prime candidate for the realization of universal topological quantum computation. We devise minimal quantum circuits to demonstrate the non-Abelian nature of the doubled Fibonacci topological order, as realized in the Levin-Wen string net model. Our circuits effectively initialize the ground state, create excitations, twist and braid them, all in the smallest lattices possible. We further design methods to determine the fusion amplitudes...

We present a systematic numerical method to compute the elementary braiding operations for topological quantum computation (TQC). Braiding non-Abelian anyons is a crucial technique in TQC, offering a topologically protected implementation of quantum gates. However, obtaining matrix representations for braid generators can be challenging, especially for systems with numerous anyons or complex fusion patterns. Our proposed method addresses this challenge, allowing for the inclu...

The fusion basis of Fibonacci anyons supports unitary braid representations that can be utilized for universal quantum computation. We show a mapping between the fusion basis of three Fibonacci anyons, $\{|1\rangle, |\tau\rangle\}$, and the two length 4 Dyck paths via an isomorphism between the two dimensional braid group representations on the fusion basis and the braid group representation built on the standard $(2,2)$ Young diagrams using the Jones construction. This corre...

Exact synthesis is a tool used in algorithms for approximating an arbitrary qubit unitary with a sequence of quantum gates from some finite set. These approximation algorithms find asymptotically optimal approximations in probabilistic polynomial time, in some cases even finding the optimal solution in probabilistic polynomial time given access to an oracle for factoring integers. In this paper, we present a common mathematical structure underlying all results related to the ...

We review the q-deformed spin network approach to topological quantum field theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. The simplest case of these models is the Fibonacci model, itself universal for quantum computation. We here formulate these braid group representations in a shape suitable for computation and algebraic work.