Experimental realization of a topologically protected Hadamard gate via braiding Fibonacci anyons Yu-ang Fan1Yingcheng Li2Yuting Hu3Yishan Li1Xinyue Long1Hongfeng Liu1 Xiaodong Yang1Xinfang Nie1Jun Li1Tao Xin1Dawei Lu1and Yidun Wan2

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Experimental realization of a topologically protected Hadamard gate via braiding Fibonacci anyons

Yu-ang Fan,1Yingcheng Li,2Yuting Hu,3Yishan Li,1Xinyue Long,1Hongfeng Liu,1

Xiaodong Yang,1Xinfang Nie,1Jun Li,1Tao Xin,1Dawei Lu,1, ∗and Yidun Wan2, †

1Shenzhen Institute for Quantum Science and Engineering and Department of Physics,

Southern University of Science and Technology, Shenzhen 518055, China

2State Key Laboratory of Surface Physics, Department of Physics, Center for Field Theory and Particle Physics,

and Institute for Nanoelectronic devices and Quantum computing, Fudan University,

Shanghai 200433, and Shanghai Qi Zhi Institute, Shanghai 200030, China

3School of Physics, Hangzhou Normal University, Hangzhou 311121, China

Topological quantum computation (TQC) is one of the most striking architectures that can realize fault-

tolerant quantum computers. In TQC, the logical space and the quantum gates are topologically protected, i.e.,

robust against local disturbances. The topological protection, however, requires rather complicated lattice mod-

els and hard-to-manipulate dynamics; even the simplest system that can realize universal TQC–the Fibonacci

anyon system–lacks a physical realization, let alone braiding the non-Abelian anyons. Here, we propose a disk

model that can realize the Fibonacci anyon system, and construct the topologically protected logical spaces

with the Fibonacci anyons. Via braiding the Fibonacci anyons, we can implement universal quantum gates

on the logical space. Our proposal is platform-independent. As a demonstration, we implement a topological

Hadamard gate on a logical qubit through a sequence of 15 braiding operations of three Fibonacci anyons with

merely 2nuclear spin qubits. The gate ﬁdelity reaches 97.18% by randomized benchmarking. We further prove

by experiment that the logical space and Hadamard gate are topologically protected: local disturbances due to

thermal ﬂuctuations result in a global phase only. Our work is a proof of principle of TQC and paves the way

towards fault-tolerant quantum computation.

Among all the schemes of quantum computation, topolog-

ical quantum computation (TQC)1stands out because of its

fault tolerance due to topological protection: In a topological

quantum computer, the logical computational space is a sub-

space of the Hilbert space of certain number of non-Abelian

anyons, whose braiding implements the logical gates; such

logical gates and logical computing spaces are topologically

protected against local disturbances, and are thus robust.

Majorana fermions have been suggested to realize TQC2–4,

but are unsatisfactory because they cannot realize for exam-

ple the Hamamard gate and hence non-universal5. The sim-

plest non-Abelian anyons that can realize universal TQC are

the Fibonacci anyons6–8. Unfortunately, physical realizations

of controllable Fibonacci anyons are still missing, let alone

more complicated anyons. This conundrum is ascribed to two

main reasons. On the one hand, a real material that bear

Fibonacci anyons is still unknown. On the other hand, lat-

tice models of Fibonacci anyons are complicated, and there

has not been any lattice model in which all quasiparticles are

Fibanocci anyons. For example, the string-net model9–12 is a

lattice model that accomadates not only Fibonacci anyons but

also anti-chiral Fibonacci anyons and their composites. Real-

izing such models with Fibonacci anyons demands numerous

degrees of freedom. The worse is, identifying and manipu-

lating these Fibonacci anyons are beyond the reach of state-

of-the-art technologies. We however ﬁnds a way out of this

conundrum.

In this paper, we ﬁrst propose a lattice model on the disk

describing a Fibonacci anyon system, in which the bound-

ary spectrum is chiral, i.e., only the Fibonacci anyons can be

∗ludw@sustech.edu.cn

†ydwan@fudan.edu.cn

exited at the boundary of the disk and then braided to im-

plement logical quantum gates on the logical qubits encoded

in the Hilbert space of the Fibonacci anyons. Realization of

our proposal is platform independent, i.e., can be done in any

controllable system of physical qubits. As a demonstration

of our proposal, we then implement a topological Hadamard

gate on a logical qubit through a sequence of 15 braiding oper-

ations of 3 boundary Fibonacci anyons with merely 2 nuclear

spin qubits. This result contrasts previous works that spends

3 qubits to realize only the ground states (no anyons) of a

Fibonacci string-net model13 and that spends 4 qubits to re-

alize only the ground states of a toric code model14,15. The

gate ﬁdelity is 97.18% by randomized benchmarking (RB).

Via purity benchmarking (PB), we found that the origin of the

inﬁdelity is the incoherent error generally caused by the de-

phasing of our physical system.

A working topological quantum computer may suffer local

disturbances due to thermal ﬂuctuations. A topological quan-

tum computer works at controlled, extremely low tempera-

ture, at which the thermal ﬂuctuations cannot produce any real

Fibonacci anyons on top of the existing Fibonacci anyons used

for computation. Then the thermal ﬂuctuations can only pro-

duce paired Fibonacci anyons, which may interfere with the

braiding of the nearby real Fibonacci anyons. Fortunately, the

logical space and gates have been argued not to be affected at

all by such disturbances6, viz topologically protected. In this

paper, we prove the topological protection by experiment: lo-

cal disturbances due to thermal ﬂuctuations result in a global

phase only.

By realizing the topological Hadamard gate on three Fi-

bonacci anyons and demonstrating the topological protection,

our work is a proof of principle of Fibonacci-anyon based

TQC.

arXiv:2210.12145v1 [quant-ph] 21 Oct 2022

Figure 1. Fibonacci anyon system and the model. a, Doubled Fibonacci topological phase with a gapped boundary. The bumps on the disk

are possible bulk excitations but irrelevant. The balls on the boundary depict boundary Fibonacci anyons. b, Trivalent lattice that discretizes

the disk. Boundary Fibonacci anyons (orange balls) reside on open edges. c, Three boundary Fibonacci anyons encoding the logical qubit. The

braid generator σ12 (σ23) braids the worldlines of the ﬁrst and the second (the second and the third) Fibonacci anyons. d, Minimal subsystem

(conisiting of two degrees of freedom jand k) that bears three boundary Fibonacci anyons.

A Fibonacci anyon system and the logical qubit

Our model describes the doubled Fibonacci topological order

with a gapped boundary. Along the gapped boundary, there

can be Fibonacci anyons (Fig. 1a). The model is deﬁned on

a disk, discretized by a trivalent lattice, as shown in Fig. 1b.

The open edges and the internal edges respectively carry the

boundary and bulk degrees of freedom, which take value in

{0,1}. The Hilbert space of the model is spanned by all possi-

ble conﬁgurations of the degrees of freedom on the edges. An

open edge taking value 1bears a boundary Fibonacci anyon.

We can braid the boundary Fibonacci anyons by moving them

around each other through the bulk. Braiding the Fibonacci

anyons can be represented by certain matrices acting on the

Hilbert space of the model.

According to Ref.6, we need in our model three neighbour-

ing boundary Fibonacci anyons to construct a logical qubit.

We can braid the three Fibonacci anyons to implement the

single-qubit logical gates. All possible braiding operations of

the three Fibonacci anyons can be generated by two basic op-

erations, σ12 and σ23 (see Fig. 1c), and their inverses. The

operation σ12 (σ23) exchanges the left (right) two Fibonacci

anyons’ worldline on top. See the Seupplemental Information

for more details of the model.

As a subspace of the Hilbert space of our model, the logi-

cal qubit is 2-dimensional and is invariant under braiding the

three boundary Fibonacci anyons. To operate on the logical

qubit, we need to express the braid generators σ12 and σ23

explicitly as matrices, which depend on the positions of the

three Fibonacci anyons on the boundary. In order to realize a

logical qubit with the smallest number of physical qubits, we

bipartite the system into two disentangled parts: a minimal

subsystem (shown in Fig. 1b) that bears three boundary Fi-

bonacci anyons, supporting a logical qubit, and the subsystem

consisting the rest degrees of freedom of the total system. The

minimal subsystem has only two degrees of freedom jand

k. In this setting, any braiding operation of the three bound-

ary Fibonacci anyons decomposes into the tensor product of a

4×4matrix acting on ∣j, k⟩and an identity matrix on the other

subsystem.

As such, the logical space in the basis of ∣j, k⟩is spanned

by ∣0⟩L=1

φ∣01⟩+√1

φ∣11⟩, and ∣1⟩L=−1

φ3/2∣01⟩+√1

φ∣10⟩+

φ2∣11⟩, where φ=1+√5

2is the golden ratio. See Methods for

the two braiding generators σ12 and σ23. Details of construct-

ing the logical qubit can be found in the Supplemental Infor-

mation. One can realize any single-qubit gate on the logical

qubit with arbitrary high precision by sequentially implement-

ing the σ12 and σ23. As an example, a Hadamard gate can be

realized by implementing the following sequence of braiding

operations16

HL=(σ12)4(σ23 )−2(σ12)2(σ23)−2(σ12)2(σ23)2×

(σ12)−2(σ23 )4(σ12)2(σ23)−2(σ12)−2(σ23)2(σ12)2.(1)

To realize a single-qubit topological quantum computer,

we shall simulate the subsystem in Fig. 1b and the logical

Hadamard gate Eq. (1). Using our model, it is straightforward

to scale up the topological quantum computer by considering

larger subsystems.

Experimental implementation of the Hadamard gate

We use the nuclear magnetic resonance (NMR) quantum

Hadamard gate17. The 13 C and 1H spins serve as two phys-

ical qubits, each being controlled by a radio-frequency (rf)

ﬁeld18,19. The total Hamiltonian is20

HNMR =πJ

2ˆσ1

zˆσ2

z+2

∑

i=1

πBi(cos φiˆσi

x+sin φiˆσi

y),(2)

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ExperimentalrealizationofatopologicallyprotectedHadamardgateviabraidingFibonaccianyonsYu-angFan,1YingchengLi,2YutingHu,3YishanLi,1XinyueLong,1HongfengLiu,1XiaodongYang,1XinfangNie,1JunLi,1TaoXin,1DaweiLu,1,andYidunWan2,1ShenzhenInstituteforQuantumScienceandEngineeringandDepartmentofPhysics,Souther...

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Experimental realization of a topologically protected Hadamard gate via braiding Fibonacci anyons Yu-ang Fan1Yingcheng Li2Yuting Hu3Yishan Li1Xinyue Long1Hongfeng Liu1 Xiaodong Yang1Xinfang Nie1Jun Li1Tao Xin1Dawei Lu1and Yidun Wan2.pdf

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Experimental realization of a topologically protected Hadamard gate via braiding Fibonacci anyons Yu-ang Fan1Yingcheng Li2Yuting Hu3Yishan Li1Xinyue Long1Hongfeng Liu1 Xiaodong Yang1Xinfang Nie1Jun Li1Tao Xin1Dawei Lu1and Yidun Wan2

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