Abstract
Quantum manybody systems with a nonAbelian topological order can host anyonic quasiparticles. It has been proposed that anyons could be used to encode and manipulate information in a topologically protected manner that is immune to local noise, with quantum gates performed by braiding and fusing anyons. Unfortunately, realizing nonAbelian topologically ordered states is challenging, and it was not until recently that the signatures of nonAbelian statistics were observed through digital quantum simulation approaches. However, not all forms of topological order can be used to realize universal quantum computation. Here we use a superconducting quantum processor to simulate nonAbelian topologically ordered states of the Fibonacci stringnet model and demonstrate braidings of Fibonacci anyons featuring universal computational power. We demonstrate the nontrivial topological nature of the quantum states by measuring the topological entanglement entropy. In addition, we create two pairs of Fibonacci anyons and demonstrate their fusion rule and nonAbelian braiding statistics by applying unitary gates on the underlying physical qubits. Our results establish a digital approach to explore nonAbelian topological states and their associated braiding statistics with current noisy intermediatescale quantum processors.
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Main
The discovery of topological order^{1} has revolutionized the understanding of quantum matter based on the Landau–Ginzburg symmetrybreaking paradigm^{2}. Different topologically ordered phases could bear exactly the same symmetries and showcase topologically distinct features, such as longrange entanglement and the emergence of quasiparticles with anyonic braiding statistics^{3,4,5,6}. They are of fundamental importance in understanding strongly correlated quantum phases of matter, and promise crucial applications in faulttolerant quantum computing as well^{7}. Owing to their intrinsic nonlocal nature, logical code spaces immune to arbitrary local perturbations can be constructed from the topological degrees of freedom of the system, and logical operations can be implemented by creating, braiding and fusing anyons. In general, the braiding of two anyons can be described by either Abelian or nonAbelian statistics, which leads to a complex phase factor or a unitary matrix acting on the degeneratestate manifold, respectively. NonAbelian anyons are quasiparticle excitations in topologically ordered systems that obey nonAbelian braiding statistics. They are the building blocks of topological quantum computing^{8}.
Realizing nonAbelian topologically ordered states and their associated nonAbelian anyons has been a longsoughtafter goal in condensedmatter physics^{8,9}. Exciting progresses have been made in both theory^{10,11,12,13,14,15} and experiment^{16,17,18,19,20,21}. Yet, the direct observation of nonAbelian exchange statistics has remained elusive so far. In recent years, notable advances have been achieved towards the fabrication of programmable quantum platforms such as superconducting circuits^{22,23,24,25}, Rydberg atomic arrays^{26}, photons^{27,28} and trapped ions^{29,30}, giving rise to unprecedented opportunities in the synthesis and exploration of increasingly complex topological quantum states^{31,32,33,34}. Along this direction, nonAbelian statistics has been recently observed by simulating the projective Ising anyons in the toriccode model^{35,36} and creating the groundstate wavefunction of nonAbelian D_{4} topological order^{37}. However, neither of the braidings of anyons realized in these experiments alone sustain a universal gate set. The Ising anyons are related to the Witten–SU(2)–Chern–Simons theory at level k = 2, where the SU(2) model is computationally universal for k = 3 or k ≥ 5 (ref. ^{38}). For the quantum double model of the finite group (including D_{4}), the gate set realized by braiding is finite and is not universal^{39}. With current noisy intermediatescale quantum processors, realizing topological orders hosting nonAbelian anyons with universal computational power demands an elaborate design of deviceadapted quantum circuits combined with the stateoftheart gate fidelity and coherence time, which is exceedingly challenging and has evaded experiment thus far.
Here we report the experimental realization of Fibonacci stringnet states^{6,40}, which are predicted to host nonAbelian Fibonacci anyons carrying universal computational power^{41} (Fig. 1a), with 27 superconducting transmon qubits. We upgrade our device by optimizing the device fabrication and controlling process, and execute efficient quantum circuits obtained by variational algorithms to prepare the desired nonAbelian ground state of the stringnet Hamiltonian. We measure the multibody vertex and plaquette operators, yielding average expectation values of 0.94 and 0.58, respectively. The topological order of the prepared states is characterized by measuring the topological entanglement entropy, whose averaged value reaches −0.82, which is well below zero (for a topologically trivial state), and −0.69 (for the Z_{2} topologically ordered toriccode state). In addition, we create two pairs of Fibonacci quasiparticle excitations by acting string operators on the prepared ground state and demonstrate their nontrivial mutual statistics by braiding them with sequences supporting universal singlequbit logic gates. We extract the characterizing monodromy matrix and the quantum dimension of the Fibonacci anyon from the measured fusion results, which unambiguously indicates that the quasiparticle excitations created in our experiment are indeed Fibonacci anyons.
Framework and experimental setup
We consider the Fibonacci stringnet model—the Levin–Wen model—which is the simplest stringnet model supporting braidinguniversal topological quantum computing^{6}. The corresponding Hamiltonian is defined on a honeycomb lattice with spins living on the edges (Fig. 1b):
where Q_{v} denotes the threebody vertex operator that constrains the string types meeting at a trivalent vertex, and B_{p} denotes the twelvebody plaquette operator that measures the ‘magnetic flux’ through a plaquette and provides the dynamics for stringnet configurations^{6}. The ground state of H is topologically ordered and satisfies 〈Q_{v}〉 = 〈B_{p}〉 = 1 for all vertices v and plaquettes p. The quasiparticle excitations are Fibonacci anyons satisfying the following fusion rule:
where 1 and τ denote the vacuum and Fibonacci anyon, respectively. They can be created and manipulated by string operators^{42} (Fig. 1b,c). Apparently, preparing the ground state of H and manipulating Fibonacci anyons pose a serious challenge due to the intricate multibody plaquette operators involved in the model. To overcome this difficulty, we optimize our device and exploit efficient quantum circuits, which are obtained through the variational unitary synthesis technique^{43}, to prepare the desired nonAbelian ground state and use the idea of digital quantum simulation to implement creations and braidings of Fibonacci anyons (Methods and Supplementary Section III).
Our experiments are performed on a flipchip superconducting quantum processor with frequencytunable transmon qubits arranged in a square lattice^{36}. We select 27 neighbouring qubits and construct a honeycomb lattice with three plaquettes out of the underlying square lattice (Extended Data Fig. 1). The qubits living on the edges of the honeycomb lattice are used for implementing the stringnet Hamiltonian H, and the other ones serve as ancillary qubits to facilitate the implementation of multiqubit string operators. Arbitrary singlequbit gates can be realized for each qubit, whereas twoqubit controlledZ gates can be implemented on an arbitrary neighbouring qubit pair connected by a tunable coupler. By optimizing the device fabrication and controlling process, we push the median lifetime of these qubits to 117 μs and the median simultaneous single and twoqubit gate fidelities to around 99.96% and 99.50%, respectively. This enables us to successfully prepare the desired nonAbelian topological ground state of H and implement the braidings of Fibonacci anyons with quantum circuits of depths up to 100. Supplementary Section III.A provides the calibration procedures and detailed parameters of the device.
Groundstate preparation
We prepare the ground state of H by utilizing the fact that all Q_{v} and B_{p} are projectors commuting with each other. Noting that the Nqubit product state 0〉^{⊗N} is an eigenstate of all Q_{v}, the ground state G〉 can be expressed as
where \({B}_{p}^{s}\) with s ∈ {0, 1} is a twelvebody plaquette operator and \(\phi =(\sqrt{5}+1)/2\) is the golden ratio. For an independent type0 string loop 0〉⊗⋯⊗0〉, the \({B}_{p}^{0}\) operator leaves the configuration unchanged, whereas the \({B}_{p}^{1}\) operator changes it to a type1 string loop 1〉⊗⋯⊗1〉 according to the fusion rule 1 × τ = τ (ref. ^{6}). Thus, the projector B_{p} for isolated string loops acting on the initial state 0〉^{⊗N} can be implemented by randomly choosing one qubit from the plaquette p, preparing it onto the state \(\frac{1}{\sqrt{1+{\phi }^{2}}}(\left\vert 0\right\rangle +\phi \left\vert 1\right\rangle )\) first with a singlequbit gate \({U}_{{\rm{S}}}=\frac{1}{\sqrt{1+{\phi }^{2}}}\left(\begin{array}{cc}1&\phi \\ \phi &1\end{array}\right)\) and then successively applying controlledNOT (CNOT) gates on the rest of the qubits, with the chosen qubit being the control qubit. Furthermore, we use the F moves to entangle different isolated loops. The ground state can be prepared by creating isolated loops and entangling them in the honeycomb lattice layer by layer. Such an approach is efficient, in the sense that the circuit depth scales only linearly with the number of plaquettes (Methods)^{36,44}.
The quantum circuit for the stepbystep preparation of a threeplaquette ground state is sketched in Fig. 2a, which is composed of singlequbit U_{S} gates, CNOT gates and Fmove gates. The Fmove gates can be further decomposed into multiqubitcontrolled unitary gates and CNOT gates (Fig. 2b). In our experiments, further compilations are required to fit the circuit to the nearestneighbour geometry of our quantum device with native gates (that is, arbitrary singlequbit gates and the twoqubit controlledZ gate). However, direct decomposition of the fivequbit F move is expensive and would result in a circuit with a depth of around 200 to prepare the ground state, which is impractical to reliably implement with a system size as large as 27 qubits for the stateoftheart superconducting processors. We elude this dilemma by exploiting a variational approach^{43} (Methods) to efficiently implement the three and fivequbit Fmove operations. The process infidelity between the synthetic unitary U and target unitary V, which is defined as \(1\frac{ \,{{\mbox{Tr}}}\,({U}^{{\dagger} }V) }{{4}^{n}}\), where n is the qubit number, is optimized to be below 10^{−5}. We note that this variational approach is device adapted and can substantially suppress the circuit depth for implementing F moves. Its scalability is also assured by the fact that F moves act locally and we only need to variationally approximate the F moves up to five qubits.
With this greatly simplified implementation of F moves, we first prepare the ground state of H step by step (Fig. 2a). We measure the expectation values of Q_{v} and B_{p} after each step, with the results shown in Extended Data Fig. 2. Although all the Q_{v} operators are diagonal in the computational basis and hence can be directly measured in the experiment, the B_{p} operators involve 99,328 twelvebody Pauli terms in decomposition and require 290 twelvebody measurements under different Pauli bases. The average values of Q_{v} and B_{p} after preparing the threeplaquette ground state are 0.88 and 0.36, respectively. For the preparation of the threeplaquette ground state, we can further simplify the circuit to a depth of 53 by directly targeting the final state instead of the whole unitary during the variational search, which can generate a state with an infidelity to the target state as low as 10^{−5} in theory. We prepare the ground state with this further simplified circuit as well. The measured expectation values of Q_{v} and B_{p} are displayed in Fig. 2c, with average values of 0.94 and 0.58, respectively. These apparently largerthanzero values indicate that the nonAbelian topological state prepared in our experiment indeed has a large overlap with the ideal ground state of H, showing the efficiency and effectiveness of our approaches. In the following, we use the prepared ground state through the further simplified circuit to study the exotic properties of the Fibonacci stringnet model, including distinct topological entanglement entropy and braiding statistics of Fibonacci anyons.
Topological entanglement entropy
To characterize the topological order of the prepared ground state G〉, we measure its topological entanglement entropy, which is a universal constant reflecting the topological properties of entanglement that survive at arbitrarily long distances^{45,46}. We deliberately choose three subregions A, B and C (Fig. 3a), and the topological entanglement entropy (denoted as S_{topo}) can then be obtained through
where AB indicates the union of A and B and S_{I}(I = A, B, C, AB, BC, AC, ABC) denotes the von Neumann entanglement entropy of a subsystem I: S_{I} = –Tr(ρ_{I}lnρ_{I}), where ρ_{I} is the reduced density matrix. For the stringnet model considered in our experiment, it is necessary to map the wavefunction to a new lattice with two qubits per boundary edge so that the partitioning can be implemented in a symmetric way^{46}. From the perspective of topological quantum field theory, S_{topo} is directly related to the total quantum dimension D of the medium by S_{topo} = –lnD (refs. ^{45,46}). For the Fibonacci stringnet model, we have \(D=1+{d}_{\tau }^{\,2}\), where d_{τ} = ϕ is the quantum dimension of a Fibonacci anyon.
Directly measuring S_{topo} requires quantum state tomography in general, which is resource consuming and impractical for the system size considered in this work. Alternatively, one can measure the secondorder Rényi entropy, from which S_{topo} can be estimated up to an exponentially small deviation for the Fibonacci stringnet model^{47}. In our experiment, we adopt this approach and exploit the recently developed randomized measurement method to attain S_{topo} (refs. ^{33,48,49}). We extend the ground state to a new lattice by copying the three qubits (labelled by Q_{(7,11)}, Q_{(7,13)} and Q_{(5,11)}) on the common boundary edges of the three plaquettes to the neighbouring free qubits (Q_{(9,13)}, Q_{(5,13)} and Q_{(5,9)}) with CNOT gates; therefore, each common edge is associated with two qubits and can be symmetrically separated into different subsystems^{46} (Methods and Supplementary Section I.G). The numbers in the subscript of Q denote the row and column indices of the corresponding qubits (Extended Data Fig. 1).
Our results are summarized in Fig. 3b,c. In Fig. 3b, we plot the distributions of the measured entanglement entropies of all the subsystems involved, with qubit numbers ranging from three to eleven. Ideally, the entanglement entropy of a subsystem scales linearly with its boundary, which is a reminder of the arealaw entanglement^{50} satisfied by the ground state G〉. In our experiment, all the measured entanglement entropies are slightly above the predicted values, which is consistent with numerical estimates considering the control and decoherence errors obtained during the calibration procedures (Supplementary Section III). The nine extracted S_{topo} estimates are also slightly above the predicted value (Fig. 3c). The mean value of the measured S_{topo} is −0.82, which is significantly lower than zero (for the topologically trivial state) and –ln2 ≈ 0.69 (for the Z_{2} topologically ordered state). This provides strong evidence for the Fibonacci topological order of the ground state G〉.
Braiding statistics
The topological order realized above supports a coveted type of quasiparticle—the Fibonacci anyons—whose braiding statistics can give rise to universal topological quantum computation^{8}. To demonstrate the nontrivial braiding statistics of Fibonacci anyons, we create two pairs of them from vacuum living on two plaquettes (Fig. 4a). We then braid them following different sequences by the corresponding string operators. After braiding, we fuse them pairwise and measure the fusion outcomes to detect their braiding statistics. We encode a logical qubit into four Fibonacci anyons as \(\left\vert \bar{0}\right\rangle =\left\vert {\left(\tau \times \tau \right)}_{{{{\bf{1}}}}},{\left(\tau \times \tau \right)}_{{{{\bf{1}}}}}\right\rangle\) and \(\left\vert \bar{1}\right\rangle =\left\vert {\left(\tau \times \tau \right)}_{\tau },{\left(\tau \times \tau \right)}_{\tau }\right\rangle\), and denote the braiding operations of the first and middle two anyons as σ_{1} and σ_{2}, respectively (Fig. 4b). We note that σ_{1} and σ_{2} are unitary logical gates and their matrix representation can be calculated by the F and R moves (Methods and Supplementary Section I.B).
Starting with the prepared ground state, we create two pairs of Fibonacci anyons labelled as τ_{1,2,3,4} by acting on two short type1 open strings. In the logical space, we initialize the system into state \(\left\vert \bar{0}\right\rangle\). We consider five different braiding sequences (Fig. 4c) and plot the corresponding measured fusion results in Fig. 4d(i). (1) Without braiding: as shown in Fig. 4d(i), we measure a probability of 0.81 and 0.04 for both pairs fusing to 1 and τ, respectively, which confirms the theoretical prediction that anyons created in pairs from vacuum will annihilate back into the vacuum without braiding. (2) Braiding of the middle two Fibonacci anyons once: this will change the fusion output for both pairs, resulting in a superposition state in the logical space as \({\sigma }_{2}\left\vert \bar{0}\right\rangle ={\phi }^{1}{{\rm{e}}}^{4\uppi{\rm{i}}/5}\left\vert \bar{0}\right\rangle +{\phi }^{1/2}{{\rm{e}}}^{3\uppi{\rm{i}}/5}\left\vert \bar{1}\right\rangle\). In our experiment, the measured probabilities of the two pairs after braiding fusing to 1 and τ are 0.32 and 0.56, respectively. This agrees with the theoretical prediction and verifies the nonAbelian fusion rule in equation (2); (3) and (4) preparation and verification of a logical eigenstate of σ_{2} through braidings: from the Yang–Baxter equation^{51,52}, \({\sigma }_{1}{\sigma }_{2}\left\vert \bar{0}\right\rangle\) is an eigenstate of σ_{2}. This is verified by our experimental result that the difference between the fusion results before and after implementing an additional σ_{2} on \({\sigma }_{1}{\sigma }_{2}\left\vert \bar{0}\right\rangle\) is negligible (Fig. 4d(iii),(iv)); (5) braiding of the middle two Fibonacci anyons twice, which provides information about the monodromy matrix M that characterizes the mutual statistics of Fibonacci anyons from the perspective of modular tensor category theory^{40}. The elements of M can be written in the form of a logical observable as \({M}_{\tau \tau }=\left\langle \bar{0}\right\vert {\sigma }_{2}{\sigma }_{2}\left\vert \bar{0}\right\rangle\), where M_{11}, M_{1τ} and M_{τ1} equal 1 since the braiding with vacuum 1 does not change the fusion results. From the experimental result shown in Fig. 4d(v), we obtain that M_{ττ} = −0.39, which agrees well with the theoretical value of −1/ϕ^{2} ≈ −0.38. The measured quantum dimension of the Fibonacci anyon is ϕ_{exp} = 1.60, very close to the ideal value of d_{τ} = ϕ ≈ 1.618. This gives a piece of clear evidence that the quasiparticle excitations we created in the experiment are indeed Fibonacci anyons. We note that a logical Hadamard gate has recently been implemented by simulating braiding sequences of boundary Fibonacci anyons with two nuclear spin qubits^{53}.
We mention that the braidings carried out in our experiment involve no Hamiltonian dynamics of quasiparticle excitations. As a result, they are not endowed with topological protection that naturally arises from an energy gap separating the manybody degenerate ground states from the lowlying excited states. This is distinct from conventional protocols for braiding anyons^{8}, and therefore, our experiment is more of a quantum simulation of braiding Fibonacci anyons in this sense. This also explains the evident small deviations between the experimentally measured fusion results after braidings and the ideal theoretical predictions. Without topological protection, inevitable experimental imperfections including gate errors and limited coherence time would cause a sizable infidelity for the final states after braidings. To leverage the Fibonacci anyons in our experiment for topologically protected quantum computing, an active error correction procedure such as the Fibonacci Turaev–Viro code^{54} must be enforced during the braiding process. We leave this interesting and important topic for future study.
Conclusion and outlook
In summary, we have experimentally prepared the ground state of the Fibonacci stringnet model with nonAbelian topological order on a programmable superconducting quantum processor. We demonstrated the creation, braiding and fusion of Fibonacci anyons by applying appropriate string operators on the prepared ground state. Unlike Isingtype anyons^{35,36} and those related to the D_{4} topological order^{37}, the Fibonacci anyons demonstrated in our experiment support universal topological quantum computing. Combined with the potential inclusion of the error correction procedure^{54} in the future, our results pave an alternative path towards faulttolerant quantum computation.
The controllability of the superconducting platform and the effectiveness of our variational approach in simplifying the quantum circuits demonstrated in our experiment open up several new avenues for future studies of other exotic topologically ordered states of matter, as well as their related nonAbelian quasiparticle excitations with peculiar braiding statistics. In particular, it would be interesting and important to implement the generalized stringnet models that break tetrahedral^{55} or timereversal symmetry^{40}, admit symmetryenriched topological orders^{56} and others described by unitary fusion categories with fusion multiplicities^{57}. Experimental realizations of such topologically ordered nonAbelian states would not only deepen our understanding of these unconventional phases of matter but also provide valuable guidance for potential applications.
Methods
Fixedpoint wavefunction
The Hamiltonian in equation (1) of the stringnet model is designed to capture the most essential fixedpoint wavefunctions, which are superpositions of various stringnet configurations. These configurations are characterized by the geometry and the types of an ensemble of strings. The fixedpoint wavefunction captures the universal properties of the stringnet condensed phases in (2+1) dimensions, which can describe all the socalled ‘doubled’ topological phases. Here we present the exact groundstate wavefunction in the stringnet picture and the corresponding quantum state simulated on physical qubits. Denoting the wavefunction as Φ, it is uniquely specified by the following four local constraints^{6}:
where the shaded regions represent arbitrary stringnet configurations. d_{a} is the quantum dimension of string type a and the sixindex tensor F has a onetoone correspondence to the doubled topological phases. Since we only consider the selfdual model in this work, all the string configurations discussed here are unoriented. The wavefunction Φ is precisely the ground state of the Hamiltonian in equation (1).
According to these local constraints, the general value of Φ can be exactly calculated for any stringnet configurations. For a given geometry g, the wavefunction Φ becomes a function of string types {s}. For example,
where Φ(vacuum) = 1 following the notation from another work^{40}. One can also calculate the amplitudes of different string types on two independent loops:
From these two examples, we see that the wavefunction Φ can be recognized as a function to represent the linear relations between different stringnet configurations. Once the geometry of the configuration is determined, it becomes a function of string types.
In the quantum circuit scheme, we simulate the linear relations described by symbol F, which corresponds to multicontrolled unitary gates (Fig. 2b). From equation (5d), the F move changes the type of one string according to its four connected strings, which is a fivequbit gate. We can use a simplified quantum circuit to realize the F move when there is some prior information on the stringnet configuration (Fig. 2). Here we denote the quantum circuit corresponding to the complete and simplified F move as C_{F} for brevity, whereas a detailed description can be found in Supplementary Section II.B. Now we give the quantum state that simulates the state Φ with the geometry g = 1 loop. For the Fibonacci stringnet model, Φ_{1}(s_{1} = 0) = d_{0} = 1 and Φ_{1}(s_{1} = 0) = d_{1} = ϕ according to equation (6). The corresponding normalized quantum state reads^{60,61}
where \({U}_{{\rm{S}}}=\frac{1}{\sqrt{1+{\phi }^{2}}}\left(\begin{array}{cc}1&\phi \\ \phi &1\end{array}\right)\). More precisely, the state Φ under the geometry of one isolated loop is
where 〈0G_{1}〉 on the denominator is for the consistence with Φ(vacuum) = 1 (ref. ^{40}).
Similarly, we can simulate the wavefunction Φ of two independent loops with G_{2}〉 = U_{S}0〉 ⊗ U_{S}0〉. Now, we consider a more complex geometry of the two connected loops:
The corresponding quantum state in the quantum circuit scheme is
where G_{2}〉 corresponds to the configuration of two isolated string loops a and b, G_{3}〉 corresponds to the configuration of two string loops a and b connected by string j and C_{F} is the quantum circuit corresponds to \({F}_{ab0}^{bac}\) in equation (10).
A simplified stringnet wavefunction representation under the geometry shown in Fig. 2 can be expressed as
In the quantum circuit scheme, the initialization of an independent polygon can be realized by implementing U_{S} first, and then entangling the remaining strings of this polygon by CNOT gates controlled by the qubits to which U_{S} applies. We call this operation as a ‘copy’ since this operation prepares multiple qubits to the same quantum state corresponding to the same string type. To merge the separated hexagons, we use the type0 strings to connect them and use F moves to obtain the desired geometry, similar to the process shown in equation (10). As shown in Fig. 2, the first F move \({F}_{ab0}^{ba{i}_{1}}\) and the second F move \({F}_{bc0}^{cb{i}_{2}}\) are realized by a threequbit gate since there are two repetitive indexes for each of them. The last F move \({F}_{{i}_{1}{i}_{2}b}^{ca{i}_{3}}\) is realized by a fivequbit gate as the general case. The number of edges changes in equation (12), where we add/remove auxiliary qubits to/from the corresponding quantum circuit accordingly. The decomposed circuits of the other multiqubit gates corresponding to the F move and the methods to remove qubits are described in Supplementary Section II.B.
Fixedpoint Hamiltonian
An exactly solvable lattice spin Hamiltonian has been introduced^{6} in the form of equation (1) with the fixedpoint wavefunction Φ as the ground state. In this Hamiltonian, the Q_{v} operator is defined as
where the wavefunction ijk〉_{v} represents the types of three strings meeting at vertex v, and the tensor δ_{ijk} corresponds to the fusion rules for specific anyons. For the Fibonacci anyon, valid fusion rules are
which gives δ_{ijk} = 1 if ijk ∈ {000, 011, 101, 110, 111} and δ_{ijk} = 0 otherwise^{54}.
Meanwhile, B_{p} corresponds to the local constraints in equations (5a)–(5d) that uniquely specify the wavefunction capturing the properties of topologically ordered states. It is a sum of closed string operators describing particle and antiparticle pairs created from vacuum, moved along the edges of a plaquette (Fig. 1b), and annihilated back to vacuum. In the Fibonacci stringnet model, B_{p} is defined as
where s ∈ {0, 1} represents the string types. \({B}_{p}^{s}\) changes the state on the six edges of the plaquette p controlled by the state on the six outer links of p. The explicit algebraic form of \({B}_{p}^{s}\) is presented in the ‘String operators’ section as the smallest closed string operator along the edge of one plaquette, which describes the process of creating a pair of types anyons, moving around this plaquette and fusing to vacuum.
String operators
The quasiparticle excitations live at the endpoints of the string operators. In Fig. 1c, we illustrate how the F and R moves extend the string operator and turn its direction. Here we give the explicit algebraic form to create, move and fuse these quasiparticle excitations in the stringnet picture. In the main text, we have mentioned that we use the tailed stringnet picture^{42} where the tails represent the quasiparticle excitations located at the endpoints of the string operator. We also use this picture to conveniently describe the creation and fusion of these excitations.
As shown in Extended Data Fig. 3a, the creation of a pair of types excitations from vacuum can be described as adding a short types open string. Since one can erase or add the null (vacuum) strings at will^{40,52}, the types open string is connected to the string net by the null string. Then, we use one F move to turn this configuration into a tailed string net where the endpoints of the string operator are well defined. The fusion of these excitations can be implemented by connecting the tails with F move (Extended Data Fig. 3b). Under this framework, the algebraic form of the string operator is the same as that in ref. ^{6} for moving quasiparticles. The creation and fusion are defined near the endpoints of the string operator and exhibit some ambiguity, which is not important for our purposes since it does not affect the braiding statistics of the excitations^{40}. For the closed string operators, the creation and fusion operation will introduce a constant factor related to the quantum dimension of types string, as discussed in more detail subsequently.
We consider the closed string operator \({B}_{p}^{s}\), which can be regarded as creating a pair of types excitations from vacuum, winding them around in this plaquette, and then annihilating them to the vacuum. As shown in Extended Data Fig. 4, we create a pair of types excitations at string a with \({F}_{aa{a}^{{\prime} }}^{\,ss0}\). Then, we move the tail on the left around this plaquette with \({F}_{s{a}^{{\prime} }{b}^{{\prime} }}^{\;gba}\)⋯\({F}_{s{f}^{{\prime} }{a}^{{\prime} }}^{\,laf}\). Finally, we annihilate these two excitations to vacuum with \({F}_{{a}^{{\prime} }s0}^{\,s{a}^{{\prime} }a}\). According to the normalization convention^{6}, \({F}_{aa{a}^{{\prime} }}^{\,ss0}=\frac{{v}_{{a}^{{\prime} }}}{{v}_{s}{v}_{a}}{\delta }_{sa{a}^{{\prime} }}\) and \({F}_{{a}^{{\prime} }s0}^{\,s{a}^{{\prime} }a}=\frac{{v}_{a}}{{v}_{s}{v}_{{a}^{{\prime} }}}{\delta }_{sa{a}^{{\prime} }}\), where v represents the square root of the quantum dimension d. The product of these two terms is a constant factor 1/d_{s}, which is eliminated by equation (5b) considering the fact that \({B}_{p}^{s}\) create a types closed loop. The types string operator on this plaquette can be expressed as
where \({B}_{p}^{s}\) does not change the types of the six outgoing strings connected to the hexagon.
As the endpoints of the string operator, the tails always have a definite string type that matches the type of the corresponding string operator. Consequently, we do not attach physical qubits to the tails since there is no degree of freedom for their types. However, the tails representing the fusion results have multiple values for nonAbelian anyons and require to be captured by physical qubits (Fig. 4 and Extended Data Fig. 3). For example, in the implementation of (closed) string operator \({B}_{p}^{s}\), the first movement of excitation is implemented by \({F}_{s{a}^{{\prime} }{b}^{{\prime} }}^{\;gba}\). Although it has six indexes, the value of s is predetermined as the type of the simple string operator and does not occur in the quantum circuit scheme. The most complicated part in the circuit implementation of simple string operators is to apply fourqubit gates, different from the cases of preparing the ground state and the projective measurement of the plaquette operator B_{p} where fivequbit gates are involved^{44,62}. A more detailed description to implement these multiqubit gates is given in Supplementary Section II.
Fusion space
NonAbelian anyons have multiple fusion outputs and can be used for constructing the topologically protected logical qubits. In this work, we use four Fibonacci anyons with the vacuum total charge to encode one logical qubit. The measurement results of the logical qubit can be obtained by measuring the fusion outcomes of the first or last two anyons. The fusion results of the first and last two anyons should be the same according to charge conservation^{36}. The quantum gates implemented on logical qubits are realized by the braiding operators, whose matrix representations are associated with the encoding scheme. A common calculation method is through the fusion tree notation^{63}.
The braiding operator σ_{1} in our encoding scheme of \(\left\vert \bar{0}\right\rangle =\left\vert {\left(\tau \times \tau \right)}_{{{{\bf{1}}}}},{\left(\tau \times \tau \right)}_{{{{\bf{1}}}}}\right\rangle\) and \(\left\vert \bar{1}\right\rangle =\left\vert {\left(\tau \times \tau \right)}_{\tau },{\left(\tau \times \tau \right)}_{\tau }\right\rangle\) can be calculated by the R matrix of Fibonacci theory as
Furthermore, the braiding operator σ_{2} is calculated by the F and R matrices of Fibonacci theory as
According to the matrix representations of the F and R move, \({\sigma }_{1}=\left(\begin{array}{cc}{{\rm{e}}}^{4\uppi{\rm{i}}/5}&0\\ 0&{{\rm{e}}}^{3\uppi{\rm{i}}/5}\end{array}\right)\) and \({\sigma }_{2}=\left(\begin{array}{cc}{\phi }^{1}{{\rm{e}}}^{4\uppi{\rm{i}}/5}&{\phi }^{1/2}{{\rm{e}}}^{3\uppi{\rm{i}}/5}\\ {\phi }^{1/2}{{\rm{e}}}^{3\uppi{\rm{i}}/5}&{\phi }^{1}\end{array}\right)\). In the logical space, processes (2)–(5) (Fig. 4c(ii)–(v), respectively) are expressed as
respectively.
We notice that equation (19d) describes a threestep process defining the elements M_{ab} of the monodromy matrix^{40}: (1) create two particle–antiparticle pairs \(\left(a,\bar{a},b,\bar{b}\right)\) from vacuum; (2) braid particle a around particle b; and (3) annihilate both pairs to the vacuum, which is exactly the process shown in Fig. 4c(v). In the Fibonacci stringnet model, the amplitude of the logical \(\left\vert \bar{0}\right\rangle\) measured in process (5) gives the value of M_{ττ}, denoted as \({M}_{\tau \tau }=\left\langle \bar{0}\right\vert {\sigma }_{2}{\sigma }_{2}\left\vert \bar{0}\right\rangle\). By the theory of modular tensor category^{64}, the element M_{ττ} is a real negative value taking the form of −1/d^{2}, where d is the quantum dimension of the quasiparticle excitation. In our experiment, we obtain M_{ττ} ≈ −0.39, which is the negative square root of the experimentally measured probability for the state 00〉 in Fig. 4d(v). Consequently, an experimental estimation for the quantum dimension of the Fibonacci anyon is \({d}_{\tau }=\sqrt{1/{M}_{\tau \tau }}\approx 1.60\).
Circuit implementation
The original circuits for preparing the ground state and realizing different types of F move are compiled to fit the native gate set (that is, arbitrary singlequbit gates and twoqubit CZ gates) and the layout geometry of the processor. We tackle this problem by exploiting the variational unitary synthesis technique, which can be divided into a discrete optimization part searching for the best circuit architecture and a continuous optimization part finding the best set of singlequbit rotation angles. In practice, we adopt the recently introduced CPFlow package^{43} to design the desired circuits.
The circuits for groundstate preparation and anyon braiding in this work are composed of scalable modules and can be optimized by blocks. In addition, for the groundstate preparation, we can further set the state vector of the ground state as the target and optimize the circuit as a whole, which further reduces the circuit depth. Before running the circuit, further alignments are executed to reduce the impact of decoherence errors, and Carr–Purcell–Meiboom–Gill gates are inserted to echo lowfrequency noises. The experimental circuits for preparing the ground state are explicitly displayed in Supplementary Section III.B.
Randomized measurement
In our experiment, we adopt the randomized measurement method to obtain the secondorder Rényi entropies and calculate the topological entanglement entropy^{33,48,49}. This method is achieved by applying random unitaries, which are products of singlequbit unitaries sampled from the circular unitary ensemble, to the system and measuring the final states on the computational basis. For each instance of random unitaries, we repeat the measurement many times to sample the probabilities of the bit strings. The secondorder Rényi entropy can be computed as
where N_{A} and ρ_{A} are the qubit number and density matrix of system A, respectively. Here w and w′ are the binary strings and H(w, w′) is the Hamming distance between them. P(w) denotes the probability of observing w. The average is over different instances of random unitaries in randomized measurement. During the calculation, we also use the iterative Bayesian unfolding scheme to mitigate measurement errors and alleviate undersampling bias (Supplementary Section III.D).
After preparing the ground state, we apply random unitaries to an 18qubit system and measure its final state, from which we can obtain the secondorder Rényi entropies of all the subsystems described in the main text. In practice, we find that an instance number of 1,500 and a sampling number of 300,000 for each instance are required to provide a reliable estimate of the secondorder Rényi entropy of an 11qubit subsystem. Supplementary Section III.C provides more details on the choices of the number of instances as well as the tomography verification of the randomized measurement method with small systems.
Data availability
The data presented in the figures and that support the other findings of this study are publicly available via Figshare at https://doi.org/10.6084/m9.figshare.24947646 (ref. ^{65}).
Code availability
The data analysis and numerical simulation codes for this study are publicly available via Figshare at https://doi.org/10.6084/m9.figshare.24947646 (ref. ^{65}).
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Acknowledgements
We thank M. A. Levin and X.G. Wen for enlightening discussions. The device was fabricated at the MicroNano Fabrication Center of Zhejiang University. We acknowledge support from the Innovation Program for Quantum Science and Technology (grant nos. 2021ZD0300200 and 2021ZD0302203), the National Natural Science Foundation of China (grant nos. 92065204, 12075128, T2225008, 12174342, 12274368, 12274367 and U20A2076) and the Zhejiang Provincial Natural Science Foundation of China (grant no. LDQ23A040001). H.W. is supported by the New Cornerstone Science Foundation through the XPLORER PRIZE. C.S. is supported by the Xiaomi Young Scholars Program. Z.Z.S., W.L., W.J. and D.L.D. are additionally supported by Tsinghua University Dushi Program and the Shanghai Qi Zhi Institute.
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S.X. and K.W. carried out the experiments under the supervision of C.S. and H.W. J.C. and X. Zhu. designed the device and H.L. fabricated the device, supervised by H.W. Z.Z.S. designed the quantum circuits under the supervision of D.L.D. W.L., W.J., L.W.Y., Z.Z.S. and D.L.D. conducted the theoretical analysis. All authors contributed to the experimental setup, analysis of data, discussions of the results and writing of the manuscript.
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Extended data
Extended Data Fig. 1 The layout of the 27 qubits (blue circles) used in our experiments, based on which we construct the desired honeycomb lattice.
The neighboring qubits are connected with tunable couplers denoted as bars. Each physical qubit is labeled by Q_{(i, j)} with i(j) being the row (column) index.
Extended Data Fig. 2 The measured expectation values of the vertex (Q_{v}) and plaquette (B_{p}) operators after step 1 (left), 2 (middle) and 3 (right).
A repetition number of 3000 (300,000) is used to obtain the probability distributions in the computational basis, which are corrected with iterative Bayesian unfolding methods^{58,59} to mitigate readout error for calculating 〈Q_{v}〉 (〈B_{p}〉) (Fig. 2a).
Extended Data Fig. 3 Creation and fusion of the quasiparticle excitations.
a, To create a pair of quasiparticles, we add a short string on the stringnet configuration. It can be regarded as connected to the honeycomb lattice with the vacuum string, which can be arbitrarily added and removed. One Fmove acting on the types string, the type0 connecting string, and the nearby edge can change the fattened lattice picture^{6} to the tailed stringnet picture. b, To annihilate (fuse) two quasiparticles, we detach the two tails with one Fmove to directly connect them. The string connecting these two detached tails indicates the fusion result of these two quasiparticles.
Extended Data Fig. 4 The process described by \({\mathbf{B}}_{\mathbf{p}}^{\mathbf{s}}\) as a closed string operator.
The effect of \({B}_{p}^{s}\) can be understood as creating a pair of types excitations from vacuum, moving the excitations around this plaquette, and annihilating them to vacuum. Under the tailed stringnet picture, the positions of the tails clearly reveal this process. In this figure, we only move the tail initialized on the left. However, different schemes of moving these tails along this path do not change the algebraic representation of the closed string operator \({B}_{p}^{s}\) according to Mac Lane’s coherence theorem^{52}.
Supplementary information
Supplementary Information
Supplementary Sections I–III and Figs. 1–23.
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Xu, S., Sun, ZZ., Wang, K. et al. NonAbelian braiding of Fibonacci anyons with a superconducting processor. Nat. Phys. (2024). https://doi.org/10.1038/s41567024025296
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DOI: https://doi.org/10.1038/s41567024025296
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