Abstract
Indistinguishability of particles is a fundamental principle of quantum mechanics^{1}. For all elementary and quasiparticles observed to date—including fermions, bosons and Abelian anyons—this principle guarantees that the braiding of identical particles leaves the system unchanged^{2,3}. However, in two spatial dimensions, an intriguing possibility exists: braiding of nonAbelian anyons causes rotations in a space of topologically degenerate wavefunctions^{4,5,6,7,8}. Hence, it can change the observables of the system without violating the principle of indistinguishability. Despite the welldeveloped mathematical description of nonAbelian anyons and numerous theoretical proposals^{9,10,11,12,13,14,15,16,17,18,19,20,21,22}, the experimental observation of their exchange statistics has remained elusive for decades. Controllable manybody quantum states generated on quantum processors offer another path for exploring these fundamental phenomena. Whereas efforts on conventional solidstate platforms typically involve Hamiltonian dynamics of quasiparticles, superconducting quantum processors allow for directly manipulating the manybody wavefunction by means of unitary gates. Building on predictions that stabilizer codes can host projective nonAbelian Ising anyons^{9,10}, we implement a generalized stabilizer code and unitary protocol^{23} to create and braid them. This allows us to experimentally verify the fusion rules of the anyons and braid them to realize their statistics. We then study the prospect of using the anyons for quantum computation and use braiding to create an entangled state of anyons encoding three logical qubits. Our work provides new insights about nonAbelian braiding and, through the future inclusion of error correction to achieve topological protection, could open a path towards faulttolerant quantum computing.
Similar content being viewed by others
Main
Elementary particles in three dimensions are either bosons or fermions. The existence of only two types is rooted in the fact that the worldlines of two particles in three plus one dimensions can always be untied in a trivial manner. Hence, exchanging a pair of indistinguishable particles twice is topologically equivalent to not exchanging them at all, and the wavefunction must remain the same. Representing the exchange as a matrix R acting on the space of wavefunctions with a constant number of particles, it is thus required that R^{2} = 1 (a scalar), leaving two possibilities: R = 1 (bosons) and R = −1 (fermions). Such continuous deformation is not possible in two dimensions, thus allowing collective excitations (quasiparticles) to show richer braiding behaviour. In particular, this permits the existence of Abelian anyons^{2,3,6,7,8,24,25}, in which the global phase change due to braiding can take any value. It has been proposed that there exists another class of quasiparticles known as nonAbelian anyons, in which braiding instead results in a change of the observables of the wavefunction^{4,5,24}. In other words, R^{2} does not simplify to a scalar, but remains a unitary matrix. Therefore, R^{2} is a fundamental characteristic of anyon braiding. The topological approach to quantum computation^{26} aims to leverage these nonAbelian anyons and their topological nature to enable gate operations that are protected against local perturbations and decoherence errors^{5,27,28,29,30}. In solidstate systems, primary candidates of nonAbelian quasiparticles are lowenergy excitations in Hamiltonian systems, including the 5/2 fractional quantum Hall states^{31,32}, vortices in topological superconductors^{33,34} and Majorana zero modes in semiconductors proximitized by superconductors^{35,36,37,38}. However, direct verification of nonAbelian exchange statistics has remained elusive^{39,40,41}.
We formulate the necessary requirements for experimentally certifying a physical system as a platform for topological quantum computation^{5,26}: (1) create an anyon pair; (2) verify the rules that govern the ‘collision’ of two anyons, known as the fusion rules; (3) verify the nonAbelian braiding statistics reflected in the matrix structure R^{2} and (4) realize controlled entanglement of anyonic degrees of freedom. Notably, the observation of steps (2)–(4) requires measurements of multianyon states, by means of fusion or nonlocal measurements.
The advent of quantum processors allows for controlled unitary evolution and direct access to the wavefunction rather than the parameters of the Hamiltonian. These features enable the use of local operations for efficient preparation of topological states that can host nonAbelian anyons, and—as we will demonstrate—their subsequent braiding and fusion. Moreover, these platforms allow for probing arbitrary Pauli strings through destructive multiqubit (that is, nonlocal) measurements. As the braiding of nonAbelian anyons in this platform is achieved through unitary gate control rather than adiabatic evolution of a Hamiltonian system, we note that the anyons are not quasiparticles in the sense of eigenstates that persist throughout a Hamiltonian evolution. Their movement is achieved through local operations along their paths, and they are kept spatially separated throughout the braiding. We therefore emphasize that the twodimensional braiding processes are physically taking place on the device, leading to actual nonAbelian exchange effects of local anyons in the manybody wavefunction, rather than matrix operations that simply follow the same algebra.
To realize a manybody quantum state that can host anyons, it is essential to control the topological degeneracy. A suitable platform for achieving this requirement is a stabilizer code^{42}, in which the wavefunctions are characterized by a set of commuting operators \(\{{\hat{S}}_{p}\}\) called stabilizers, with \({\hat{S}}_{p}\left\psi \right\rangle ={s}_{p}\left\psi \right\rangle \) and s_{p} = ±1. The code space is the set of degenerate wavefunctions for which s_{p} = 1 for all p. Hence, every independent stabilizer divides the degeneracy of the code space by two.
Whereas the physical layout of qubits is typically used to determine the structure of the stabilizers, the qubits can be considered to be degree j vertices (DjV; j ∈ {2, 3, 4}) on more general planar graphs (Fig. 1a)^{23}. Using this picture, each stabilizer can be associated with a plaquette p, whose vertices are the qubits on which \({\hat{S}}_{p}\) acts:
\({\hat{\tau }}_{p,v}\) is here a singlequbit Pauli operator acting on vertex v, chosen to satisfy a constraint around that vertex (Fig. 1b). An instance where s_{p} = −1 on a plaquette is called a plaquette violation. These can be thought of as quasiparticles, which are created and moved through singlequbit Pauli operators (Fig. 1a). A pair of plaquette violations sharing an edge constitute a fermion, ε. We recently demonstrated the Abelian statistics of such quasiparticles in the surface code^{43}. To realize nonAbelian statistics, one needs to go beyond such plaquette violations; it has been proposed that dislocations in the stabilizer graph—analogous to lattice defects in crystalline solids—can host projective nonAbelian Ising anyons^{9,10}. For brevity, we refer to these as ‘nonAbelian anyons’ or simply ‘anyons’ from here on.
In the graph framework introduced above, it has been shown that such dislocations are characterized as vertices of degree 2 and 3 (ref. ^{23}). Consider the stabilizer graph of the surface code^{26,44}, specifically with boundary conditions such that the degeneracy is two. Although all the vertices in the bulk are D4Vs, one can create two D3Vs by removing an edge between two neighbouring plaquettes p and q, and introducing the new stabilizer \(\hat{S}={\hat{S}}_{p}{\hat{S}}_{q}\) (Fig. 1b). Evidently, the introduction of two D3Vs reduces the number of independent stabilizers by one and thus doubles the degeneracy. This doubling is exactly what is expected when a pair of Ising anyons is introduced^{9,10}; hence, D3Vs appear as a candidate of nonAbelian anyons, and we will denote them as σ.
To be braided and fused by unitary operations, the D3Vs must be moved. Whereas the structure of the stabilizer graph is usually considered to be static, it was predicted by Bombin that the dislocations in the surface code would show projective nonAbelian Ising statistics if braided^{10}. Here, we will use a specific protocol recently proposed by Lensky et al.^{23} for deforming the stabilizer graph (and thus moving the anyons) using local twoqubit Clifford gates. To shift a D3V from vertex u to v, an edge must be disconnected from v and reconnected to u. This can be achieved by means of the gate unitary \(\exp \left(\frac{\pi }{8}[{\hat{S}}_{p}^{{\prime} },{\hat{S}}_{p}]\right)\), where \({\hat{S}}_{p}\) is the original stabilizer containing the edge and u, and \({\hat{S}}_{p}^{{\prime} }\) is the new stabilizer that emerges after moving the edge^{23}. In cases where the D3V is shifted between two connected vertices, the unitary simplifies to the form \({U}_{\pm }({\hat{\tau }}_{u}{\hat{\tau }}_{v})\equiv \exp \left(\pm i\frac{\pi }{4}{\hat{\tau }}_{u}{\hat{\tau }}_{v}\right)\), where \({\hat{\tau }}_{u}\) and \({\hat{\tau }}_{v}\) are Pauli operators acting on vertices u and v. We experimentally realize this unitary through a controlled Z (CZ) gate and singlequbit rotations (median errors of 7.3 × 10^{−3} and 1.3 × 10^{−3}, respectively; Methods).
Following these insights from Kitaev and Bombin, we now turn to our experimental study of the proposed anyons, using the protocol described in ref. ^{23}. In the first experiment, we demonstrate the creation of anyons and the fundamental fusion rules of σ and ε (Fig. 2a). In a 5 × 5 grid of superconducting qubits, we first use a protocol consisting of four layers of CZ gates to prepare the surface code ground state (Fig. 2b(i), see also ref. ^{43}). The average stabilizer value after the ground state preparation is 0.94 ± 0.04 (individual stabilizer values shown in Extended Data Fig. 3c). We then remove a stabilizer edge to create a pair of D3Vs (σ) and separate them through the application of twoqubit gates. Fig. 2b(i)–(iv) show the measured stabilizer values in the resultant graph in each step of this procedure (determined by simultaneously measuring the involved qubits in their respective bases, n = 10,000; note that the measurements are destructive and the protocol is restarted after each measurement). In Fig. 2b(v), singlequbit Z gates are applied to two qubits near the lower left corner of the grid to create adjacent plaquette violations, which together form a fermion. Through the sequential application of X and Z gates (Fig. 2b(vii)–(viii)), one of the plaquette violations is then made to encircle the right σ vertex. Crucially, after moving around σ, the plaquette violation does not return to where it started, but rather to the location of the other plaquette violation. This enables them to annihilate (Fig. 2b(viii)), causing the fermion to seemingly disappear. However, by bringing the two σ back together and annihilating them (Fig. 2b(ix)–(xi)), we arrive at a striking observation: an ε particle reemerges on two of the square plaquettes where the σ vertices previously resided.
Our results demonstrate the fusion of ε and σ. The disappearance of the fermion from step (v) to (viii) establishes the fundamental fusion rule of ε and σ:
We emphasize that none of the singlequbit gates along the path of the plaquette violation are applied to the qubits hosting the mobile σ; our observations are therefore solely due to the nonlocal effects of nonAbelian D3Vs, and exemplify the unconventional behaviour of the latter. Moreover, another fusion rule is seen by considering the reverse path (iv) → (i), and comparing it to the path (viii) → (xi). These two paths demonstrate that a pair of σ can fuse to form either vacuum (\({\mathbb{1}}\)) or one fermion (steps (i) and (xi), respectively):
The starting points of these two paths ((iv) and (viii)) cannot be distinguished by any local measurement. We therefore introduce a nonlocal measurement technique that allows for detecting an ε without fusing the σ (refs. ^{10,23,26}). The key idea underlying this method is that bringing a plaquette violation around a fermion should result in a π phase. We therefore measure the Pauli string \(\hat{{\mathcal{P}}}\) that corresponds to creating two plaquette violations, bringing one of them around the two σ, and finally annihilating them with each other (grey paths in Fig. 2c). The existence of an ε inside the σpair should cause \(\langle \hat{{\mathcal{P}}}\rangle =1\). To simplify this technique further, \(\hat{{\mathcal{P}}}\) can be reduced to a shorter string \(\hat{{{\mathcal{P}}}^{{\prime} }}\) (black paths in Fig. 2c) by taking advantage of the stabilizers it encompasses. For instance, if \(\hat{{\mathcal{P}}}\) contains three of the operators in a fourqubit stabilizer, these can be switched out with the remaining operator. Measuring \(\langle \hat{{{\mathcal{P}}}^{{\prime} }}\rangle \) in step (iv), in which the σ are separated but the fermion has not yet been introduced, gives \(\langle \hat{{{\mathcal{P}}}^{{\prime} }}\rangle =+\,0.84\pm 0.01\), consistent with the absence of fermions (Fig. 2c). However, performing the exact same measurement in step (viii), in which the σ are in the same positions, we find \(\langle \hat{{{\mathcal{P}}}^{{\prime} }}\rangle =\,0.85\pm 0.01\), indicating that an ε is delocalized across the spatially separated σ pair (Fig. 2c). This observation highlights the nonlocal encoding of the fermions, which cannot be explained with classical physics.
Having demonstrated the above fusion rules involving σ, we next braid them with each other to directly show their nonAbelian statistics. We consider two spatially separated σ pairs, A and B, by removing two stabilizer edges (Fig. 3a,b(ii)). Next, we apply twoqubit gates along a horizontal path to separate the σ in pair A (Fig. 3b(iii)), followed by a similar procedure in the vertical direction on pair B (Fig. 3b(iv)), so that one of its σ crosses the path of pair A. We then subsequently bring the σ from pairs A and B back to their original positions (Fig. 3b(v)–(viii) and (ix)–(xi), respectively). When the two σ pairs are annihilated in the final step (Fig. 3b(xii)), we observe that a fermion is revealed in each of the positions where the σ pairs resided (average stabilizer value −0.45 ± 0.06). This shows a clear change in local observables from the initial state in which no fermions were present. As a control experiment, we repeat the experiment with distinguishable σ pairs, achieved by attaching a plaquette violation to each of the σ in pair B (Fig. 3c,d; see also Extended Data Fig. 8 for stabilizer measurements through the full protocol). Moving the plaquette violation along with the σ requires a string of singlequbit gates, which switches the direction of the rotation in the multiqubit unitaries, U_{±} → U_{∓}. In this case, no fermions are observed at the end of the protocol (average stabilizer value +0.46 ± 0.04), thus providing a successful control.
Fermions can only be created in pairs in the bulk. Moreover, the fusion of two σ can only create zero or one fermion (equation (3)). Hence, our experiment involves the minimal number of bulk σ (four) needed to encode two fermions and demonstrate nonAbelian braiding. Because the fermion parity is conserved, effects of gate imperfections and decoherence can be partially mitigated by postselecting for an even number of fermions. This results in fermion detection values of −0.76 ± 0.03 and +0.79 ± 0.04 in Fig. 3b,d, respectively.
Together, our observations show the change in local observables by braiding of indistinguishable σ and constitute a direct demonstration of their nonAbelian statistics. In other words, the doublebraiding operation R^{2} is a matrix that cannot be reduced to a scalar. Specifically, it corresponds to an X gate acting on the space spanned by zero and twofermion wavefunctions.
The full braiding circuit consists of 40 layers of CZ gates and 41 layers of singlequbit gates (36 of each after ground state preparation). The effects of imperfections in this hardware implementation can be assessed through comparison with the control experiment. The absolute values of the stabilizers in which the fermions are detected in the two experiments (dashed boxes in Fig. 3b,d(xii)) are very similar (average values of −0.45 and +0.46). This is consistent with the depolarization channel model, in which the measured stabilizer values are proportional to the ideal values (±1).
We next study the prospects of using D3Vs to encode logical qubits and prepare an entangled state of anyon pairs. By doubling the degeneracy, each additional σ pair introduces one logical qubit, where the \({\left0\right\rangle }_{{\rm{L}}}\) (\({ 1\rangle }_{{\rm{L}}}\)) state corresponds to an even (odd) number of hidden fermions. The measurements of the fermion numbers in several σ pairs are not fully independent: bringing a plaquette violation around one σ pair is equivalent to bringing it around all the other pairs (due to the conservation of fermionic parity). Hence, N ≥ 2 anyons encode N/2 − 1 logical qubits. The D3Vs we have created and manipulated so far are not the only ones present in the stabilizer graph; with the boundary conditions used here, each of the four corners are also D3Vs, no different from those in the bulk^{23}. Indeed, the existence of D3Vs in the corners is the reason why a single fermion could be created in the corner in Fig. 2b(v). This is also consistent with the fact that the surface code itself encodes one logical qubit in the absence of additional D3Vs. Here we create two pairs of D3Vs, in addition to the four that are already present in the corners, to encode a total of three logical qubits.
Through the use of braiding, we aim to prepare an entangled state of these logical qubits, specifically a GHZ state on the form \(( 000\rangle + 111\rangle )/\sqrt{2}\). The definition of a GHZ state and the specifics of how it is prepared is basisdependent. In most systems, the degrees of freedom are local and there is a natural choice of basis. For spatially separated anyons, the measurement operators are necessarily nonlocal. Here we choose the basis defined as follows: for the first two logical qubits, we choose the logical \({\hat{Z}}_{{\rm{L}},i}\) operators to be Pauli strings encircling each of the bulk σ pairs, as was used in Fig. 2c (green and turquoise paths in the left column of Fig. 4a). For the logical surface code qubit, we define \({\hat{Z}}_{{\rm{L}},3}\) as the Pauli string that crosses the grid horizontally through the gap between the bulk D3V pairs, effectively enclosing four σ (red path in Fig. 4a). In this basis, the initial state is a product state.
Whereas a double braid was used to implement the operator X in Fig. 3, we now perform a single braid (Fig. 4b) to realize \(\sqrt{X}\) and create a GHZ state. We implement this protocol by bringing one σ from each bulk pair across the grid to the other side (Fig. 4c). For every anyon double exchange across a Pauli string, the value of the Pauli string changes sign. Hence, a double exchange would change \(\left000\right\rangle \) to \(\left111\right\rangle \), whereas a single exchange is expected to realize the superposition, \((\left111\right\rangle +\left000\right\rangle )/\sqrt{2}\).
To study the effect of this operation, we perform quantum state tomography on the final state, which requires measurements of not only \({\hat{Z}}_{{\rm{L}},i}\), but also \({\hat{X}}_{{\rm{L}},i}\) and \({\hat{Y}}_{{\rm{L}},i}\) on the three logical qubits. For the first two logical qubits, \({\hat{X}}_{{\rm{L}},i}\) is the Pauli string that corresponds to bringing a plaquette violation around only one of the σ in the pair (as demonstrated in Fig. 2b). Both the logical \({\hat{X}}_{{\rm{L}},i}\) and \({\hat{Z}}_{{\rm{L}},i}\) operators can be simplified by reducing the original Pauli strings (green and turquoise paths in the left column of Fig. 4c) to equivalent, shorter ones (right column). \({\hat{Z}}_{{\rm{L}},1}\) can in fact be reduced to a single \(\hat{Y}\)operator. For the logical surface code qubit, we define \({\hat{X}}_{{\rm{L}},3}\) as the Pauli string that crosses the grid vertically between the bulk D3V pairs (red path in Fig. 4a). Finally, the logical \({\hat{Y}}_{{\rm{L}},i}\)operators are simply found from \({\hat{Y}}_{{\rm{L}},i}=i{\hat{X}}_{{\rm{L}},i}{\hat{Z}}_{{\rm{L}},i}\). Measuring these operators, we reconstruct the density matrix of the final state (Fig. 4d,e), which has a purity of \(\sqrt{{\rm{Tr}}\{{\rho }^{2}\}}=0.646\pm 0.003\) and an overlap with the ideal GHZ state of \({\rm{Tr}}\{{\rho }_{{\rm{GHZ}}}\rho \}=0.623\pm 0.004\) (uncertainties estimated from bootstrapping method; resampled 10,000 times from the original data set). The fact that the state fidelity is similar to the purity suggests that the infidelity is well described by a depolarizing error channel.
In conclusion, we have realized highly controllable braiding of degree3 vertices, enabling the demonstration of the fusion and braiding rules of nonAbelian Ising anyons. We have also shown that braiding can be used to create an entangled state of three logical qubits encoded in these anyons. In other, more conventional candidate platforms for nonAbelian exchange statistics, which involve Hamiltonian dynamics of quasiparticle excitations, topological protection naturally arises from an emergent gap that separates the computational states from other states. To leverage the nonAbelian anyons in our system for topologically protected quantum computing, the stabilizers must be measured throughout the braiding protocol. The potential inclusion of this error correction procedure, which involves overheads including readout of fivequbit stabilizers, could open a new path towards faulttolerant implementation of Clifford gates, a key ingredient of universal quantum computation.
Methods
Qubit decoherence and gate characterization
The experiments are performed on a quantum processor with frequencytuneable transmon qubits and a similar design to that in ref. ^{45}. Extended Data Fig. 1a shows the measured relaxation times of the 25 qubits that were used in the experiment, with a median value of T_{1} = 21.7 μs. We also measure the dephasing time T_{2} in a Hahn echo experiment, shown in Extended Data Fig. 1b, with the same median value of 21.7 μs. We note that the equality of T1 and T2 is a coincidence and that the discrepancy between the measured decoherence rate 1/T_{2} and the relaxationlimited rate 1/(2T_{1}) is due to remnant noise not decoupled in the Hahn echo experiment.
Next, we benchmark the gates used in the experiment. Extended Data Fig. 2a,b show the cumulative distribution of the Pauli errors for single and twoqubit (CZ) gates, respectively. The median Pauli errors are 1.3 × 10^{−3} for the singlequbit gates and 7.3 × 10^{−3} for the twoqubit gates.
Readout details
Because the readout of the qubit state is imperfect, the raw data gives a somewhat incorrect representation of the actual quantum state of the system. We write the probability of readout error of state 0(1) on qubit i as p_{0(1),i}, and the readout fidelity is thus given by 1 − (p_{0,i} + p_{1,i})/2. To correct for any asymmetry between readout of the \(\left0\right\rangle \) and \(\left1\right\rangle \) states, we perform symmetrized measurements in which π pulses are applied to the qubits before the readout in half of the measurements and the recorded qubit values are inverted. The measured value of a stabilizer with actual value ⟨S⟩ = ⟨∏_{i}α_{i}⟩ (where the product runs over qubits in the stabilizer) is then:
where we made use of the fact that each qubit is measured equally often in the \(\left0\right\rangle \) and \(\left1\right\rangle \) states in the symmetrized measurements. Note the absence of the factor 1/2 compared to the expression for the readout fidelity, as perfectly incorrect readout (p_{0} = p_{1} = 1) would give a readout fidelity of 0, but a measured value of −α_{i}. To correct for the discrepancy between the measured stabilizer value and the actual stabilizer value, we measure ⟨Z_{1}. . . Z_{n}⟩ of the state \(\left00..00\right\rangle \) with the same qubits (using again symmetrized measurements) to find:
The readoutcorrected ⟨S⟩ is then found from:
Extended Data Fig. 3 shows the measured readout errors, as well as a comparison of the stabilizer values in the surface code ground state (same data as Fig. 2b(i)) before and after readout correction.
Dynamical decoupling
To mitigate the effects of qubit decoherence during the circuits, we perform dynamical decoupling on qubits that are idle for more than three layers of gates. In particular, we use the Carr–Purcell–Meiboom–Gill scheme, consisting of X pulses interspaced by a wait time of τ = 25 ns. Extended Data Fig. 4 shows an example comparison of the stabilizers in cases with and without dynamical decoupling, after braiding of anyons (41 layers of SQ gates and 40 layers of CZ gates). A clear improvement is observed, increasing the average absolute stabilizer value from 0.50 to 0.58.
Circuit details
Extended Data Fig. 5 shows the circuits used in the experiments presented in the main text. In our experiment, the twoqubit unitaries \({U}_{\pm }({\hat{\tau }}_{1}{\hat{\tau }}_{2})\) are converted to singlequbit rotations and CZ gates, as shown in Extended Data Fig. 6b. In the particular case in which a D3V is moved diagonally (Fig. 3b(iv)), we realize the unitary by including two SWAP gates (also converted to CZ gates) as the qubits are connected in a square grid (Extended Data Fig. 6c). Moreover, the threequbit unitary in Fig. 3b(viii) is equivalent to a combination of singlequbit gates, four SWAP gates and four CZ gates (Extended Data Fig. 6d), which can be further converted to singlequbit gates and 15 CZ gates. In the experimental implementation of the circuit, adjacent singlequbit gates on the same qubit are merged and performed in the layer after the most recent CZ gate (Extended Data Fig. 6e).
Numerical simulation of braiding in presence of noise
To better understand the role of errors in the experimental results in Fig. 3 of the main text, we perform a numerical simulation of the density matrix evolution subject to the braiding circuit in the presence of noise. We use the method of quantum trajectories to approximate the expectation value of stabilizers with the 25qubit density matrix. The model of noise includes T_{1} and T_{2} effects described by the singlequbit Kraus operators,
where t is the duration of the evolution, as well as additional one and twoqubit depolarizing channel error for each gate. The depolarizing channel error rate is chosen such that the combined Pauli error from T_{1}, T_{2} and depolarizing error matches the gate Pauli error measured in an independent experiment (Qubit decoherence and gate characterization). We take these values to be uniform across the chip. The expectation values of the four stabilizers that correspond to the noisefree value of −1, see light red stabilizers in Fig. 3b(xii) and Extended Data Fig. 7, have the following values (×100): (−58, −46, −34, −46) with statistical error 4. For comparison, the experimental values for the same set of stabilizers is (−52, −41, −39, −49). Our simulation results are in relatively good agreement with the measured data, suggesting that the model captures the effects of noise well. The observed discrepancies are expected to be due to inhomogeneity of the errors, which was not included in our error model. The simulations used an open source simulator qsim^{46}.
Additional braiding data
In Fig. 3, we demonstrate that no fermion appears when distinguishable σ are braided with each other. In Extended Data Fig. 8, we show the data for each step in that protocol, analogous to those shown for indistinguishable σ in the main text. Moreover, we also present an alternative braiding scheme in Extended Data Fig. 9, which requires fewer (18) CZ gates. In this case, however, pair B is not brought back together, and neither of the σ pairs are annihilated. Therefore, similar to in Fig. 2c, we measure the Pauli string corresponding to bringing a plaquette violation around pair A (grey path in Extended Data Fig. 9c), which in this case can be reduced to \(\hat{Y}\) on the qubit where the two σ overlap. We find \(\langle \hat{{\mathcal{P}}}\rangle =\langle \hat{Y}\rangle =\,0.71\,\pm \,0.01\), indicating that braiding the σ led to the creation of a fermion (Extended Data Fig. 9c). Note that we here only search for fermions in one of the σ pairs. As a control experiment, we repeat the experiment with distinguishable σ pairs, as in the main text (Extended Data Fig. 9d). In this case, we find \(\langle \hat{{\mathcal{P}}}\rangle =+\,0.71\,\pm \,0.01\), thus demonstrating that no fermion was produced. Together, these observations constitute another demonstration of nonAbelian exchange statistics of the D3Vs.
A summary of the theoretical framework
It was observed by Kitaev that fluxes of the e–m exchange symmetry are expected to host Majorana modes and therefore have the degeneracy of Ising anyons^{9}. Bombin gave a particular stabilizer configuration realizing such a flux as a fixed lattice dislocation in a square grid, showed on general grounds that if such fluxes were wellseparated and could be braided they would be projective Ising anyons, and noted that it may be possible to braid such fluxes by code deformation^{10}. A general formalism for theories realized by braiding of symmetry fluxes was described in ref. ^{47}. These constructions focus on the longdistance physics, and in practical terms^{23} gives an account of ‘microscopics’. An explicit mapping to a gauge theory shows how the anyons are localized to a single qubit, and is used to derive a simple, efficient and systematic procedure for creating, braiding and measuring the fusion outcomes of Ising anyons on general stabilizer graphs. The bridge between the microscopics and general arguments established by the gauge theory mapping allows us to fit several anyons on presentday devices, probe the full twodimensional nature of their braiding by maintaining their separation, and demonstrate braid generators that restore all local observables. For details discussions of the protocol, see ref. ^{23}.
Data availability
The data that support the findings in this study are available at https://doi.org/10.5281/zenodo.7869220.
References
Sakurai, J. J. Modern Quantum Mechanics (AddisonWesley, 1993).
Leinaas, J. & Myrheim, J. On the theory of identical particles. Nuovo Cim. B. 37, 1–23 (1977).
Wilczek, F. Quantum mechanics of fractionalspin particles. Phys. Rev. Lett. 49, 957–959 (1982).
Wilczek, F. Fractional Statistics and Anyon Superconductivity (World Scientific, 1990).
Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Sarma, S. D. NonAbelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083–1159 (2008).
Camino, F. E., Zhou, W. & Goldman, V. J. e/3 Laughlin quasiparticle primaryfilling ν = 1/3 interferometer. Phys. Rev. Lett. 98, 076805 (2007).
Bartolomei, H. et al. Fractional statistics in anyon collisions. Science 368, 173–177 (2020).
Nakamura, J., Liang, S., Gardner, G. C. & Manfra, M. J. Direct observation of anyonic braiding statistics. Nat. Phys. 16, 931—936 (2020).
Kitaev, A. Anyons in an exactly solved model and beyond. Ann. Phys. 321, 2–111 (2006).
Bombin, H. Topological order with a twist: Ising anyons from an Abelian model. Phys. Rev. Lett. 105, 030403 (2010).
Hormozi, L., Zikos, G., Bonesteel, N. E. & Simon, S. H. Topological quantum compiling. Phys. Rev. B 75, 165310 (2007).
You, Y.Z. & Wen, X.G. Projective nonabelian statistics of dislocation defects in a \({{\mathbb{Z}}}_{N}\) rotor model. Phys. Rev. B 86, 161107 (2012).
Barkeshli, M., Jian, C.M. & Qi, X.L. Twist defects and projective nonAbelian braiding statistics. Phys. Rev. B 87, 045130 (2013).
Barkeshli, M. & Qi, X.L. Topological nematic states and nonAbelian lattice dislocations. Phys. Rev. X 2, 031013 (2012).
von Keyserlingk, C. W., Burnell, F. J. & Simon, S. H. Threedimensional topological lattice models with surface anyons. Phys. Rev. B 87, 045107 (2013).
Teo, J. C., Roy, A. & Chen, X. Unconventional fusion and braiding of topological defects in a lattice model. Phys. Rev. B 90, 115118 (2014).
Zheng, H., Dua, A. & Jiang, L. Demonstrating nonAbelian statistics of majorana fermions using twist defects. Phys. Rev. B 92, 245139 (2015).
Teo, J. C., Hughes, T. L. & Fradkin, E. Theory of twist liquids: gauging an anyonic symmetry. Ann. Phys. 360, 349–445 (2015).
Brown, B. J., Laubscher, K., Kesselring, M. S. & Wootton, J. R. Poking holes and cutting corners to achieve clifford gates with the surface code. Phys. Rev. X 7, 021029 (2017).
Zhu, G., Hafezi, M. & Barkeshli, M. Quantum origami: transversal gates for quantum computation and measurement of topological order. Phys. Rev. Res. 2, 013285 (2020).
Benhemou, A., Pachos, J. K. & Browne, D. E. Nonabelian statistics with mixedboundary punctures on the toric code. Phys. Rev. A 105, 042417 (2022).
Tantivasadakarn, N., Verresen, R. & Vishwanath, A. The shortest route to nonAbelian topological order on a quantum processor. Preprint at https://arxiv.org/abs/2209.03964 (2022).
Lensky, Y. D., Kechedzhi, K., Aleiner, I. & Kim, E.A. Graph gauge theory of mobile nonAbelian anyons in a qubit stabilizer code. Preprint at https://arxiv.org/abs/2210.09282 (2022).
Stern, A. Anyons and the quantum Hall effect: a pedagogical review. Ann. Phys. 323, 204–249 (2008).
Harle, N., Shtanko, O. & Movassagh, R. Observing and braiding topological Majorana modes on programmable quantum simulators. Nat. Commun. 14, 2286 (2023)
Kitaev, A. Y. Faulttolerant quantum computation by anyons. Ann. Phys. 303, 2–30 (2003).
Freedman, M. H. P/np, and the quantum field computer. Proc. Natl Acad. Sci. USA 95, 98–101 (1998).
Pachos, J. K. Introduction to Topological Quantum Computation (Cambridge Univ. Press, 2012).
Stern, A. & Lindner, N. Topological quantum computation—from basic concepts to first experiments. Science 339, 1179–1184 (2013).
Field, B. & Simula, T. Introduction to topological quantum computation with nonAbelian anyons. Quantum Sci. Technol. 3, 045004 (2018).
Moore, G. & Read, N. Nonabelions in the fractional quantum Hall effect. Nucl. Phys. B 360, 362–396 (1991).
Willett, R. L., Pfeiffer, L. N. & West, K. W. Alternation and interchange of e/4 and e/2 period interference oscillations consistent with filling factor 5/2 nonAbelian quasiparticles. Phys. Rev. B 82, 205301 (2010).
Read, N. & Green, D. Paired states of fermions in two dimensions with breaking of parity and timereversal symmetries and the fractional quantum Hall effect. Phys. Rev. B 61, 10267 (2000).
Ivanov, D. A. NonAbelian statistics of halfquantum vortices in Pwave superconductors. Phys. Rev. Lett. 86, 268–271 (2001).
Lutchyn, R. M., Sau, J. D. & Das Sarma, S. Majorana fermions and a topological phase transition in semiconductorsuperconductor heterostructures. Phys. Rev. Lett. 105, 077001 (2010).
Oreg, Y., Refael, G. & von Oppen, F. Helical liquids and majorana bound states in quantum wires. Phys. Rev. Lett. 105, 177002 (2010).
Mourik, V. et al. Signatures of majorana fermions in hybrid superconductorsemiconductor nanowire devices. Science 336, 1003–1007 (2012).
NadjPerge, S. et al. Observation of majorana fermions in ferromagnetic atomic chains on a superconductor. Science 346, 602–607 (2014).
Banerjee, M. et al. Observation of halfinteger thermal hall conductance. Nature 559, 205–210 (2018).
Kasahara, Y. et al. Majorana quantization and halfinteger thermal quantum Hall effect in a Kitaev spin liquid. Nature 559, 227–231 (2018).
Bonderson, P., Kitaev, A. & Shtengel, K. Detecting nonAbelian statistics in the ν = 5/2 fractional quantum Hall state. Phys. Rev. Lett. 96, 016803 (2006).
Gottesman, D. Stabilizer Codes and Quantum Error Correction (California Institute of Technology, 1997).
Satzinger, K. et al. Realizing topologically ordered states on a quantum processor. Science 374, 1237–1241 (2021).
Wen, X.G. Quantum orders in an exact soluble model. Phys. Rev. Lett. 90, 016803 (2003).
Arute, F. et al. Quantum supremacy using a programmable superconducting processor. Nature 574, 505–510 (2019).
Isakov, S. V. et al. Simulations of quantum circuits with approximate noise using qsim and cirq. Preprint at https://arxiv.org/abs/2111.02396 (2021).
Barkeshli, M., Bonderson, P., Cheng, M. & Wang, Z. Symmetry fractionalization, defects, and gauging of topological phases. Phys. Rev. B 100, 115147 (2019).
Acknowledgements
Y.L. and E.A.K. acknowledge support by a New Frontier Grant from Cornell University’s College of Arts and Sciences. E.A.K. acknowledges support by the National Science Foundation under grant no. OAC2118310, the Ewha Frontier 1010 Research Grant and the Simons Fellowship in Theoretical Physics award no. 920665. E.A.K. performed a part of this work at the Aspen Center for Physics, which is supported by the National Science Foundation grant no. PHY160761.
Author information
Authors and Affiliations
Consortia
Contributions
Y.D.L., K.K., E.A.K. and I.A. developed the underlying theory. T.I.A., Y.D.L., K.K., E.A.K., I.A. and P.R. developed the experiment. T.I.A. performed the experiment and analysed the data. I.K.D., A.B., S.H., A.M., X.M. and A.O. provided assistance with calibration. T.I.A., Y.D.L., K.K., E.A.K., I.A. and P.R. wrote the manuscript. T.I.A., Y.D.L., K.K. and P.R. wrote the Methods. All authors contributed to revising the manuscript. All authors contributed to the experimental and theoretical infrastructure to enable the experiment.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Nature thanks Bernard van Heck and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Extended data figures and tables
Extended Data Fig. 1 Qubit relaxation (T_{1}) and coherence (T_{2}) times.
a,b, Cummulative distributions of T_{1} (a) and T_{2} (b), where the latter is measured using a Hahn echo sequence. Dashed lines indicate the median values of 21.7 μs for both measures. Insets: T_{1} and T_{2} plotted against qubit number.
Extended Data Fig. 2 Gate errors.
a,b, Cummulative distributions of the Pauli error for singlequbit (a) and twoqubit CZ (b) gates. We find median error values of 1.3 × 10^{−3} and 7.3 × 10^{−3} for the singlequbit and CZ gates, respectively.
Extended Data Fig. 3 Readout benchmarking and correction.
a, Histogram of readout error, with a median value of 2.0% (dashed vertical line). Inset: readout error plotted against qubit number. b,c, Stabilizer values of the surface code ground state before (b) and after (c) readout correction.
Extended Data Fig. 4 Dynamical decoupling.
a,b, Stabilizer values without (a) and with (b) dynamical decoupling, after D3V braiding. Dynamical decoupling improves the average absolute stabilizer value from 0.50 to 0.58.
Extended Data Fig. 5 Circuit details.
Circuits used for the fusion experiment (a), the fullbraid experiment (b), and the halfbraid experiment (c), shown in Figs. 2–4, respectively, in the main text. Turqoise and gray boxes denote dynamical decoupling and phased XZgates, respectively. In the fullbraid experiment (b), we include five singlequbit rotations to permute \(\hat{X}\), \(\hat{Y}\) and \(\hat{Z}\) of the three stabilizers touching the moving D3V in steps VVIII and IXXI, as well as three Hadamardgates to return all stabilizers to the original \(\hat{Z}\hat{X}\hat{X}\hat{Z}\)form in XII. See Extended Data Fig. 6 for the circuit used for ground state preparation, as well as details on how the multiqubit unitary gates used to move anyons are decomposed into CZgates.
Extended Data Fig. 6 Ground state preparation and CZdecompositions.
a, Schematic showing the circuit used for preparation of the ground state of the surface code. The protocol is the same as that shown in ref. ^{43}, except with the inclusion of Hadamard gates on alternating qubits in the final step, since we use symmetrized stabilizers on the form \(\hat{Z}\hat{X}\hat{X}\hat{Z}\). b, The unitary needed to move a D3V between two neighboring vertices is realized in the experiment through the use of one CZgate and singlequbit rotations. c, When D3Vs are moved diagonally, we include two SWAP gates, requiring three CZgates each. d, Main: the threequbit unitary used in step VIII in Fig. 3 is equivalent to a combination of singlequbit gates, 4 SWAPgates and 4 CZgates. Right dashed box: decomposition of a SWAPgate into CZgates. e, Adjacent singlequbit gates are merged and shifted left to the nearest CZgate.
Extended Data Fig. 7 Simulation of braiding in the presence of noise.
a, Simulation results. b, Experimental data (same as in step XII in Fig. 3b). We observe relatively good agreement between the simulation and the experimental results, except some discrepancies that are attributed to inhomogeneity of the errors.
Extended Data Fig. 8 Braiding distinguishable D3Vs.
a, Braiding schematic of worldlines. b, Stepbystep depiction of stabilizers as the two σ are braided, analogous to that in Fig. 3, but with distinguishable σ.
Extended Data Fig. 9 Alternative protocol for braiding σ.
a, Schematic displaying the braiding process of the two σpairs. b, Experimental demonstration of braiding, displaying the values of the stabilizers throughout the process. Two σpairs, A and B, are created from the vacuum \({\mathbb{1}}\), and one of the D3Vs in pair A is brought to the right side of the grid. Next, a σ from pair B is moved to the top, thus crossing the path of the first σ, before bringing the σ from pair A back again to complete the braid. The diagonal σ move performed in step VI is achieved by including two SWAPgates, corresponding to 6 additional CZgates. The yellow triangles represent the locations of the σ, while the brown and green lines represent the paths of σ from pair A and B, respectively. The average absolute stabilizer value is 0.93 ± 0.06 and 0.77 ± 0.09 in the first and last step, respectively. c, After braiding the σ, we search for hidden fermions by measuring the Pauli string \(\hat{{\mathcal{P}}}\) (left panels), which here is equivalent to \(\hat{Y}\) on the qubit where the two σ overlap. The measurement yields \(\langle \hat{{\mathcal{P}}}\rangle =\langle \hat{Y}\rangle =\,0.71\,\pm \,0.01\), indicating creation of a fermion. Right: worldlines of braiding process, including nonlocal measurement based on plaquette violation loop. d, Same as c, but after braiding two distinguishable σ, achieved by applying the inverse twoqubit gates when moving the σ in pair B. The measurement yields \(\langle \hat{Y}\rangle =+\,0.71\,\pm \,0.01\), indicating no fermion creation.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Google Quantum AI and Collaborators. NonAbelian braiding of graph vertices in a superconducting processor. Nature 618, 264–269 (2023). https://doi.org/10.1038/s41586023059544
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/s41586023059544
This article is cited by

Experimental error suppression in CrossResonance gates via multiderivative pulse shaping
npj Quantum Information (2024)

Logical quantum processor based on reconfigurable atom arrays
Nature (2024)

NonAbelian topological order and anyons on a trappedion processor
Nature (2024)

Anyons go universal
Nature Physics (2024)

Topological matter created on a quantum chip produces quasiparticles with computing power
Nature (2024)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.