Non-Abelian braiding of graph vertices in a superconducting processor

Abstract

Indistinguishability of particles is a fundamental principle of quantum mechanics¹. 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 non-Abelian anyons causes rotations in a space of topologically degenerate wavefunctions^4-8. Hence, it can change the observables of the system without violating the principle of indistinguishability. Despite the well-developed mathematical description of non-Abelian anyons and numerous theoretical proposals^9-22, the experimental observation of their exchange statistics has remained elusive for decades. Controllable many-body quantum states generated on quantum processors offer another path for exploring these fundamental phenomena. Whereas efforts on conventional solid-state platforms typically involve Hamiltonian dynamics of quasiparticles, superconducting quantum processors allow for directly manipulating the many-body wavefunction by means of unitary gates. Building on predictions that stabilizer codes can host projective non-Abelian Ising anyons^9,10, we implement a generalized stabilizer code and unitary protocol²³ 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 non-Abelian braiding and, through the future inclusion of error correction to achieve topological protection, could open a path towards fault-tolerant quantum computing.

Fig. 1. Deformations of the surface code.

a, Stabilizer codes are conveniently described in a graph framework. Through deformations of the surface code graph, a square grid of qubits (crosses) can be used to realize more generalized graphs. Plaquette violations (red) correspond to stabilizers with s_p = −1 and are created by local Pauli operations. In the absence of deformations, plaquette violations are constrained to move on one of the two sublattices of the dual graph in the surface code, hence the two shades of blue. b, A pair of D3Vs (yellow triangles) appears by removing an edge between two neighbouring stabilizers, ${\hat{S}}_1$ and ${\hat{S}}_2$, and introducing the new stabilizer, $\hat{S}={\hat{S}}_1{\hat{S}}_2$. A D3V is moved by applying a two-qubit entangling gate, $\exp \left(\frac{\pi }{8}\left[{\hat{S}}^{\prime },\hat{S}\right]\right)$. In the presence of bulk D3Vs, there is no consistent way of chequerboard colouring, hence the (arbitrarily chosen) grey regions. The top right shows that in a general stabilizer graph, ${\hat{S}}_p$ can be found from a constraint at each vertex, where {τ₁, τ₂} = 0.

Fig. 2. Demonstration of the fundamental fusion rules of D3Vs.

a, The braiding worldlines used to fuse ε and σ. b, Expectation values of stabilizers at each step of the unitary operation after readout correction (see Extended Data Fig. 3 for details and individual stabilizer values). We first prepare the ground state of the surface code (step (i); average stabilizer value of 0.94 ± 0.04, where the uncertainty is one standard deviation). A D3V (σ) pair is then created (ii) and separated (iii)–(iv), before creating a fermion, ε (v). One of the plaquette violations is brought around the right σ (vi)–(viii), allowing it to annihilate with the other plaquette violation (viii). The fermion has seemingly disappeared, but re-emerges when the σ are annihilated ((xi); stabilizer values −0.86 and −0.87). The path (v) → (viii) demonstrates the fusion rule, σ × ε = σ. The different fermion parities at the end of the paths (viii) → (xi) and (iv) → (i) show the other fusion rule, $\sigma \times \sigma =\mathbb{1}+\epsilon$. Yellow triangles represent the positions of the σ. The brown and red lines denote the paths of the σ and the plaquette violation, respectively. Red squares (diamonds) represent X (Z) gates. Upper left shows a table of two-qubit unitaries used in the protocol. Each stabilizer was measured n = 10,000 times in each step. c, A non-local technique for hidden fermion detection: the presence of a fermion in a σ-pair can be deduced by measuring the sign of the Pauli string $\hat{\mathcal{P}}$ corresponding to bringing a plaquette violation around the σ-pair (grey path). $\hat{\mathcal{P}}$ is equivalent to the shorter string $\widehat{{\mathcal{P}}^{\prime }}$ (black path). Measurements of $\widehat{{\mathcal{P}}^{\prime }}$ in steps (viii) (top) and (iv) (bottom) give values of −0.85 ± 0.01 and +0.84 ± 0.01, respectively. This indicates that there is a hidden fermion pair in the former case, but not in the latter, despite the stabilizers being the same.

Fig. 3. Braiding of non-Abelian anyons.

a, Wordline schematic of the braiding process. b, Experimental demonstration of braiding, showing the values of the stabilizers throughout the process. Two σ pairs, A and B, are created from the vacuum $\mathbb{1}$, and one of the σ 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 pair A, before bringing σ pairs A and B back together to complete the braid. In the final step, two fermions appear in the locations where the σ pairs resided, constituting a change in the local observables. The diagonal σ move in step (iv) requires two SWAP gates (three CZ gates each) and a total of ten CZ gates. The three-qubit unitary in step (viii) requires four SWAP gates and a total of 15 CZ gates. In the full circuit, a total of 40 layers of CZ gates are applied (Methods). The yellow triangles represent the locations of the σ; the brown and green lines represent the paths of σ from pairs A and B, respectively. The four red stabilizers in (xii) have a mean value of −0.45 ± 0.06, where the uncertainty is one standard deviation. Each stabilizer was measured n = 10,000 times in each step. c, As a control experiment, we perform the same braid as in a, but with distinguishable σ by attaching a plaquette violation to the σ in pair B (represented with purple triangles). d, Same as b, but using distinguishable σ (only showing steps (i), (iv) and (xii)). In contrast to b, no fermions are observed in step (xii).

Fig. 4. Entangled state of anyon-encoded logical qubits by means of braiding.

a, Logical operators of the three logical qubits encoded in the eight anyons (other basis choices are possible). The coloured curves in the left column denote plaquette violation paths, before reduction to shorter, equivalent Pauli strings measured in the experiment (right column). b, Worldline schematic of the single exchange used to realize an entangled state of the logical qubits. c, Single exchange of the non-Abelian anyons, showing measurements of the stabilizers throughout the protocol. Yellow triangles represent the locations of the σ, whereas brown and green lines denote their paths. The average stabilizer values are 0.95 ± 0.04 and 0.88 ± 0.05 (one standard deviation) in the first and last step, respectively. Each stabilizer was measured n = 20,000 times in each step. d,e, Real (d) and imaginary (e) parts of the reconstructed density matrix from the quantum state tomography. $\mathrm{Re}\left(\rho \right)$ has clear peaks in its corners, as expected for a GHZ state on the form $\left(∣000⟩+∣111⟩\right)/\sqrt{2}$. The overlap with the ideal GHZ state is $\mathrm{Tr}\left\{{\rho }_{\mathrm{GHZ}}\rho \right\}=0.623\pm 0.004$, where the uncertainty is one standard deviation determined from bootstrapping.

Extended Data Fig. 1. Qubit relaxation (T₁) and coherence (T₂) times.

a,b, Cummulative distributions of T₁ (a) and T₂ (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: T₁ and T₂ plotted against qubit number.

Extended Data Fig. 2. Gate errors.

a,b, Cummulative distributions of the Pauli error for single-qubit (a) and two-qubit CZ (b) gates. We find median error values of 1.3 × 10⁻³ and 7.3 × 10⁻³ for the single-qubit 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 full-braid experiment (b), and the half-braid experiment (c), shown in Figs. 2–4, respectively, in the main text. Turqoise and gray boxes denote dynamical decoupling and phased XZ-gates, respectively. In the full-braid experiment (b), we include five single-qubit rotations to permute $\hat{X}$, $\hat{Y}$ and $\hat{Z}$ of the three stabilizers touching the moving D3V in steps V-VIII and IX-XI, as well as three Hadamard-gates 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 multi-qubit unitary gates used to move anyons are decomposed into CZ-gates.

Extended Data Fig. 6. Ground state preparation and CZ-decompositions.

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. , 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 CZ-gate and single-qubit rotations. c, When D3Vs are moved diagonally, we include two SWAP gates, requiring three CZ-gates each. d, Main: the three-qubit unitary used in step VIII in Fig. 3 is equivalent to a combination of single-qubit gates, 4 SWAP-gates and 4 CZ-gates. Right dashed box: decomposition of a SWAP-gate into CZ-gates. e, Adjacent single-qubit gates are merged and shifted left to the nearest CZ-gate.

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, Step-by-step 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 σ-pairs. b, Experimental demonstration of braiding, displaying the values of the stabilizers throughout the process. Two σ-pairs, 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 SWAP-gates, corresponding to 6 additional CZ-gates. 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 $⟨\hat{\mathcal{P}}⟩=⟨\hat{Y}⟩=-0.71\pm 0.01$, indicating creation of a fermion. Right: world-lines of braiding process, including non-local measurement based on plaquette violation loop. d, Same as c, but after braiding two distinguishable σ, achieved by applying the inverse two-qubit gates when moving the σ in pair B. The measurement yields $⟨\hat{Y}⟩=+0.71\pm 0.01$, indicating no fermion creation.