Qutrit toric code and parafermions in trapped ions

Abstract

The development of programmable quantum devices can be measured by the complexity of many-body states that they are able to prepare. Among the most significant are topologically ordered states of matter, which enable robust quantum information storage and processing. While topological orders are more readily accessible with qudits, experimental realizations have thus far been limited to lattice models of qubits. Here, we prepare and measure a ground state of the $Z_3$ toric code state on 24 qutrits (obtained by encoding one qutrit into two qubits) in a trapped ion quantum processor with fidelity per qutrit exceeding 96.5(3)%. We manipulate two types of defects which go beyond the conventional qubit toric code: a parafermion, and its bound state which is related to charge conjugation symmetry. We further demonstrate defect fusion and the transfer of entanglement between anyons and defects, which we use to control topological qutrits. Our work opens up the space of long-range entangled states with qudit degrees of freedom for use in quantum simulation and universal error-correcting codes.

Fig. 1. Concept.

We start with 56 trapped ¹⁷¹Yb⁺ ions in a quantum charge-coupled device and algorithmically encode pairs of qubits into qutrits. This allows us to prepare ground states of ${\mathbb{Z}}_3$ toric codes on tori of up to 6 × 4 qutrits. We conduct experiments to study the relationship between the anyons and the topological defects of this system, namely parafermion (shaded yellow) and charge conjugation defects (hatch pattern).

Fig. 2. Preparation of qutrit toric code.

a Square lattice on a torus with qutrits on the vertices. b Qutrits are initialized in the $\mid 0\left(\right.⟩$ state, satisfying B =  + 1 (visually represented by the intense turquoise color of type-B plaquettes, in contrast to the faded green color of type-A plaquettes, which do not satisfy the A =  + 1 condition at this stage). c Preparation of one of the type-A plaquettes. One of the qutrits is initialized in the $\mid +\left(\right.⟩$ state. This qutrit is used as a control, and we apply C$\mathcal{X}$ or C${\mathcal{X}}^{†}$ gates to other qutrits in the plaquette. This leads to satisfying A =  + 1, indicated by a bright green color on the right-hand side of the arrow. A small square in one corner of the plaquette indicates the control qutrit. The number within the square denotes the order in which the corresponding stabilizer is prepared. d Expectation values of projectors ${\mathrm{\Pi }}_{\bullet }^1$ obtained by measuring qutrits in the $\mathcal{X}$ and $\mathcal{Z}$ basis. The maximum error in estimating the expectation values is 0.022. The mean energy density, 〈H〉/24≥ − 1, is found to be  − 0.945(3). e Mean expectation values of projectors for the logical $\mathcal{X}$ and $\mathcal{Z}$ operators in two directions on the torus closely match the theoretical predictions: $⟨{\mathrm{\Pi }}_{{\mathcal{Z}}_{\mathrm{hori}}}^1⟩=⟨{\mathrm{\Pi }}_{{\mathcal{Z}}_{\mathrm{vert}}}^1⟩=1$ and $⟨{\mathrm{\Pi }}_{{\mathcal{X}}_{\mathrm{hori}}}^1⟩=⟨{\mathrm{\Pi }}_{{\mathcal{X}}_{\mathrm{vert}}}^1⟩=\frac{1}{3}$. All error bars in this work denote one standard error on the mean.

Fig. 3. Creation of and braiding around parafermion defects.

Any plaquette containing an anyon is colored black, with the value of $\max \left({\mathrm{\Pi }}_{\bullet }^{\omega },{\mathrm{\Pi }}_{\bullet }^{\overline{\omega }}\right)$ displayed. An arrow within each plaquette indicates the direction specified by the arg 〈stabilizer〉, where the stabilizer could be A_p, B_p, or any of the defect stabilizers. The arrow’s direction serves as a visual cue to distinguish anyons from their conjugates. a, b A pair of defects is inserted into the ground state by measuring the middle qutrit in the $\mathcal{X}\mathcal{Z}$-basis and performing feed-forward based on the measurement outcome. c A sketch illustrating the braiding experiment in steps (d–g). A pair of charges, e and $ē$, is created by applying a $\mathcal{Z}$ operator, which toggles the eigenvalues of the neighboring green $\left(\mathcal{X}\right)$-type plaquettes. Charge $ē$ remains fixed, while e is dragged through the defect pair and emerges as m on the other side of the defect pair, signaled by the fact a blue ($\mathcal{Z}$-type) plaquette is now excited. The maximum estimation error is 0.022.

Fig. 4. Creation and braiding of CC defects and their relation to parafermions.

a Ground state of the ${\mathbb{Z}}_3$ toric code with a CC defect pair. The endpoints of the thick line, representing the CC construction circuit U, correspond to high-weight stabilizers A₀ and B₅ (defined in Supplementary Fig. 9c). These stabilizers are marked by a hatching pattern with a ' × ' symbol, and their values are omitted for clarity as they label the internal state of the defects which is not locally accessible. b A sketch of the braiding experiment in (c–f). A flux pair $m-\overline{m}$ is created. $\overline{m}$ is transmuted into m by commuting it through the CC defect line, and it is then fused with the fixed m anyon at the top right corner through a sequence of four steps (c–f), by applying ${\mathcal{X}}_1$, ${\mathcal{X}}_2^{†}$, ${\mathcal{X}}_3^{†}$, and ${\mathcal{X}}_4^{†}$. g Outline of the braiding experiment in (h) where the CC defect pair is fused. This is achieved by applying the same circuit U used in (a) (see Supplementary Note 5). The dashed line shows the path (as implemented by U) taken to fuse the defect pair.The altered state of the CC defect pair is revealed as a flux m at one endpoint. i A sketch of the braiding experiment in (j). We prepare the ground state and create two parafermion defect pairs. The m flux from the pair $m-\overline{m}$ created in the second plaquette from the top, leftmost corner remains fixed, while its partner $\overline{m}$ anyon is commuted through two parafermion defect pairs. The resulting m is then fused with the pinned m to give a single $\overline{m}$ anyon.

Fig. 5. Entanglement transfer from anyons to CC defects for initializing topological qutrits.

a A sketch of the different steps, with intermediate states, involved in moving a charge anyon around defects. The braiding followed by measuring an ancilla transfers a Bell state of charge anyons into an entangled logical state of CC defects. b Results for the final step as depicted in (a). We create ${\mathbb{Z}}_3$ ground state with two pairs of defects, labeled 1 and 2 (cf. Supplementary Fig. 10). Defect pair 1, marked with the  × -hatch pattern, extends between points B₀ and A₇. Defect pair 2, indicated by the /-hatch pattern, has endpoints B₁ and A₈. The orange loop represents a braid that stabilizes the prepared topological qutrit state; a black border on the solid circle indicates the application of ${\mathcal{X}}^{†}$, while its absence indicates $\mathcal{X}$. The expectation values of the projectors ${\mathrm{\Pi }}_{B_0}^1$ and ${\mathrm{\Pi }}_{B_1}^1$ for the non-local stabilizers are 0.931(15) and 0.927(15), respectively. We measure the expectation values of ${\mathrm{\Pi }}_{A_7}^1$, ${\mathrm{\Pi }}_{A_8}^1$, and ${\mathrm{\Pi }}_{A_7{A}_8}^1$ for different ancilla outcomes ($∣0_a⟩$, $∣1_a⟩$, and $∣2_a⟩$). For ${\mathrm{\Pi }}_{A_7}^1$, the measured values are 0.44(5), 0.40(5), and 0.39(5), respectively. Similarly, for ${\mathrm{\Pi }}_{A_8}^1$, the values are 0.44(5), 0.42(5), and 0.36(5). Finally, ${\mathrm{\Pi }}_{A_7{A}_8}^1$ yields values of 0.81(4), 0.78(4), and 0.74(4) for the respective ancilla states. This is a manifestation of the fact that, although the outcomes for each individual defect pair are random, they are jointly in an entangled state.