The standard surface code is defined on a square lattice with (data) qubits located on vertices. Stabilizers are associated with faces and boundaries. Aquamarine faces and boundaries indicateZ-type stabilizers, as in equation (6). Red faces and boundaries indicateX-type stabilizers, as in equation (5). The surface code with boundaries encodes a single logical qubit defined by its logical Pauli-X and Pauli-Z operators. These operators are defined on paths connecting opposite boundaries of the lattice and act as products ofX- andZ-operators, respectively, along the paths. Here, two representative logical operators are drawn as products of Pauli-operators within the dashed rectangles. Red indicates Pauli-X operators and green indicates Pauli-Z operators. The two operators anti-commute at the crossing drawn in yellow.

Lattice surgery (LS) enables measurement-based implementations of logic gates and logical state teleportation. LS operations are logical joint measurements of the form\({M}_{P\mathop{P}\limits^{ \sim }}=({\mathbb{I}}\pm P\mathop{P}\limits^{ \sim })/2\) where\(P,\mathop{P}\limits^{ \sim }\) are Pauli operators. Moreover, the protocols make use of single-qubit measurements of the formM_P = (\({\mathbb{I}}\) ± P)/2. Thick lines indicate logical qubits in the circuit model and double lines represent classical bits indicating measurement outcomesm_i = 0, 1. Pauli corrections need to be applied which are conditioned on the measurement outcomes as\({P}_{{\rm{L}}}^{{m}_{i}}\). The symbol⊕ represents an XOR-gate between classical bits.a, Measurement-based implementation of a logical CNOT-gate between arbitrary control and target qubits requiring an auxiliary qubit in the |+_L⟩ state.b, Measurement-based teleportation protocol for state teleportation between two logical qubits usingX-type LS.c, Measurement-based implementation of a logical Hadamard gateH based on the teleportation protocol.

Surface code LS betweenZ-type andX-type boundaries implementing logical joint measurements\({M}_{{\rm{XX}}}^{\pm }=({\mathbb{I}}\pm {X}_{{\rm{L}}}^{{\rm{A}}}{X}_{{\rm{L}}}^{{\rm{B}}})/2\) (a) and\({M}_{{\rm{ZZ}}}^{\pm }=({\mathbb{I}}\pm {Z}_{{\rm{L}}}^{{\rm{A}}}{Z}_{{\rm{L}}}^{{\rm{B}}})/2\) (b), respectively.A, Encoded. The two initial surface codes are defined on 2 × 2 lattices whereX-stabilizers are associated with orange faces andZ-stabilizers with aquamarine faces in accordance with equation (1). Logical operators are products of Pauli operators connecting opposite boundaries as in equation (2).a,Z-type encoded. The two surface codes are arranged such that they are aligned along theirZ-type boundary.b, X-type encoded. The surface codes are aligned along theirX-type boundary.B, Merged. Treating the two codes as a single (asymmetric) surface code, (merging) stabilizers along the boundaries are measured. The merged code encodes a single logical qubit corresponding to the logical Pauli operators\({X}_{{\rm{L}}}^{{\rm{M}}},{Z}_{{\rm{L}}}^{{\rm{M}}}\).a,Z-type merged. Merging stabilizers (indicated in red) are chosen such that their product is\({X}_{{\rm{L}}}^{{\rm{A}}}{X}_{{\rm{L}}}^{{\rm{B}}}\).b,X-type merged. Merging stabilizers (indicated in green) are chosen such that their product is\({Z}_{{\rm{L}}}^{{\rm{A}}}{Z}_{{\rm{L}}}^{{\rm{B}}}\).C, Split. In order to split the merged code while preserving the eigenstate of the joint logical operator, the boundary stabilizers of the original code are measured. These operators anti-commute with the merging stabilizers and thus project onto the individual codes. Since the boundary operators commute with individual logical operators, the resulting state remains an eigenstate of the joint logical operator.a,Z-type split. MeasuringZ-stabilizers along the boundary (indicated in green) preserves the eigenstate of\({X}_{{\rm{L}}}^{{\rm{A}}}{X}_{{\rm{L}}}^{{\rm{B}}}\) while projecting onto the individual codes.b,X-type split. MeasuringX-stabilizers along the boundary (indicated in red) preserves the eigenstate of\({Z}_{{\rm{L}}}^{{\rm{A}}}{Z}_{{\rm{L}}}^{{\rm{B}}}\) while projecting onto the individual codes.

Bell state generation via lattice surgery along theX-type boundary between two surface code qubits through a logical joint measurement\({M}_{{\rm{ZZ}}}^{+}\propto {\mathbb{I}}+{Z}_{{\rm{L}}}^{{\rm{A}}}{Z}_{{\rm{L}}}^{{\rm{B}}}\). Post-selected measurements are presented in light coloured bars.A, Encoded. Two logical qubits (a) are encoded with average stabilizer values (b) of⟨|S_i|⟩ = 0.813(4). We observe raw and post-selected state fidelities (c) of \( {\mathcal F} (|{0}_{{\rm{L}}}^{{\rm{A}}}\rangle )=93.3(5)|98.7(2){\rm{ \% }}\) for logical qubit A and\( {\mathcal F} (|{0}_{{\rm{L}}}^{{\rm{B}}}\rangle )=92.4(5)|97.9(3){\rm{ \% }}\) for logical qubit B.B, Merged. The two separated logical qubits are merged (a) into a single logical qubit by measuring the stabilizer\({S}_{7}^{{\rm{M}}}\) using auxiliary qubit A₁ as syndrome qubit. Thereby, the code space is extended in the vertical direction and the new logical operator\({X}_{{\rm{L}}}^{{\rm{M}}}={X}_{{\rm{L}}}^{{\rm{A}}}{X}_{{\rm{L}}}^{{\rm{B}}}\) is formed. As the data show, the stabilizer\({S}_{7}^{{\rm{M}}}\) is indeed created. The average stabilizer values (b) are⟨|S_i|⟩ = 0.719(5) and logical state fidelities (c) are\( {\mathcal F} (|{+}_{{\rm{L}}}^{{\rm{M}}}\rangle )=76.2(8)|93.1(6){\rm{ \% }}\).C, Split. The single logical qubit is again split into two logical qubits (a) along the same boundary they have been initially merged through. We measure the stabilizer\(\bar{S}{}_{6}^{{\rm{M}}}\) by using auxiliary qubit A₂ as syndrome qubit to perform the splitting and obtain average stabilizer values (b) of⟨|S_i|⟩ = 0.763(5). The fidelity (c) of the generated state to the logical Bell state is\( {\mathcal F} (|{\psi }_{{\rm{L}}}^{+}\rangle )=63.9(2.8)|78.0(2.7){\rm{ \% }}\). Note that measuring the merging stabilizer\({S}_{7}^{{\rm{M}}}\) = \({Z}_{{\rm{L}}}^{{\rm{A}}}{Z}_{{\rm{L}}}^{{\rm{B}}}\) directly projects onto a joint eigenstate of the logicalZ-operators such that the splitting becomes redundant. Nevertheless, the general procedure as described in Methods requires the measurement ofX-stabilizers along the boundary which is why it is still included here.

This file contains additional details on the performed experiment.