Creating Entangled Logical Qubits in the Heavy-Hex Lattice with Topological Codes

Designs for quantum error correction depend strongly on the connectivity of the qubits. For solid-state qubits, the most straightforward approach is to have connectivity constrained to a planar graph. Practical considerations may also further restrict the connectivity, resulting in a relatively sparse graph such as the heavy-hexagonal ("heavy-hex") architecture of current IBM Quantum devices. In such cases, it is hard to use all qubits to their full potential. Instead, in order to emulate the denser connectivity required to implement well-known quantum error-correcting codes, many qubits remain effectively unused. In this work, we show how this bug can be turned into a feature. By using the unused qubits of one code to execute another, two codes can be implemented on top of each other, allowing easy application of fault-tolerant entangling gates and measurements. We demonstrate this by realizing a surface code and a Bacon-Shor code on a 133-qubit IBM Quantum device. Using transversal controlled-X (cx) gates and lattice surgery, we demonstrate entanglement between these logical qubits with code distance up to d=4 and five rounds of stabilizer-measurement cycles. The nonplanar coupling between the qubits allows us to simultaneously measure the logical XX, YY, and ZZ observables. With this, we verify the violation of Bell's inequality for both the d=2 case with postselection featuring a fidelity of 94% and the d=3 instance using only quantum error correction.