how to implement multi-qubit logical operators in the surface code,b

Implementing multi-qubit logical operators in the surface code involves encoding and manipulating multiple logical qubits within the surface code framework. Here are some insights from the search results:

Dividing the Hilbert Space: A method has been proposed that involves dividing the Hilbert space into reduced Hilbert spaces with individual logical qubits and synthesizing these reduced Hilbert spaces through a mathematical collaboration between classical bits and logical quantum states. This approach supports at least 20 logical qubits in a surface code with a code distance of 3[1].

Synthesizing Related Logical Qubits for Lattice Surgery: To perform logical CNOT in the surface code, one approach involves generating one Hilbert space by synthesizing related logical qubits for lattice surgery. This method supports large-scale logical qubits in the surface code using a classical computer[1].

Subsystem Surface Code: A simplified version of the surface code, known as the subsystem surface code, allows for the encoding of multiple logical qubits on a planar lattice with punctured holes. This code inherits many favorable properties of the standard surface code, such as efficient decoding and high error threshold[2].

Logical Qubit Behavior Model: A logical qubit behavior model (LQBM) has been proposed to calculate the state of the logical qubit based on a surface code after a logical qubit operation. This model can simulate the surface code and calculate the output state of the logical qubit after the logical operation without performing a complex process of error syndrome measurement[4].

These approaches and models demonstrate the ongoing research and development in implementing multi-qubit logical operators in the surface code, aiming to enable the encoding and manipulation of multiple logical qubits within the fault-tolerant framework of the surface code.

Citations: [1] https://iopscience.iop.org/article/10.1088/2058-9565/accec5/meta [2] https://scipost.org/SciPostPhysLectNotes.49/pdf [3] https://iopscience.iop.org/article/10.1088/2058-9565/accec5/pdf [4] https://link.springer.com/article/10.1007/s11128-023-04044-8 [5] https://clelandlab.uchicago.edu/pdf/fowler_et_al_surface_code_submit_3po.pdf

lattice surgery

twist deformations

how to implement multi-qubit logical operators in the surface codeb