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Generation of logical zero states encoded with a quantum error-correctingcode is the first step for fault-tolerant quantum computation, but requiresconsiderably large resource overheads in general. To reduce such overheads, wepropose an efficient encoding method for the distance-three, nine-qubit surfacecode and show its fault tolerance. This method needs no measurement, unlikeother fault-tolerant encoding methods. Moreover, this is applicable to aone-dimensional qubit array. Observing these facts, we experimentallydemonstrate the logical zero-state encoding of the surface code using asuperconducting quantum computer on the cloud. We also experimentallydemonstrate the suppression of fast dephasing due to intrinsic residualinteractions in this machine by a dynamical decoupling technique dedicated forthe qubit array. To extend this method to larger codes, we also investigate theconcatenation of the surface code with itself, resulting in a distance-nine,81-qubit code. We numerically show that fault-tolerant encoding of this largecode can be achieved by appropriate error detection. Thus, the proposedencoding method will provide a new way to low-overhead fault-tolerant quantumcomputation.

Measurement-free fault-tolerant logical zero-state encoding of the distance-three nine-qubit surface code in a one-dimensional qubit array