Research ArticlePHYSICS

A fault-tolerant non-Clifford gate for the surface code in two dimensions

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Science Advances  22 May 2020:
Vol. 6, no. 21, eaay4929
DOI: 10.1126/sciadv.aay4929


  • Fig. 1 The three-dimensional surface code in spacetime.

    (A) The topological cluster-state model is a three-dimensional model that propagates quantum information over time with only a two-dimensional array of live qubits at any given moment. We show a gray plane of live qubits that propagates in the direction of the time arrow. (B) Gray loops show the connectivity of plaquette measurements that returned the −1 outcome. An arbitrary state is initialized fault-tolerantly by initializing the system with an encoded two-dimensional fixed-gauge surface code on the gray face at the left of the image. (C) The boundary configurations of the three copies of the surface code are required to perform a local transversal controlled-controlled-phase gate. The first code has rough boundaries on the top and the bottom of the lattice. The middle (right) code has rough boundaries on the left and right (front and back) sides of the lattice. The orientation of the boundaries determines the time direction in which we can move the planes of live qubits.

  • Fig. 2 The non-Clifford gate shown in spacetime.

    Two codes traveling in different temporal directions cross. The third code is omitted as it can run in parallel with one of the two shown. Live qubits of the spacetime history are shown on light gray planes. The transversal gate is applied in the cubic region in the middle. It will be applied on the qubits shown at the dark gray plane where the two-dimensional arrays of qubits are overlapping.

  • Fig. 3 Just-in-time gauge fixing.

    The spacetime diagram of an error on the dual qubits of the topological cluster state where time travels upward. The gray area shows the two-dimensional area of live qubits at a given moment. At the point where an error is found on the left diagram, it is unlikely that the defects should be paired because of their separation. We therefore defer matching the defects to a later time after more information emerges as decoding progresses, as in the middle figure. After enough time, the most likely outcome is that the defects we found in the left figure should be paired. The error we introduce fills the interior of the error and the chosen correction.

  • Fig. 4 A lattice geometry for a surface code.

    (A) A unit cell is composed of four primal cubes and four dual cubes configured as shown with primal and dual cubes shown in white and gray, respectively. (B) A Pauli-X “star” operator supported on a primal cube. (C) A plaquette operator supported on the corner of a dual cube. (D) A smooth boundary stabilizer. (E) A rough boundary stabilizer.

  • Fig. 5 Microscopic details of the gauge-fixing procedure.

    One period of the gauge-fixing process for the models undergoing the controlled-controlled-phase gate. Time progresses between the figures from the left to the right from time t to t + 1 via an intermediate step at time t + 1/2. (A) The lattice above shows the code moving from left to right through the spacetime volume of the controlled-controlled-phase gate, marked by the black cube, and (B) the lower figures show a code moving upward through the black cubic region. The live surface codes are overlapping at all points in time. The figures at the left show a two-dimensional surface code. In the middle figures, we produce a thin layer of three-dimensional surface code by adding additional qubits and measuring the plaquette operators that are supported on the displayed qubits. The gauge-fixing correction is made before transversal controlled-controlled-phase gates are applied. Once the controlled-controlled-phase gates are applied, qubits are measured destructively to recover the system at the right of the figure.

  • Fig. 6 A two-dimensional layout for the non-Clifford gate.

    The progression of the controlled-controlled-phase gate. (A) Qubits are copied onto the stacked arrays of qubits from other surface codes using lattice surgery. (B) The thick black qubits are passed under the other two arrays and controlled-controlled-phase gates are applied transversally between the three arrays where the qubits overlap. (C) and (D) show later stages in the dynamics of the gate.

  • Fig. 7 Error correction with just-in-time gauge fixing.

    Not to scale. The diagram sketches the proof of a threshold for the controlled-controlled-phase gate. (A) An error described by the chunk decomposition acting on the qubits included on the spacetime of the controlled-controlled-phase gate. See Lemma 1. The image shows connected components of the error contained within black boxes. Errors are shown at two length scales. One error at the larger length scale is shown to the top right of the image. (B) After just-in-time gauge fixing is applied, errors are spread by a constant factor of the size of the connected components. This is shown by the gray regions around each of the initial black errors. (C) Given a sufficiently large Q, the spread is not problematic since smaller untethered spread errors are far away from other components of equal or greater size. They are therefore easily dealt with by the renormalization group decoder. Small components of the error that lie close to a larger error will be neutralized with the larger error close to its boundary.

  • Fig. 8 Fixing the gauge of the two-dimensional surface code.

    (A) A single measurement error on a face at the beginning of the controlled-controlled-phase gate will introduce a defect that may not be paired for a time that scales like the size of the system; this may introduce a macroscopic error after gauge fixing. (B) We can determine where errors have occurred on the plaquettes of the initial face by looking at the defects before we begin the controlled-controlled-phase gate. (C) We decode, or prefix the initial face, before we begin the controlled-controlled-phase gate to determine the locations of measurement errors on the initial face.

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