Surface codes: Towards practical large-scale quantum computation

TL;DR

The concept of the stabilizer, using two qubits, is introduced, and the single-qubit Hadamard, S and T operators are described, completing the set of required gates for a universal quantum computer.

Abstract

This article provides an introduction to surface code quantum computing. We first estimate the size and speed of a surface code quantum computer. We then introduce the concept of the stabilizer, using two qubits, and extend this concept to stabilizers acting on a two-dimensional array of physical qubits, on which we implement the surface code. We next describe how logical qubits are formed in the surface code array and give numerical estimates of their fault-tolerance. We outline how logical qubits are physically moved on the array, how qubit braid transformations are constructed, and how a braid between two logical qubits is equivalent to a controlled-NOT. We then describe the single-qubit Hadamard, S and T operators, completing the set of required gates for a universal quantum computer. We conclude by briefly discussing physical implementations of the surface code. We include a number of appendices in which we provide supplementary information to the main text.

Figures (35)

• Figure 1: (Color online)(a) A two-dimensional array implementation of the surface code. Data qubits are open circles$(\mathrm{O})$, measurement qubits are filled circles $(\bullet)$, with measure-Z qubits colored green (dark) and measure-X qubits colored orange (light). Away from the boundaries, each data qubit contacts four measure qubits, and each measure qubit contacts four data qubits; the measure qubits perform four-terminal measurements. On the boundaries, the measure qubits contact only three data qubits and perform three-terminal measurements, and the data qubits contact either two or three measure qubits. The solid line surrounding the array indicates the array boundary. (b) Geometric sequence of operations (left), and quantum circuit (right) for one surface code cycle for a measure-Z qubit, which stabilizes $\hat{Z}_{a} \hat{Z}_{b} \hat{Z}_{c} \hat{Z}_{d}$. (c) Geometry and quantum circuit for a measure-X qubit, which stabilizes $\hat{X}_{a} \hat{X}_{b} \hat{X}_{c} \hat{X}_{d}$. The two identity $\hat{I}$ operators for the measure-Z process, which are performed by simply waiting, ensure that the timing on the measure-X qubit matches that of the measure-Z qubit, the former undergoing two Hadamard $\hat{H}$ operations. The identity operators come at the beginning and end of the sequence, reducing the impact of any errors during these steps.
• Figure 2: (Color online) Schematic evolution of measurement outcomes (filled circles with ± signs), over a segment of the 2D array. Time progresses moving up from the array at the bottom of the figure, with measurement steps occurring in each horizontal plane. Vertical heavy red (gray) lines connect time steps in which a measurement outcome has changed, with the spatial correlation indicating an$\hat{X}$ bit-flip error, a $\hat{Z}$ phase-flip error, a $\hat{Y}=\hat{Z} \hat{X}$ error, and temporal correlation a measurement ( $M$ ) error, which is sequential in time.
• Figure 3: (Color online) A square 2D array of data qubits, with X boundaries on the left and right, and Z boundaries on the top and bottom. The array has 41 data qubits, but only 40$\hat{X}$ and $\hat{Z}$ stabilizers. A product chain $\hat{X}_{L}=\hat{X}_{1} \hat{X}_{2} \hat{X}_{3} \hat{X}_{4} \hat{X}_{5}$ of $\hat{X}$ operators connects the two X boundaries, commutes with all the array stabilizers and changes the array state from the quiescent state $|\psi\rangle$ to $\left|\psi_{X}\right\rangle=\hat{X}_{L}|\psi\rangle$ with the same measurement outcomes as $|\psi\rangle$. A second product chain $\hat{Z}_{L}=\hat{Z}_{6} \hat{Z}_{7} \hat{Z}_{3} \hat{Z}_{8} \hat{Z}_{9}$ connects the two Z boundaries and commutes with the array stabilizers; it changes the array state from $|\psi\rangle$ to $\left|\psi_{Z}\right\rangle=\hat{Z}_{L}|\psi\rangle$. The operator chains $\hat{X}_{L}$ and $\hat{Z}_{L}$ anti-commute. A modification of the $\hat{X}_{L}$ chain to the chain $\hat{X}_{L}^{\prime}=\hat{X}_{1} \hat{X}_{10} \hat{X}_{11} \hat{X}_{12} \hat{X}_{3} \hat{X}_{4} \hat{X}_{5}$ generates a quiescent state $\left|\psi_{X^{\prime}}\right\rangle=X_{2,10,11,12}\left|\psi_{X}\right\rangle$, related to $\left|\psi_{X}\right\rangle$ by the result of the measurement $X_{2,10,11,12}= \pm 1$ of the encircled measure-X qubit (outlined in black).
• Figure 4: (Color online) (a) Numerical simulations of surface code error rates, and how these error rates scale with the distance$d$ of the array. Per-step error rates $p$ less than the per-step threshold error rate of $p_{\text{th }}=0.57 \%$ (dashed line) yield surface code logical error rates $P_{L}$ that vanish rapidly with increasing $d$. This threshold corresponds to roughly a $4 \%$ error rate for the entire surface code measurement cycle. (b) Estimated error rates, using statistical arguments given in main text, for array distances $d=3,7,11,25$ and 55 . Note the estimate gives a logical error rate $P_{L}$ and a per-step threshold that are qualitatively similar to those from the more precise simulations.
• Figure 5: (Color online) (a) An example where two measure-Z qubits report errors in a single row of a 2D array, marked by "E"s. This error report could be generated by (b) two$\hat{X}$ errors appearing in the same surface code cycle on the 2nd and 3rd data qubit from the left, or (c) three $\hat{X}$ errors appearing in the other three data qubits in the row.