High-threshold universal quantum computation on the surface code

We present a comprehensive and self-contained simplified review of the quantum computing scheme of Raussendorf et al. [Phys. Rev. Lett. 98, 190504 (2007); N. J. Phys. 9, 199 (2007)], which features a two-dimensional nearest-neighbor coupled lattice of qubits, a threshold error rate approaching 1%, natural asymmetric and adjustable strength error correction, and low overhead arbitrarily long-range logical gates. These features make it one of the best and most practical quantum computing schemes devised to date. We restrict the discussion to direct manipulation of the surface code using the stabilizer formalism, both of which we also briefly review, to make the scheme accessible to a broad audience.

Figure 1

Basic layout of surface code data qubits, each represented by a circle. A data qubit is located at the center of each edge of a square lattice. The square lattice is a guide for the eye only; it does not represent interactions.

Figure 2

(Color online) Effect of single bit flips $\left(X\right)$ and phase flips $\left(Z\right)$ on the surface code—adjacent face and vertex stabilizers are made negative, respectively.

Figure 3

(Color online) Surface code suffering from multiple errors (indicated by filled dots) and an incorrect syndrome measurement (indicated by dashed circle).

Figure 4

(Color online) (a) Locations in space and time, indicated by filled dots, where and when the reported syndrome is different from that in the previous time step. Note that this is not a three-dimensional physical structure, just a three-dimensional classical data structure. (b) Optimal matching highly likely to lead to a significant reduction of the number of errors if bit-flips are applied to the spacelike edges.

Figure 5

(Color online) Examples of smooth and rough boundaries including a chain of $X$ errors ending in a smooth boundary without changing the sign of any stabilizers, and a chain of $Z$ errors ending in a rough boundary, also without leaving any evidence of its presence. $X$ and $Z$ stabilizers are represented by shaded shapes.

Figure 6

(Color online) Surface code with one additional degree of freedom introduced by not enforcing the stabilizer associated with one face (shaded). This face or a region of such faces is called a smooth defect. The degree of freedom can be phase flipped by any ring of $Z$ operators encircling the defect and bit flipped by any chain of $X$ operators connecting the defect to a smooth boundary.

Figure 7

(Color online) Surface code with one degree of freedom introduced via the measurement $M_X$ of four qubits in the $X$ basis and removal of five stabilizers. Note that four new three term $X$ stabilizers are created with not necessarily positive sign (indicated by shaded triangles).

Figure 8

(Color online) Smooth qubit comprised of two smooth defects. $Z_L$ corresponds to any ring of $Z$ operators around either defect. $X_L$ corresponds to any chain of $X$ operators connecting the two defects.

Figure 9

(Color online) Initializing a smooth qubit in the $|{+}_L⟩$ state. After preparing a region of qubits each in the $|+⟩$ state, every $X$ stabilizer on the boundary of the region and every $Z$ stabilizer outside the dashed regions is measured. Negative eigenvalues are treated as errors and corrected.

Figure 10

(Color online) Initializing a rough qubit in the $|{+}_L⟩$ state via $Z$ basis measurements $M_Z$ and ignoring stabilizers (shaded). $X_L$ is any ring of $X$ operators around either defect. $Z_L$ is any chain of $Z$ operators linking the two defects.

Figure 11

Example of measurement of a smooth qubit in the $Z_L$ basis in the absence of errors. Note that the measurements around every face have even parity whereas the parity of any path of measurements encircling either defect is odd. The figure thus corresponds to the measurement result $|1_L⟩$.

Figure 12

(Color online) (a) Smooth defect and surface in the $+1$ eigenstate of $Z_L$. (b) After measuring the center qubit in the $X$ basis, the shape of the $Z_L$ operator is deformed. (c) Measuring and possibly correcting the indicated $Z$ stabilizer using a bit flip on the center qubit completes the movement of the defect.

Figure 13

(Color online) (a) Smooth defect and surface in the $+1$ eigenstate of $X_L$. (b) After measuring the center qubit in the $X$ basis, it is possible that three term $X$ stabilizers and $X_L$ stabilizers with negative sign are created (potential locations indicated by shaded triangles). (c) All signs can be corrected by applying the appropriate single-qubit $Z$ operators and chains of $Z$ operators. (d) Measuring and possibly correcting the indicated $Z$ stabilizer using a bit flip on the center qubit completes the movement of the defect.

Figure 14

(Color online) (a) Movement of a large smooth defect via many measurements in the $X$ basis. Many pairs of three term $X$ stabilizers with negative sign are likely to be created (indicated by shaded triangles). (b) After several rounds of error correction, it becomes exponentially unlikely that three term $X$ stabilizers with negative sign remain that were generated in the measurement round. New chains of errors on the boundary can occur, but these will be corrected during normal error correction after the size of the defect is reduced to complete the movement.

Figure 15

(Color online) (a) Surface containing a smooth qubit in the $+1$ eigenstate of $X_L$ and a rough qubit. The lower smooth defect has been braided around the upper rough defect using $X$ measurements. Note that is not possible to complete the braiding in one step as a ring of $X$ measurements corresponds to measurement of the rough qubit in the $X_L$ basis. (b) Via correction of many $Z$ stabilizers, the $X_L$ operator is dragged around the upper rough defect. (c) Additional $X$ measurements extend the defect back to its original position. (d) Further correction of $Z$ stabilizers returns the defects to their original positions but the surface is now in the $+1$ eigenstate of both the smooth and rough $X_L$ operators.

Figure 16

(Color online) (a) Surface containing a smooth defect and a rough defect in the $+1$ eigenstate of $Z_L$. The lower smooth defect has been braided around the upper rough defect using $X$ measurements, deforming the shape of the rough $Z_L$ operator. (b) By first correcting many $Z$ stabilizers and then performing further $X$ measurements, the smooth defect can be extended back to its original position. (c) A final round of $Z$ stabilizer correction returns the defects to their original configuration but with the state of the surface changed. (d) The $Z_L$ operator shown in part (c) is equivalent to the tensor product of smooth and rough $Z_L$.

Figure 17

(Color online) Smooth qubits are represented by black lines and rough qubits by lighter lines. (a) Circuit equivalent to $Z^{M_X}$ on the target qubit followed by CNOT between the control and target qubit followed by $X^{M_Z}$ on the target qubit. (b) Schematic representing the initialization, braiding, and measurement of defects in a surface code to implement (a). Time runs from left to right, and the surface code should be imagined oriented vertically and into and out of the page. (c) Simplified schematic equivalent to (b).

Figure 18

Surface code fragment and numbered qubits used to assist the visualization of the discussion of Sec. 6A.

Figure 19

(a) Encoding circuit for the seven-qubit Steane code. (b) Distillation circuit for the $|Y⟩=|0⟩+i|1⟩$ state.

Figure 20

(a) Encoding circuit for the 15-qubit Reed-Muller code. (b) Distillation circuit for the $|A⟩=|0⟩+e^{i\pi /4}|1⟩$ state.

Figure 21

(a) Circuit performing the single-qubit unitary $X^{M_Z}{R}_Z\left[{\left(-1\right)}^{M_Z}\theta \right]$ given an appropriate ancilla state. (b) Circuit performing the single-qubit unitary $Z^{M_X}{R}_X\left[{\left(-1\right)}^{M_X}\theta \right]$ given an appropriate ancilla state.

Figure 22

(Color online) A smooth qubit isolated from a larger piece of surface code using a ring of $Z$ measurements so that logical Hadamard can be applied by local transversal Hadamard gates. Shaded triangles in the upper part of the figure represent three term $Z$ stabilizers of negative sign. See text for details.

Figure 23

Circuit showing how an additional syndrome qubit (top line of each figure) is used to measure (a) $Z$ stabilizers and (b) $X$ stabilizers.

Figure 24

(Color online) Syndrome measurement typically involves six gates: syndrome initialization, CNOT with the four surrounding data qubits (fewer on boundaries), and finally syndrome readout.

Figure 25

(Color online) Logical $Z\left(X\right)$ error detection involves checking if the parity of $Z\left(X\right)$ operators along any of the vertical (horizontal) lines of qubits is odd. Above, we show this in an example of a logical $Z$ error.

Figure 26

(Color online) A plot of average time until failure versus the physical error rate $p$. A threshold is observed at $p\approx 6.0\times {10}^{-3}$ where the curves of different lattice sizes cross. The case where no error correction is used (single qubit) is represented by a dashed line.

Figure 27

(Color online) Potential $X$ error chains (dashed lines) around a rough defect and consequent minimum separation from long straight smooth boundaries and two types of corners such that $X$ error chains beginning and ending on the smooth boundary are not more likely than an $X$ error ring around the rough defect.

Figure 28

(a) A rough qubit ready to be sent to a separate piece of surface code. (b) Remote gates are used to join the two surfaces together. (c) A sequence of measurements is used to move the rough qubit. After the necessary correction associated with completing the movement, the long-range gates can be discontinued to separate the two pieces of surface code once more.