Chapter 13 - Topological Quantum Error Correction

Surface Code

Introduction

Surface codes are topological quantum error-correcting codes in which we can think of qubits being arranged on a 2-D lattice of qubits with only nearest neighbor interactions. This in practice may prove to be a very useful feature, since for many systems interacting qubits that are close to each other is substantially less difficult than ones that are further apart. We can think of physical qubits as being arranged on the edges of a lattice as shown in Figure 1. An example of the surface code are toric code and planar code, the main difference between both of them is the boundary condition. In the toric code, the boundaries are periodic whereas in the case of the planar code, the boundaries are not periodic. In the toric code, the qubits are arranged on a lattice which can be thought of as spread over a surface of a torus, and in a planar code case we think of the data qubits as living on a simple 2-D plane and ancilla qubits on the faces and the intersections.

Figure 1A two-dimensional array implementation of the surface code. Data qubits are open circles, measurements (ancilla) qubits are filled circles.The yellow area is to measure-Z qubits while the green area is to measure-X qubits.

The stabilizer generators for the surface code are the tensor products of Z on the four data qubits around each face, and the tensor products of X on the four data qubits around each intersection. Neighbouring stabilizers share two data qubits ensuring that adjacent X and Z stabilizers commute. The qubit Z eigenstate are called the ground state $\left\vert {g}\right\rangle$ and the excited state $\left\vert {e}\right\rangle$. The ground state is the +1 eigenstate of Z, with $Z\left\vert {g}\right\rangle =+\left\vert {g}\right\rangle$, and the excited state is the -1 eigenstate, with $Z\left\vert {e}\right\rangle =-\left\vert {e}\right\rangle$. It is tempting to think of the qubit as a kind of quantum transistor, with the ground state corresponding to "off" and the excited state to "on". However, in distinct contrast to a classical logical element, a qubit can exist in a superposition of its eigenstate, $\left\vert {\psi }\right\rangle =\alpha \left\vert {g}\right\rangle +\beta \left\vert {g}\right\rangle$, so a qubit can be both "off" and on" at the same time. A measurment $M_{Z}$ of the qubit will however return only one of two possible measurement outcomes,+1 with the qubit state projected to $\left\vert {g}\right\rangle$, or -1 with the qubit state projected to $\left\vert {e}\right\rangle$ .

A planar code has four boundaries, two that are called “smooth” and two that are called “rough”. Smooth boundaries have four-term Z stabilizer generators, and three-term X stabilizer generators, whereas rough boundaries have four-term X stabilizer generators and three-term Z stabilizer generators. A planar code, with two rough and two smooth boundaries can encode a single logical qubit (as in Figure 2). Also look at http://arxiv.org/abs/1208.0928 it is an excellent reference as it represents a comprehensive review of the surface code, also written for the absolute beginner.

Figure 2Examples of smooth and rough boundaries. This figure has been copied with a permission from the authors of Ref.2

Syndrome Extraction and Error Detection

Detecting errors involves measuring check operators, and observing which ones give a value of -1 (due to anti-commuting with errors). This information helps us guess where errors occurred. In practice, of course, errors do not have to occur on their own, and often one can observe multiple instances next to each other. In these cases, the error operators form error chains throughout the lattice. Since only the ends of such error chains anti-commute with the check operators, determining where errors occurred often involves guessing the most likely scenario. In the planar case, the chains connect opposite boundaries of the same type (either left to right, or top to bottom), and in the toric case, chains that span all the way across a given dimension of the lattice. They turn out to the change the encoded, logical state of the qubit and hence are called logical errors. Two examples are shown Figures 3 and 4. $Z_{L}$ is a chain of Z operators that connects two rough boundaries, and $X_{L}$ chain of X operators that connects two smooth ones.

Figure 3Examples of error syndromes on the Surface code (planar and toric). The state is initialized to the +1 eigenstate of all stabilizers.Shaded qubits indicate locations of X errors. This figure has been copied with a permission from the authors of Ref.5 Figure 4A planar surface code in which a logical Z (X) error is a chain of Z (X) operators that spans the whole lattice, and connects rough (smooth) boundaries.This figure has been copied with a permission from the authors of Ref.5