We can now proceed as for the toric code. a chain in the dual lattice, which has zero boundary in the dual lattice. Similarly, the blue line represents a co-chain, i.e. In the relative version of the chain complex, its boundary is zero, hence this is the equivalent to a closed loop in the toric geometry. Its boundary would be in the (removed) rough boundary. Note that this one-chain is “open-ended”. Here we have removed the rough boundary at the left and right of the lattice and the same for the dual lattice. ![]() In the literature, the lattice and its dual are typically visualized differently, namely with all edges that do not carry qubits removed, as in the following diagram. a path perpendicular to the edges of the lattice L. Similarly, a co-chain is a path in the dual lattice, i.e. the edges that carry qubits, we obtain a subgroup which is known to topologists as the group of one chains relative to the rough boundary – again I will strive to keep this post free from too much algebraic topology and refer the reader to my notes for more details and a more precise description.Īs for the toric code, every such relative one-chain c gives raise to a operators X c and Z c, obtained by letting X respectively Z act on any qubit crossed by the chain. If we only allow the edges that are not part of the rough boundary, i.e. a set of edges, is an element of a group, the group of one chains. Again, some edges of the dual lattice carry qubits and others do not, so we have a smooth and a rough boundary as well (note that the smooth boundary of L gives raise to the rough boundary of the dual lattice and vice versa).Īgain a path along the edges, i.e. This dual lattice is indicated by the dashed lines in our diagram. We again place qubits on the edges of the lattice, but for the boundary, we only place qubits (again indicated by black dots) on the edges which are part of the smooth boundary.Īs for the case of a toric code, we can again create a second lattice called the dual lattice whose vertices are the center points of the faces of L and whose edges are perpendicular to the edges of L. The solid lines form a lattice L with two types of boundary – a “smooth” boundary at the top and the bottom and a “rough boundary” left and right (ignore the dashed lines for a moment, we will explain their meaning in a second). The solution found in is to use two types of boundaries. So we need some a more sophisticated boundary condition. What happens if we simply drop it? A short calculation using a bit of algebraic topology shows that in this case, the code space collapses to a one-dimensional code space which is not enough to hold a logical qubit. If we want to use this code in planar setting, it is clear that we somehow have to modify the boundary conditions. Recall that the starting point for the toric code was a lattice L with periodic boundary conditions so that the geometry of the lattice becomes toroidal, which gave us a four-dimensional code space. Luckily, a version of the toric codes that works well in a planar geometry exists – the surface code. In reality, qubits are more likely to be arranged in a planar geometry. If the parity checks for the current surface code cycle are the same as the previous one, we conclude that the state of the data qubits involved in the parity checks has not changed due to any erroneous operation.In my previous post on quantum error correction, we have looked at the toric code which is designed for a rather theoretical case – a grid of qubits on a torus. We present pseudo-threshold and threshold values for the proposed surface code design for asymmetric error channels in the presence of various degrees of asymmetry of Pauli X ^, Y ^ $\text$ stabilisers, that is, their outcome can predict the presence of errors without perturbing the system. A surface code design is being introduced, where d x( d z) represents the distance of the code for bit (phase) error correction, motivated by the fact that the severity of bit flip and phase flip errors in the physical quantum system is asymmetric. Surface codes are quantum error correcting codes typically defined on a 2D array of qubits. IET Generation, Transmission
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