Overhead for code switching

The parameters for the lattices of the 4.8.8 (square-octagon) and 6.6.6 (hexagonal) two-dimensional color codes on triangular boundaries are given in Table5.3. Note that for all distances d > 3, the number of vertices v for the hexagonal lattice is larger than that of the square-octagon lattice. The number of facesf is also larger for the hexagonal lattice.

For overhead calculations, the number of data qubits required isv, and the number of measurement qubits required depends on how one chooses to measure stabilizers. If there is one measurement qubit per stabilizer generator, then 2f data qubits are

required, whereas if a single measurement qubit is used to measure bothX andZ stabilizers on the same face one after the other, then onlyf measurement qubits are required.

Two-dimensional Lattice parameters 6.6.6 Hex. lattice 4.8.8 lattice

verticesv 3(d2−1)/4 + 1 (d−1)(d+ 3)/2 + 1 facesf 3(d2−1)/8 (d−1)(d+ 3)/4

Three∗-dimensional Lattice parameters

B.C.C lattice Doubled Color code lattice verticesv d(d+ 1)2/2 (d3+ 5d2−d−9)/4

facesf −− −−

volumesc −− −−

Table 5.2: The number of verticesv, facesf and (where relevant) volumescof the different lattices we consider as a function of (odd) distanced. In two-dimensions, we consider the 4.8.8 (square-octagon) and 6.6.6 (hexagonal) lattices used for the color code. In three-dimensions, we consider the construction by Bombin, and the psuedo-two-dimensional lattice proposed by Bravyi and Cross Sergey Bravyi and Cross, 2015. Note that, like the three-dimensional color code, the Bravy-Cross flattened doubled color code (and the other flattened color code schemes Jochym- O’Connor and Bartlett,2015; Jones, Brooks, and Harrington,2015) have overhead which is cubic in the distancev ∼constant×d3.

To estimate the overhead required for code switching with color codes to produce an encoded magic state of a desired quality (with logical error rate at mostpD), we consider the following steps:

1. Generate an encoded |+i state in a two-dimensional hexagonal color code of distance d. The error associated with this state can be estimated from

p1(d) =α(p/pT)dusing the appropriate threshold parameters.

2. Prepare a distance-dthree-dimensional color code (with hexagonal boundary) using ancilla qubits (or a flattened version of it) by measuring the appropri- ate stabilizers and applying correction operations. The code should have a boundary sheet missing (in the shape of the distanced, two-dimensional color code).

3. Apply the appropriate measurements to the two-dimensional color code and the three-dimensional color code such that they fuse resulting in an encoded

|+_i in the three-dimensional color code. We call the error associated with this statep3(d).

4. Apply the transverse T gate, resulting in the encoded |Ti in the three- dimensional color code. We call the error associated with this statep4(d). 5. Apply the appropriate measurements to "un-fuse" the two-dimensional code,

leaving the encoded|Ti in the distance d two-dimensional color code. We call the error associated with this statep5(d).

As the protection from noise improves with distance, each pi(d) monotonically decreases with d. To achieve the target state with the required accuracy with as low overhead as possible, we seek the smallest distance for which p5(d) ≤ pD. Then the overheadN(d)is the maximum number of physical qubits required at any time in steps (1) – (5). The most costly step involves the three-dimensional color code. The precise number of qubits required depends on the details. Taking the BCC lattice and assuming one measurement qubit per stabilizer generator results in

N(d) =d(d+ 1)2₋₁ .

Each step can only reduce the quality of the encoded information, and therefore

p5(d) ≥ p4(d) ≥ p3(d) ≥ p1(d). To estimate a lower bound on the overhead required for code switching, we make can consider the (unrealistic) case where steps (2) – (5) are perfect. In this case

p5(d) =p1(d) = α(p/pT)d, (5.12) and we see that the lowest distance isd=dD, such thatpD =α(p/pT)dD/2(increased sufficiently such that it is an odd integer). See figure 5.8. As steps (2) – (5) are expected to introduce a significant amount of additional noise to the system, we expect this lower bound to be far from tight.

A more accurate estimate of code-switching overhead

Ideally, to obtain a good estimate of the code-switching overhead, one would need to implement a fault-tolerant decoder for the three-dimensional color code given circuit level noise (which is not yet well understood). Then one could simulate the two-dimensional code, and three dimensional bulk code in equilibrium, and then could find a threshold for the gadget which switches from one to the other. This would not account for the errors introduced by the transverseT gate, which cannot

10-17 ₁₀-13 ₁₀-9 ₁₀-5

p

N

Figure 5.8: A lower bound on the qubit overhead for forming a magic state with logical error rate below pL using code switching. In reality the actual overhead is expected to be significantly greater since here we assume that no additional noise is introduced by the switching process. The blue points are for the three dimensional color code, and the yellow points are for the doubled color code, where in each case we assume that there is one measurement qubit for each stabilizer generator for the code. [Note one would actually need more measurement qubits for the doubled color code, since in that case one should measure the (strictly larger) number of gauge generators as the stabilizer generators are not all local].

be easily simulated classically, but the (Clifford) phase gate could be applied instead to give an estimate of the behavior.

For the psuedo-three-dimensional color codes of Bravyi-Cross and Jochym-O’Connor- Bartlett, one would expect that there is no threshold leading to worse performance above some distance for a given error rate.

Alternatively, tighter bounds than that we have given could be achieved by keeping most of the three dimensional color code implementation perfect, but adding in extra noise associated with some of steps (2) – (5).