Aportaciones científicas

Previous analysis has been made for color code thresholds Ashley M Stephens, 2014; Hector Bombin, Duclos-Cianci, and Poulin, 2012; D. S. Wang et al.,2009; Landahl, Jonas T Anderson, and Rice, 2011; Andrist et al., 2011; Ohzeki, 2009; Katzgraber, H Bombin, and MA Martin-Delgado,2009; Delfosse,2014; Sarvepalli and Raussendorf,2012. In Table5.1we summarize the threshold constants found using efficient decoders. Two conspicuous omissions from this list are phenomeno- logical and circuit-level analysis for the color code on the hexagonal lattice. We suppose that a number of papers focus on the 4.8.8 lattice rather than the hexagonal lattice since it was not known during the time of their writing how to implement the full Clifford group of logical gates transversely in the latter (theSgate was known only for the color code on the 4.8.8 lattice). As we showed in Chapter4, it is possible to implement the logicalSgate transversely by applyingSto one of the bipartition subsets of qubits in the color code lattice, andS†to the other.

Code capacity threshold

The numerical simulation to find the threshold in the code capacity setting is very simple. Errors are applied to each qubit with probabilityp (i.e. X, Y and Z are each applied with probabilityp/3), then the decoder is applied, and the net pauli (consisting of the error and correction paulis) is identified. The weight of theXpart and theZwill both be even if the decoder succeeded, but not otherwise. This process is repeated many times to estimate the probability of success, which is plotted for different distances (see figure 5.3). The threshold appears to be around 12.2 %, which is a little smaller that the threshold of 13.1 % Delfosse,2014calculated for the hexagonal lattice with the Delfosse decoder on the torus.

Phenomenological threshold

For the phenomenological threshold, the system is run for 10,000 consecutive time steps, with errors applied to each qubit in each step with probabilityp(i.e. X, Y andZ are each applied with probabilityp/3), and the exact stabilizer outcomes are

calculated given what errors have been applied, and then these outcomes are each flipped with probabilitypbefore being fed to the decoder.

The decoder is as almost as described in Section5.2. The number of highlighted vertices that would occur during 10,000 time steps would make the matching pro- hibitively slow. Instead, after each time step, the decoder is run, and connected components of highlighted edges are recorded. After a connected component is observed d times, it is fixed, and the highlighted vertices in the component are removed from the history. This essentially implements a soft window through time to achieve a sub optimal matching but much more quickly.

Logical errors are identified by copying the system after each timestep, and running an additional round with no measurement errors. This returns the system to the codespace, and one can identify if the net effect of errors and the correction operation is a stabilizer or a logical operator. The mean number to time steps before a flip is calculated for the entire run and inverted to give an estimate of the logical error probabilitypL. Repeating the procedure ten times gives a more accurate estimate of pL along with an indication of its accuracy (via the standard deviation). See figure5.4.

Thresholds with efficient decoders

Code Code capacity Phenomen. Circuit 6.6.6 13.1 %∗,[12.2%] [4.2 % ] [0.3 % ] 4.8.8 13.1 %∗∗ 3.12 %∗∗∗∗ 0.143 %∗∗∗∗ 4.4.4.4 (Kitaev) 15.46 %∗∗∗ 4.40 %∗∗∗ 1.1 %∗∗∗∗∗

Table 5.1: Threshold probability for 6.6.6 (hexagonal lattice), 4.8.8 (square-octagon lattice) color codes and the 4.4.4.4 (square lattice) surface code. Only those results found using efficiently implementable decoders are given. Our results presented in this paper are provided in square parenthesis. Note the circuit level threshold of

0.143%by Stephens uses a partial cat-state preparation, which trades extra qubits in order to achieve a higher threshold. (*) Delfosse, 2014, (**) Hector Bombin, Duclos-Cianci, and Poulin, 2012, (***) C. Wang, Harrington, and John Preskill, 2003, (****) Ashley M Stephens,2014, (*****) A. G. Fowler, A. M. Stephens, and Groszkowski,2009. All thresholds are given for depolarizing channels with strength

p(if originally given for bit-flip channel, the threshold is multiplied by3/2). This rescaling actually overestimates the threshold for phenomenological noise since the measurement errors should not be rescaled, just the qubit errors. There can be no general formula to take this into account unfortunately.

0 5 10 15 20 7.5 8.5 9.5 10.5 11.5 12.5 13.5 14.5 d=9 d=11 d=13 d=15 d=17 d=19

p

p_L

Figure 5.3: Logical error rate pL versus physical error ratepunder code capacity (perfect measurement) noise. This indicates the threshold value to be around 12.2 % for the color code. More data is needed to obtain a more accurate estimate.

Circuit level threshold

To measure stabilizers, one can consider using a single measurement qubit placed in each face of the lattice to measure both theXand theZstabilizer associated with the face, one after the other. This requires that the order of the CNOT gates avoids any qubit being involved in two gates at once. Alternatively, two measurement qubits in each face allow for two CNOT gates to be applied per time-step in each face (one associated with theX, and the other with theZ measurement).

First we analyze the case of consecutiveXandZ measurements, which is expected to have worse performance as it involves more steps in which errors can occur. The schedule for the CNOT gates is simply clockwise beginning from above. The analysis of the threshold is precisely the same as for the phenomenological case (we used the same decoder). During each time step, an error is applied to an idle qubit with probabilityp(p/3for each ofX, Y and Z). Measurements are flipped with probabilityp, and preparation of a state is replaced by the "flipped" state with probabilityp. The control gates are followed by each of the fifteen non-trivial two qubit paulis with probabilityp.

0.005 0.015 0.025 0.035 0.045 0.055 0.065 0.03 0.035 0.04 0.045 0.05 d=5 d=7 d=9 d=21 d=11

p

p_L

Figure 5.4: Logical error ratepLversus physical error ratepunder phenomenological noise. This indicates the threshold value to be around 4.2 % for the color code. More data is needed to obtain a more accurate estimate.

Circuit schedules

An alternative to measuringXandZtype stabilizers one after the other is to use two measurement qubits per face and to implement a circuit to measure bothX andZ stabilizers simultaneously. Again, it is essential that each qubit is involved in at most one gate per time step, and there is an additional condition given in Landahl, Jonas T Anderson, and Rice,2011: "any stabilizer generator for an error-free input state (including ancilla syndrome qubits) must propagate to an element of the stabilizer group for an error-free output state." For example, we must check that anXoperator applied to anX measurement qubit (which stabilizes the |+ipreparation state) at the beginning of the schedule must propagate through the circuit to give a stabilizer. If a schedule satisfies these conditions, we say it isvalid. Note that any schedule which involves measuringXandZsequentially (i.e. in each round, all CNOT gates are applied before all control-Z gates) is valid.

A schedule for the color code can be specified by three pairs of integers for each physical qubit. Each pair corresponds to one of the three faces in contact with the qubit (for boundary qubits there can be fewer than three pairs). For a given pair, the first of the two integers specifies when a CNOT gate is applied between that qubit and the X-measurement qubit at the center of the corresponding face, and

-0.01 0.01 0.03 0.05 0.07 0.09 0.11 0.13 0.15 0 0.001 0.002 0.003 0.004 0.005 d=5 d=7 d=9 d=11 Power (d=11) Poly. (d=7) Poly. (d=5) -0.01 0.01 0.03 0.05 0.07 0.09 0.11 0.13 0.15 0 0.001 0.002 0.003 0.004 0.005 D=5 d=7 d=9 d=11 Poly. (d=7) Power (d=11) Poly. (D=5)

p

p

p

p

Figure 5.5: Logical error rate pL versus physical error rate p under circuit-level noise. The logical error rate is calculated separately forXandZtype errors to point out that there is some asymmetry. This indicates the threshold value to be around 0.3 % for the color code. More data is needed to obtain a more accurate estimate.

the second specifies when a control-Z gate is applied between that qubit and the

Z-measurement qubit. We focus on uniform schedules on regular lattices, such that only the pair for each data qubit in the unit cell of the lattice is specified, and the others are given by following the tiling.

Some uniform schedules that are valid for the color code are shown in figure 5.6. For the hexagonal lattice, the schedule for two qubits need to be specified to fix the schedule for the lattice. The approximate size of the search space for the hexagonal lattice for all lengthlschedules is(l!)2, such that exhaustive search can be completed quickly for lengths l = 6,7. Unfortunately there are no schedules of length six which satisfy all these constraints for the hexagonal lattice. However, there are

763inequivalent schedules of length seven (where we consider two schedules to be equivalent if mapped by a symmetry of the lattice or by interchangingXandZ). A pair of integers for each of four qubits are needed to specify a uniform schedule for the square-octagon lattice. An exhaustive search for all lengthlschedules involves checking approximately(l!)4cases, which was computationally unfeasible forl = 8. Some valid schedules will perform better than others. One generally expects short schedules to outperform long schedules since there are fewer opportunities for errors to occur per round of syndrome measurements. Two schedules of the same length can have different performance since they propagate errors differently. To compare the performance of valid schedules of length seven, one could consider the average total number of errors that result from a single error in the circuit. To calculate

(a)

(b)

4,5

3,4 6,7

2,3

5,6

1,2

4,5

1,2

4,5

6,7

2,3

5,6

3,4

1,2

1,5 2,6

3,7 4,8

1,5

3,7

1,5 2,6

1,2

2,1

3,4

4,3

5,6

6,5

7,8

8,7

6,5

8,7

5,6

7,8

Figure 5.6: a) and b) Schedules for CNOT operations in the color code. The minimum number of steps for a valid schedule is 7 for the hexagonal lattice, and 8 for the 4.8.8 lattice. An exhaustive search for the hexagonal lattice yields 763 inequivalent valid schedules of length seven.

this, we should pick out two adjacent data sites in the bulk, and apply all single location errors to locations which involve the two qubits, propagate through the remainder of the circuit until measurement, and sum with the appropriate weight (the relative probability of that error occurring). As of yet, we have not performed such an analysis, and we hope there could be improvement on the0.3%circuit level threshold if an optimized schedule is used.