Table 1 Decoding graph edges

From: Demonstrating multi-round subsystem quantum error correction using matching and maximum likelihood decoders

Edge e

Q(e)

First-order edge flip probability \({\tilde{p}}_{e}\)

Num.

(\({z}_{0}^{0}\), b)

1

44/15pcx + 6pid + 3pinit + 2pidm

0.039

(\({z}_{0}^{0}\), \({z}_{1}^{0}\))

2

44/15pcx + 14/3pid + 3pinit + 2pidm

0.038

(\({z}_{1}^{0}\), b)

3

44/15pcx + 4pid + 3pinit + 2pidm

0.037

(\({z}_{0}^{0}\), \({z}_{0}^{1}\))

\({{\emptyset}}\)

88/15pcx + 4/3pid + 2pinit + 2pmeasure

0.061

(\({z}_{1}^{0}\), \({z}_{0}^{1}\))

2

8/5pcx

0.016

(\({z}_{1}^{0}\), \({z}_{1}^{1}\))

\({{\emptyset}}\)

88/15pcx + 4/3pid + 2pinit + 2pmeasure

0.061

(\({z}_{0}^{1}\), b)

1

56/15pcx + 28/3pid + 4pidm

0.055

(\({z}_{0}^{1}\), \({z}_{1}^{1}\))

2

56/15pcx + 22/3pid + 4pidm

0.053

(\({z}_{1}^{1}\), b)

3

56/15pcx + 28/3pid + 4pidm

0.055

(\({z}_{0}^{1}\), \({z}_{0}^{2}\))

\({{\emptyset}}\)

88/15pcx + 4/3pid + 2pmeasure + 2preset

0.061

(\({z}_{1}^{1}\), \({z}_{0}^{2}\))

2

8/5pcx

0.016

(\({z}_{1}^{1}\), \({z}_{1}^{2}\))

\({{\emptyset}}\)

88/15pcx + 4/3pid + 2pmeasure + 2preset

0.061

(\({z}_{0}^{2}\), b)

1

44/15pcx + 14/3pid + 2pidm + 3pmeasure

0.040

(\({z}_{0}^{2}\), \({z}_{1}^{2}\))

2

44/15pcx + 4pid + 2pidm + 3pmeasure

0.039

(\({z}_{1}^{2}\), b)

3

44/15pcx + 20/3pid + 2pidm + 3pmeasure

0.042

(\({x}_{0}^{0}\), \({x}_{2}^{0}\))

4

4/3ph + 8/5pcx + 8/3pid + pinit + 2/3pidm + pmeasure

0.021

(\({x}_{0}^{0}\), \({x}_{3}^{0}\))

5

2ph + 12/5pcx + 2pid + 3pinit + 2/3pidm

0.028

(\({x}_{0}^{0}\), b)

1

10/3ph + 4pcx + 16/3pid + 4pinit + 4/3pidm + pmeasure

0.049

(\({x}_{0}^{0}\), \({x}_{0}^{1}\))

\({{\emptyset}}\)

8/3ph + 16/15pcx + pinit + 2pmeasure + preset

0.012

(\({x}_{0}^{0}\), \({x}_{2}^{1}\))

4

8/15pcx

0.005

(\({x}_{0}^{0}\), \({x}_{3}^{1}\))

5

8/15pcx

0.005

(\({x}_{1}^{0}\), \({x}_{3}^{0}\))

6

4/3ph + 16/15pcx + 10/3pid + pinit + 2/3pidm + pmeasure

0.016

(\({x}_{1}^{0}\), b)

3

4/3ph + 8/5pcx + 8/3pid + 2pinit + 2/3pidm

0.020

(\({x}_{1}^{0}\), \({x}_{1}^{1}\))

\({{\emptyset}}\)

4/3ph + 8/15pcx + pmeasure + preset

0.006

(\({x}_{1}^{0}\), \({x}_{3}^{1}\))

6

8/15pcx

0.005

(\({x}_{2}^{0}\), b)

7

4/3ph + 16/15pcx + 10/3pid + 2pinit + 2/3pidm

0.016

(\({x}_{2}^{0}\), \({x}_{2}^{1}\))

\({{\emptyset}}\)

4/3ph + 8/15pcx + pmeasure + preset

0.006

(\({x}_{3}^{0}\), b)

8

10/3ph + 52/15pcx + 22/3pid + 4pinit + 4/3pidm + pmeasure

0.046

(\({x}_{3}^{0}\), \({x}_{3}^{1}\))

\({{\emptyset}}\)

8/3ph + 16/15pcx + pinit + 2pmeasure + preset

0.012

(\({x}_{0}^{1}\), \({x}_{2}^{1}\))

4

2/3ph + 28/15pcx + 8/3pid + 4/3pidm + pmeasure

0.025

(\({x}_{0}^{1}\), \({x}_{3}^{1}\))

5

4/3ph + 8/3pcx + 2pid + 4/3pidm + 2preset

0.032

(\({x}_{0}^{1}\), b)

1

2ph + 68/15pcx + 20/3pid + 8/3pidm + pmeasure + 2preset

0.058

(\({x}_{0}^{1}\), \({x}_{0}^{2}\))

\({{\emptyset}}\)

8/3ph + 16/15pcx + 2pmeasure + 2preset

0.012

(\({x}_{0}^{1}\), \({x}_{2}^{2}\))

4

8/15pcx

0.005

(\({x}_{0}^{1}\), \({x}_{3}^{2}\))

5

8/15pcx

0.005

(\({x}_{1}^{1}\), \({x}_{3}^{1}\))

6

2/3ph + 4/3pcx + 10/3pid + 4/3pidm + pmeasure

0.020

(\({x}_{1}^{1}\), b)

3

2/3ph + 28/15pcx + 10/3pid + 4/3pidm + preset

0.025

(\({x}_{1}^{1}\), \({x}_{1}^{2}\))

\({{\emptyset}}\)

4/3ph + 8/15pcx + pmeasure + preset

0.006

(\({x}_{1}^{1}\), \({x}_{3}^{2}\))

6

8/15pcx

0.005

(\({x}_{2}^{1}\), b)

7

2/3ph + 4/3pcx + 10/3pid + 4/3pidm + preset

0.020

(\({x}_{2}^{1}\), \({x}_{2}^{2}\))

\({{\emptyset}}\)

4/3ph + 8/15pcx + pmeasure + preset

0.006

(\({x}_{3}^{1}\), b)

8

2ph + 4pcx + 22/3pid + 8/3pidm + pmeasure + 2preset

0.054

(\({x}_{3}^{1}\), \({x}_{3}^{2}\))

\({{\emptyset}}\)

8/3ph + 16/15pcx + 2pmeasure + 2preset

0.012

(\({x}_{0}^{2}\), \({x}_{2}^{2}\))

4

4/3ph + 8/5pcx + 8/3pid + 2/3pidm + 2pmeasure

0.021

(\({x}_{0}^{2}\), \({x}_{3}^{2}\))

5

2ph + 12/5pcx + 2pid + 2/3pidm + pmeasure + 2preset

0.028

(\({x}_{0}^{2}\), b)

1

10/3ph + 4pcx + 20/3pid + 4/3pidm + 3pmeasure + 2preset

0.052

(\({x}_{1}^{2}\), \({x}_{3}^{2}\))

6

4/3ph + 16/15pcx + 10/3pid + 2/3pidm + 2pmeasure

0.017

(\({x}_{1}^{2}\), b)

3

4/3ph + 8/5pcx + 10/3pid + 2/3pidm + pmeasure + preset

0.022

(\({x}_{2}^{2}\), b)

7

4/3ph + 16/15pcx + 8/3pid + 2/3pidm + pmeasure + preset

0.016

(\({x}_{3}^{2}\), b)

8

10/3ph + 52/15pcx + 6pid + 4/3pidm + 3pmeasure + 2preset

0.046

  1. Edge data for the decoding graph in Fig. 1c, d correcting X (Z)-errors. Here \({z}_{s}^{t}\) (\({x}_{s}^{t}\)) indicates the sthZ (X)-stabilizer at time t as in Fig. 1c, d. If edge e is chosen by the matching decoder, a Pauli X (Z) is applied to qubit Q(e) if it is not \({{\emptyset}}\). Numeric values from the optimization in Section “IBM_Peekskill and experimental details” are provided in the last column.
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