Surface Codes: Towards Practical Large-Scale Quantum Computation

• p. 41 Table V

  rough/primal qubit

• p. 16 Sect. VIII

  A single X-cut qubit is called a rough defect or a primal defect

• p. 16 Sect. VIII, last paragraph

  Double Z-cut (double X-cut) qubits are called smooth (rough) qubits or dual (primal) qubits

Only the authors can say what is right here.

Judge's note: Checked the p.41 table and the p.16 text. In the main text, a single cut is a 'defect' and a double cut is a 'qubit'. Table V pairs 'rough/primal qubit' and 'smooth/dual qubit' with single-cut qubits. The 'double rough/smooth defect' rows map to (double) cut qubits, which is consistent. The paper's own terminology mapping disagrees between the two places.

Second check: another model agreed

• p. 2 Section I

  It takes a minimum of thirteen physical qubits to implement a single logical qubit.

• p. 12 Sect. VII A

  nq = (2d−1)2

• p. 2 Sect. I Background, paragraph 4

  It takes a minimum of thirteen physical qubits to imple-

Only the authors can say what is right here.

Judge's note: Checked both crops. The p.2 sentence says 'physical qubits'. On p.12 the total number of data and measurement qubits is defined as n_q=(2d−1)^2. A d=3 planar array has 13 data and 12 measure qubits. I found no statement anywhere in the text that explains a 13-qubit count (grep for 'thirteen' or 'minimum' turns up nothing). Only the authors can say whether 'thirteen' was meant to count data qubits only.

Second check: another model agreed

• p. 52 Appendix M

  a distance d2 = 34 code yields PL2 ≈3 × 10−19

• p. 53 Appendix M

  a d1 = 8 code with PL1 ≈3 × 10−15

• p. 2 Section I

  we assume an error rate approximately one-tenth the threshold rate

• p. 11 Eq. (11)

  PL ∼= 0.03 (p/pth)de

Only the authors can say what is right here.

Judge's note: On p.11 the paper gives pth = 0.57% and Eq. (11), PL ≈ 0.03(p/pth)^de. With d = 34 (de = 17) and p/pth = 0.1, Eq. (11) gives 3e-19, which matches the printed value. With pth = 0.57%, it gives about 4e-15. On p.2 the paper cites literature thresholds of about 1% [22, 23], which may explain the 'one-tenth' wording, but Appendix M never says which value it uses. The d1 = 8 discrepancy is treated under C044.

Second check: another model agreed

• p. 3 Section I

  takes a surface code area of about 800,000 physical qubits, and takes about 100 µs

• p. 53 Appendix M

  an AA factory can produce two |AL⟩ states every 500 × 200 ns

Judge's note: Compared p.3 with p.53. Appendix M says that one state uses about half of the 800,000-qubit × 500-cycle space-time volume, so a factory yields two states per 100 µs. That gives about 1144 factories, which the paper rounds to 1200, times 8×10^5 ≈ 10^9. Read as a rate, the main-text sentence (one state per 800,000 qubits per 100 µs) would need about twice as many. The sentence can also be read as the latency of a single state, so this is a wording inconsistency.

Second check: another model agreed

• p. 10 Footnote 11

  | . . .+−++ . . .⟩, then |ψ⟩must also include the state where these qubits are in | . . . −+ −−. . .⟩

Only the authors can say what is right here.

Judge's note: The p.10 footnote shows ± labels. Appendix A (p.40) defines |±⟩=(|g⟩±|e⟩)/√2 as X̂ eigenstates, so X̂|±⟩=±|±⟩ and X̂ cannot exchange them. The footnote's argument works only if '+' and '−' stand for computational-basis (g/e) labels. The notation clashes with the paper's own definition.

Second check: another model agreed

• p. 45 Appendix F step 3

  have to wait d/4 (rounded up)

• p. 23 Sect. XIII A

  we then wait an additional d −1 surface code cycles

Only the authors can say what is right here.

Judge's note: Compared p.45 step 3 with p.23. Step 4 of the appendix (wait d cycles) matches the main text. The intermediate wait of ⌈d/4⌉ cycles in step 3 is not counted in the main text's 1 + d total. Step 3 also says no wait is needed for d = 3, although ⌈3/4⌉ = 1. Only the authors can say whether the intermediate wait applies.

Second check: another model agreed

• p. 37 Sect. XVI B, first sentence

  are X̂L rotations of |gL⟩ by 90◦and 45◦

• p. 34 Eq. (46)

  |AL⟩= 1/√2 (|gL⟩+ eiπ/4|eL⟩)

• p. 37 Fig. 31 caption

  the central data qubit is rotated using R̂Z(θ)

Judge's note: Worked by hand: an X̂ rotation of |g⟩ by Bloch angle 45° gives cos(π/8)|g⟩ − i sin(π/8)|e⟩, which has unequal amplitudes and is not equal to Eq. (46). By 90° it gives (|g⟩ − i|e⟩)/√2, which is |Y⟩ only up to the sign of i. The Fig. 31 caption on p.37 prepares |+L⟩ and rotates it with R̂Z(θ) to |gL⟩+e^{iθ}|eL⟩. The sentence misdescribes the states; the equations themselves are correct.

Second check: another model agreed

• p. 53 Appendix M, second column, improved-error-rate paragraph

  d1 = 8 code with PL1 ≈ 3 × 10−15

Only the authors can say what is right here.

Judge's note: Eq. (11) with p/pth = 0.1 reproduces the base-case PL2. At p = 10⁻⁴ with pth = 1%, d = 8 (de = 4) gives 3e-10, and with pth = 0.57% it gives about 3e-9. Neither is 3e-15. 1800 × 8 × 3e-15 ≈ 4.3e-11 matches the printed output rate, so the printed PL1 is the source. Appendix M does not name its scaling model, so only the authors can say which value they intended.

Second check: another model agreed

• p. 53 Appendix M, last paragraph

  4000×3200 ≈12 million qubits

Judge's note: The crop and page 53 show "4000×3200 ≈ 12 million qubits" and then "about 130 million physical qubits". 4000 × 3200 = 12.8 million. Writing 12 million truncates the product rather than rounding it. The total is still roughly right (120 + 12.8 ≈ 133, written as about 130), so this is a minor imprecision in an order-of-magnitude estimate.

Second check: another model agreed

• p. 3 Section I, paragraph on gate times

  this requires more qubits one keeps the overall execution time constant

• p. 3 Sect. I Background, paragraph 3

  however this requires more qubits one keeps the overall execution time constant

• p. 3 Sect. I, last paragraphs

  however this requires more qubits one keeps the

Judge's note: The page 3 image (left column) shows the sentence exactly as quoted. The word 'if' is missing.

Second check: none (judge only)

• p. 18 Section X intro

  In this section we described two methods to

Judge's note: The p.18 image shows the sentence as quoted. It introduces the section that follows, so the present tense is meant.

Second check: none (judge only)

• p. 25 Sect. XIII C

  Byproduct operators may be appear due to any sign

Judge's note: Confirmed in the p.25 image.

Second check: none (judge only)

• p. 27 Fig. 20 caption (c)

  error-corrected outcomes Ze s1, . . . , Ze s8

• p. 27 Fig. 20 caption (e) and panel (c) labels

  (Zf s1 . . . Zf s8)

Judge's note: Page 27 image checked: panel (c) of Fig. 20 labels the stabilizers Z^f_s1…Z^f_s8, and the caption's (−1)^pZ formula uses Z^f_s1…Z^f_s8. Appendix H (text lines ~7000–7013) also calls the post-move values Z^f. The superscript e in caption (c) is inconsistent with these.

Second check: none (judge only)

• p. 41 Appendix A, before Table V

  We have compiled these names changes

Judge's note: The text appears as printed on the p.41 page image.

Second check: none (judge only)

• p. 44 Appendix E

  We first define the the logical operator

• p. 44

  the main text. We first define the the logical operator ˆZ

Judge's note: The duplicated word is visible in the p.44 crop.

Second check: code check

• p. 45 Appendix F, end

  byproduct (byproduct) operators

• p. 45 End of Appendix F

  for the byproduct (byproduct) operators can be found in Ref.

• p. 45

  byproduct (byproduct) operators can be found in Ref.

Judge's note: Visible on the p.45 page image. The parenthesis was presumably meant to give an alternative term.

Second check: none (judge only)

• p. 47 Appendix I, text after Eq. (I2)

  generates an operator X̂′′L2 that comprises the original X̂L1 chain

• p. 47 Appendix I, after Eq. (I2)

  generates an operator ˆX′′L2 that comprises the

• p. 47 Appendix I, before Eq. (I3)

  generates an operator ˆX′′ L2 that comprises the original ˆXL1 chain

Judge's note: On the p.47 image, Eq. (I2) ends with X̂″L1 ÎL2, and Eq. (I3) begins with X̂″L1 ÎL2. The sentence between them calls the same operator X̂″L2. That is a subscript slip, because L2 refers to the second (X-cut) qubit.

Second check: none (judge only)

• p. 49 Appendix J, step 2

  stabilizers shown outline in Fig. 27b

Judge's note: The crop of p.49 shows 'Ẑ^s_L stabilizers shown outline'. Step 8 on the same page uses 'the outlined Ẑ stabilizers'. This is wording only.

Second check: none (judge only)

• p. 50 Appendix K, step 3

  (see Eq. 52))

• p. 50

  using ˆRZ(θ) (see Eq. 52))

Judge's note: The p.50 image shows two closing parentheses. Eq. (52) does define R̂Z(θ), so the pointer itself is correct.

Second check: none (judge only)

• p. 52 Appendix M, second column

  120d2PL ≈ 1 × 10−15 < PA

Judge's note: The p.52 image shows 120d2PL, with no '2' on PL. The preceding sentence defines the quantity as 120d2PL2, and the number agrees with PL2 = 3e-19.

Second check: none (judge only)

• p. 52 Appendix M, second-stage distillation footprint

  16×14500 ≈2.4×105

Judge's note: The p.52 image shows '16×14500 ≈ 2.4×10^5'. This is a slight over-rounding. The 2.2 × 10⁸ space-time volume is essentially unchanged by it (about 2.15e8 with 2.32e5).

Second check: none (judge only)

• p. 53 Reference list [13]

  anyons,” (1997), arXiv:quant-ph/9707021.

• p. 53 Reference list [14]

  A. Y. Kitaev, Annals of Physics 303, 2 (2003)

Judge's note: In the reference list on p.53, both entries cite arXiv quant-ph/9707021. Listing a preprint and its journal version as two separate entries is a minor duplication.

Second check: none (judge only)

• p. 54 Reference [37]

  A new quantum ripply-carry addition circuit

• p. 54

  A new quantum ripply-carry addition circuit,”

Judge's note: The crop of page 54 shows "ripply-carry" in reference [37], so this is not an extraction artefact. "Ripply" is an evident misspelling of the standard term "ripple-carry".

Second check: none (judge only)