An Entangling Gate for Dual-Rail Erasure Qubits

• p. 1 Abstract

  remaining Pauli errors below 0.1%

• p. 3 Gate benchmarking

  we infer a bound of 0.12(1)% error per CZ gate

• p. 3 Gate benchmarking, IRB

  per CZ gate of 0.108(5)%

• p. 2 Introduction (continued)

  at or below 0.1% per

Only the authors can say what is right here.

Judge's note: Read pages 1-3. The abstract wording is as quoted. The introduction (p. 2) makes a narrower and weaker claim: "residual dephasing errors are at or below 0.1% per gate, as characterized by ... QST and ... IRB". The results give 0.029(6)% (N up to 15), 0.12(1)% (called a conservative bound) and 0.108(5)% (IRB). Both headline figures exceed 0.1%, the IRB one by about 1.6 of its quoted uncertainty. The abstract may mean the low-depth figure or the Z-error components (0.039%, 0.0112%, 0.007%), but it does not say so. Overstatement by a small margin, so S3; the authors should say which figure the abstract refers to.

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• p. 3 Gate benchmarking, repeated-CZ paragraph

  a total erasure probability per CZ gate of 0.53(2)%

• p. 3 Gate benchmarking, IRB paragraph

  erasure probability of 0.69(2)% per gate

• p. 1 Abstract

  low erasure rates of approximately 0.5% per gate

• p. 1 Abstract

  approximately 0.5% per gate

Only the authors can say what is right here.

Judge's note: Page 3 image, right column: both statements are printed as quoted. The abstract says "approximately 0.5% per gate" and Table 1 "~0.5%". The 0.69(2)% figure appears in a sentence about IRB of "our CZ gate"; it may be per Clifford or per interleaved step, including single-qubit gates (about 0.1% erasure each, p. 3), which would make it a different quantity, but the article does not say. The details are deferred to Supplementary Information section IV, which is not among the files. An unresolved ambiguity, not a demonstrated error.

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• p. 2 The SWS CZ gate, noise-structure paragraph

  experiences increased loss (with probability

• p. 5 Table 1, Erasure row, 'Total cost per CZ gate'

  ~ 0.5%

• p. 2 The SWS CZ gate, last paragraph

  (with probability t/T1 ≈ 1%)

• p. 5 Table 1, Erasure row

  π/(χbc T1c) ~ 0.5%

Only the authors can say what is right here.

Judge's note: Checked crops C004-b and C004-c and pages 2 and 5. Text: t ≈ π/χbc; loss t/T1 ≈ 1% and dephasing t/Tφ ≈ 0.1% for the control, 0.1% and 0.01% for the target. Table 1: erasure π/(χbc T1^c) ~ 0.5%, dephasing π/(χbc Tφ^c) ~ 0.1%, typical asymmetry 10:1. The dephasing entries agree, the erasure entries do not. The table footnote says values "reflect both the typical and measured performance", and 0.5% matches the measured 0.53(2)%, so the table value may be the measured one or an average over control states (only |01> populates the coupler); neither is stated. Both are order-of-magnitude estimates, hence minor.

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• p. 3 Gate benchmarking, right column

  extract an end-to-end circuit state fidelity of 87.5(1)% and state purity

• p. 3 Fig. 2b legend

  103 Gates

• p. 3 Gate benchmarking, paragraph on N = 103

  end-to-end circuit state fidelity of 87.5(1)%

• p. 3 Fig. 2b legend

  103 Gates = 87.41(9)%

Only the authors can say what is right here.

Judge's note: Crop C009-b shows the legend "1 Gate F = 99.60(1)%" and "103 Gates F = 87.41(9)%". The text on p. 3 says "show the corresponding QST results in Fig. 2b and extract an end-to-end circuit state fidelity of 87.5(1)%". The N = 1 value is the same in both places, so these are the same quantity; 87.41(9) rounds to 87.4(1), not 87.5(1). No other analysis set is described in the article. The difference is 0.09 points and the derived bound of 0.12(1)% per gate comes out the same either way, so minor; only the authors can say which value is right.

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• p. 4 Fig. 3 caption, after panel b description

  For both results, a strong asymmetry between the control and target erasures is observed

• p. 4 Fig. 3 caption, panel b

  between the control and target erasures is observed

• p. 4 Fig. 3 legend, b

  For both results, a strong asymmetry between the control and target erasures is observed.

Judge's note: Page 4 image: panel a plots erasure fraction, panel b plots Z-error fraction (control Z, target Z, ZZ, total dephasing). The sentence follows the description of panel b and says "both results", so "erasures" misdescribes panel b; "errors" is evidently meant. The data are unaffected.

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• p. 5 Table 1, bit-flip row

  ≲10−6

• p. 4 Fig. 3 legend, d

  2.8(4) × 10−6

Judge's note: Crop C004-b and page 4 caption checked. 2.8 × 10^-6 is above 10^-6, so "≲10^-6" is loose against the measured control |01> value. Elsewhere the paper speaks of the "order of 10^-6" and "the level of about 10^-6", and the table footnote says values reflect both typical and measured performance, so the entry reads as an order-of-magnitude figure. A minor consistency point.

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• p. 5 Fig. 4 legend, c

  after applying N CZ gates on the target qubit

Judge's note: Page 5 image: the circuit in panel c shows the repeated CZ gates acting on both the control and target lines, with process tomography on the target only. The main text says "perform N repeated CZ gates followed by process tomography on just the target qubit". "On the target qubit" for the CZ gates is loose wording; the meaning is clear from the figure.

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• p. 6 Discussion and conclusion, second paragraph

  ten below the respective erasure and dephasing thresholds

• p. 8 Methods, delayed erasure checks

  we extract a threshold of 5.6%

• p. 8 Methods, first section

  with a threshold close to 1%

Only the authors can say what is right here.

Judge's note: Read the Discussion crop on page 6 and both Methods crops on page 8, then searched the article text for every use of "threshold" and "dephasing". The only thresholds given are 5.6% for erasures with delayed checks (Methods and Extended Data Fig. 2 legend), 5.13% quoted from ref. 10, about 25% for phenomenological erasure noise in the introduction, and close to 1% for two-qubit depolarizing noise. None is described as a dephasing threshold. The erasure half of the statement fits the Methods rates (0.4% + 0.1% against 5.6%). For dephasing the reader has to guess that the roughly 1% depolarizing threshold is meant, or that the figure comes from the manuscript in preparation; the paper does not say. A question for the authors about which dephasing threshold is meant, not an error in a number.

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• p. 1 Introduction, second paragraph

  bit flips can be greatly suppressed relative to phase flips then allows

• p. 1 Introduction, paragraph 2

  bit flips can be greatly suppressed relative to phase flips then allows us to concatenate these physical qubits

Judge's note: Page 1 image, left column, second paragraph: the wording is as quoted; "then allows" has no subject. Not an extraction artefact.

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• p. 1 Introduction, paragraph 3

  resembles a Pauli error at known spacetime location

Judge's note: Page 1 image, left column, third paragraph: printed as quoted. Cosmetic.

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• p. 8 Methods, right column, first paragraph

  erasure qubitsthat pos

• p. 8 Methods, Leakage propagation, 2nd paragraph

  erasure qubitsthat pos-sess this ‘leakage skips subsequent gates’ property

• p. 4 CZ gate noise structure, last paragraph

  as being in the codespace in the final.

• p. 1 Introduction, stabilized cat qubits paragraph

  bit flips can be greatly suppressed relative to phase flips then allows

Judge's note: Read the crops for pages 8, 4 and 1. "qubitsthat" is fused in print in mid-line, with the hyphenated break coming later at "pos-sess", so it is not an extraction artefact. On page 4 the sentence ends "in the codespace in the final." and the next sentence starts "Once again". On page 1 the sentence has no subject for "then allows". All three are real. The fourth item in this cluster, "one of the qubit is erased" on page 6, is the same defect as C016 and is left to that cluster.

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• p. 2 The SWS CZ gate, second paragraph (dispersive interaction)

  where χbc/2π = −1.51 MHz in our system

• p. 2 The SWS CZ gate, third paragraph (sideband interaction)

  gac/2π = 4.23 MHz

• p. 2 The SWS CZ gate, dispersive interaction

  Hint/h = χbc b1† b1 c† c

• p. 2 The SWS CZ gate, sideband interaction

  H/ħ = gac/2 (a2† c + a2 c†)

Judge's note: Crop C005-a shows an italic h without a bar in the first equation and a barred ħ in the second; the text layer agrees ("h" and "ħ"). χbc is quoted as χbc/2π and the phase is written φe = χbc·t_wait, both of which need H/ħ. With that convention t = 1/(2 × 1.51 MHz) ≈ 0.33 µs, which fits a gate of about 500 ns. A notation slip with an evident fix; no number is affected.

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• p. 2 The SWS CZ gate, parametric sideband interaction sentence

  similar to the interaction used to affect single-qubit gates in dual-rails

Judge's note: Page 2 image, right column: printed as quoted. The sense is "bring about" single-qubit gates, so "effect".

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• p. 4 CZ gate noise structure, last paragraph

  a leakage error as being in the codespace in the final.

• p. 4 CZ gate noise structure, bit-flip paragraph

  as being in the codespace in the final.

Judge's note: Page 4 image, left column, end of the 'CZ gate noise structure' section: the sentence ends at "in the final." and the next sentence starts "Once again". Not an extraction artefact.

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• p. 6 Discussion and conclusion, first paragraph

  one of the qubit is

• p. 6 Discussion, first paragraph

  in the event that one of the qubit is erased before or during the gate

Judge's note: Read the crop of PDF page 6: the line ends "one of the qubit is" with no plural s, and the next line begins "erased before or during the gate". The parallel sentence on page 2 has "one of the qubits suffers". Plain number-agreement slip.

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• p. 6 Reference 4

  Collaborators.Suppressing quantum errors

• p. 6 Reference 4

  Google Quantum AI and Collaborators.Suppressing quantum errors

Judge's note: Read the crop of the reference list on PDF page 6. Reference 4 is set "Collaborators.Suppressing" with no space; reference 3 directly above has "Collaborators. Quantum error correction" with the space. Visible in the image, so not an extraction artefact.

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• p. 6 Reference 35

  different ‘colour’ states Nat. Phys. 7, 599–603 (2011)

• p. 6 Reference 35

  different ‘colour’ states Nat. Phys. 7, 599–603 (2011).

Judge's note: Read the crop of the reference list on PDF page 6. Reference 35 runs "‘colour’ states Nat. Phys." with no full stop after the title; the neighbouring entries (34, 36, 37 and others) all end the title with a full stop before the journal.

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• p. 8 Methods, first paragraph

  simulations results and methodology

• p. 8 Methods, first paragraph

  More complete simulations results and methodology

Judge's note: Read the crop of PDF page 8, left column: the text is "simulations results and methodology". A plural noun used as a modifier; cosmetic.

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• p. 8 Author contributions

  A. Agrawal, A.N., K.S.C, J.O.Y. and J.D.T. contributed to data

Judge's note: Read the crop of the author contributions on PDF page 8. The line reads "A.N., K.S.C, J.O.Y."; the same initials appear as "K.S.C." in the other four places in the same paragraph.

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• p. 10 Extended Data Fig. 1 legend

  the error rate per CZ gate are chosen to be the same as the values reported in Fig. 3

Judge's note: Read the crop of the legend on PDF page 10. The sentence reads "the error rate per CZ gate are chosen to be the same as the values"; "values" shows the plural "error rates" was intended.

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