Topological Quantum Memory
2002 · paper · Computer science
Analyzes surface codes, topological quantum error-correcting codes on a two-dimensional qubit array, and error-recovery protocols for them. It finds a phase transition at a nonzero critical error rate, below which encoded information can be protected arbitrarily well, and estimates the threshold.
Checked: PDF from arxiv.org, fetched 4 Oct 2026. Page numbers are those of that PDF.
4 findings
Substance Eq. (9) prints the phase-error term as "p(1 − p)ZρX" where the channel requires "p(1 − p)ZρZ".
- p. 10 Sec. IV A, eq. (9)
+ p(1 −p)ZρX + p2Y ρY
- p. 10 Eq. (9)
p(1 −p)ZρX + p2Y ρY
Judge's note: The page image shows ZρX, so this is not an extraction artefact. The surrounding text describes independent X and Z errors, and the other terms follow the pattern IρI, XρX, YρY. ZρX is not a Pauli-channel term.
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Substance Eqs. (30)/(31) "Prob(η) = ∏(1 − p)^{1−η_ℓ} p^{η_ℓ}" use η as a 0/1 exponent, but η_ℓ ∈ {1, −1} by definition (eq. 24/28).
- p. 15 Sec. IV E, eqs. (30) and (31)
Prob(η) = ∏(1 − p)1−ηℓpηℓ
Judge's note: In the text layer (line 1529) η_ℓ ∈ {1, −1}, with η = −1 on error links, which occur with probability p. Plugging these values into the formula as printed: η=1 gives p, not 1−p, and η=−1 gives (1−p)^2/p, not p. The exponents would need to be written as (1∓η)/2. The p.15 image confirms that the exponents are printed as 1−η and η.
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Wording "the phase boundary of the spin system in the JH–JV plane can be translated into an upper bound on the storage accuracy threshold in the p–q plane" — the convergence region of the bound in eq. (55) gives a lower bound on the threshold.
- p. 21 Sec. V C, text before eq. (69)
can be translated into an upper bound on the storage accuracy threshold in the p–q plane
- p. 21 after Eq. (68)
translated into an upper bound on the storage accuracy threshold in the p–q plane
- p. 21
bound on the storage accuracy threshold in the p–q plane,
Only the authors can say what is right here.
Judge's note: Seen in the p.21 crop. Eq. (55) is an upper bound on the failure probability. Where series (68) converges (the disordered phase), that bound goes to zero, which guarantees recovery and so bounds the threshold from below. Sec. V (p.18 opening, text after eq. 64 on p.20) and the introduction (p.2) describe this argument as giving a "lower bound on the accuracy threshold". The word "upper" contradicts that.
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Wording After eq. (111) the text says 'apply a CNOT with the data as its control and the ancilla as its target, the effect on the data is the same as Λ(i) = P', but the gate that kicks back the phase from a Y eigenstate is the Λ(iY) of eq. (110).
- p. 33 Sec. IX A 3, after eq. (111)
apply a CNOT with the data as its control and the ancilla as its target, the effect on the data is the same as Λ(i) = P
Only the authors can say what is right here.
Judge's note: My working: with Y|±>=±|±>, iY|±>=±i|±>, so Λ(iY) applied with the data as control gives the data phase Λ(±i). Λ(X) (CNOT) does not keep a Y eigenstate invariant. For example, X(|0>+i|1>)/√2 = i(|0>-i|1>)/√2, which maps |+> to i|->, so the ancilla becomes entangled and P is not implemented. Eq. (110) builds Λ(iY) out of CNOTs and H, so the stated result (P is implementable) still holds. The step as printed names the wrong gate, and the later phrases 'use a CNOT and an ancilla' have the same loose wording.
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20 minor (typos and the like; most checked by the judge only)
Reference or label Ref. [24] gives the identifier 'cond-mat/00010143 (2000)', which has eight digits, while every other old-style arXiv number in the list has seven.
- p. 38 Reference [24]
cond- mat/00010143 (2000)
- p. 38 Reference [24]
mat/00010143 (2000)
- p. 34 Sec. IX B
much better fidelity [37,38]
- p. 34 Sec. X
was discussed in Ref. [38]
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: the correct identifier, probably cond-mat/0010143. The page image of p.38 shows the eight-digit number. Every other identifier in the list, e.g. cond-mat/0007254 and quant-ph/9705031, has seven digits. The point raised about [38] being cited for both purification and defect-interaction recovery is not a defect: Dennis's paper may cover both, and nothing in the files contradicts it.
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Reference or label Ref. [33] gives de Gennes, Scaling Concepts in Polymer Physics, as '(Cornell University Press, Ithaca, N. Y., 1970)'; the year may be wrong.
- p. 39 Reference [33]
(Cornell University Press, Ithaca, N. Y., 1970)
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: that the book was published in 1979. The page image of p.39 shows 1970. Nothing in the files settles the year.
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Wording Eq. (43) "pc = .1100" is said to agree "with eq. (41) within statistical errors", but eq. (41) is ".1094 ± .0002", a gap of three times the quoted error.
- p. 17 Sec. IV F, after eq. (43)
which agrees with eq. (41) within statistical errors
- p. 17 eq. (41)
pc = .1094 ± .0002
- p. 17 Sec. IV F, eq. (41)
pc = .1094 ± .0002 .
Only the authors can say what is right here.
Judge's note: Checked eqs. (41) and (43) and the following sentence on the p.17 image. The next sentence hedges with "at least approximately". Even so, "within statistical errors" overstates the agreement for the printed uncertainty. The authors should say whether the ±.0002 is meant as one standard deviation.
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Wording 'if we wish to reach a fidelity that is exponentially small in L' should refer to an infidelity (error) that is exponentially small in L.
- p. 34 Sec. IX B, paragraph on overhead
if we wish to reach a fidelity that is exponentially small in L
Judge's note: Page 34, left column. The preceding paragraph says purification drives the fidelity 'rapidly toward one', so the quantity meant to be exponentially small is 1 − fidelity. The wording says the opposite of what is intended.
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Wording 'encoding |ψ>_anc on a small planar sheet of qubits, with a fidelity below the purification threshold' conflicts with the stated requirement that the initial states be 'not too noisy'.
- p. 34 Sec. IX B
with a fidelity below the purification threshold
Judge's note: Page 34, left column. The previous paragraph says purification needs initial states that are not too noisy, i.e. fidelity above the threshold, or error below it. 'Fidelity below the purification threshold' literally states the opposite. It is most likely shorthand for 'error below threshold', but as printed it is contradictory.
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Typo Missing article: "concluding that error recovery procedure is sure to succeed" should read "that the error recovery procedure".
- p. 2 Introduction, right column, top
concluding that error recovery procedure is sure to succeed
Judge's note: Seen on p.2, right column, top. The article is missing.
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Typo Misspellings "occured" (p.6 "the errors actually occured on the other chain", Fig. 9 caption, p.24) and "occuring" (p.12, p.24, p.27) should be "occurred"/"occurring".
- p. 6 Sec. III A
the errors actually occured on the other chain
- p. 11 Fig. 9 caption
Links where errors occured are darkly shaded
- p. 27 Sec. VII C
two-link hooks occuring with probabilities
- p. 6 Sec. III A
actually occured on the other chain
Judge's note: The crop for p.6 shows "occured". Elsewhere the paper spells it correctly ("have occurred" on p.12), so the spelling is also inconsistent.
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Typo "suppressed by the Boltzman factor exp(−∆/kT)" — should be "Boltzmann" (spelled correctly on p.36).
- p. 10 Sec. III D, last paragraph
suppressed by the Boltzman factor exp(−∆/kT )
- p. 36 Sec. X C
Boltzmann probability dis- tribution
- p. 10 Sec. III D, last paragraph
Boltzman factor exp(−∆/kT )
Judge's note: Seen in the p.10 crop.
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Typo 'circuits of Fig. (17).' uses a parenthesised figure number, unlike 'Fig. 17' elsewhere, and the Fig. 9 caption ends '...both have the same boundary' with no period.
- p. 30 Sec. VIII C
circuits of Fig. (17).
- p. 11 Fig. 9 caption
both have the same boundary
Judge's note: Both are visible in the crops from p.30 and p.11. On p.30 the same column has 'Fig. 17' and 'Fig. 19' without parentheses. The Fig. 9 caption has no closing period. Both points are cosmetic.
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Typo "qubit errors on horizonal links" (p.12) and "those and only those horizonal links" (p.21) — should be "horizontal".
- p. 12 Sec. IV C
errors on horizonal links
- p. 21 Sec. VI A
only those horizonal links
- p. 21 Sec. VI A
those and only those horizonal links
- p. 12 Sec. IV C
qubit errors on horizonal links
Judge's note: The p.12 crop shows "horizonal". The same sentence has "horizontal links" spelled correctly a few lines later.
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Typo "introducing an isolated electric charge has a infinite cost in free energy" — should be "an infinite".
- p. 16 Sec. IV E, right column
has a infinite cost in free en- ergy
- p. 16 Sec. IV E
has a infinite cost in free en-
Judge's note: Seen in the p.16 crop, right column.
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Typo "an effective way use the error syndrome" — missing "to" ("an effective way to use").
- p. 18 Sec. V A, first sentence
an effective way use the error syndrome
Judge's note: Seen in the p.18 crop, Sec. V A, first sentence.
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Typo Eq. (63) prints "(87.8)^−1 = .0113", but 1/87.8 = 0.01139, which rounds to .0114.
- p. 20 Eq. (63)
˜p, ˜q < (87.8)−1 = .0113
- p. 20 Eq. (64)
p, q < .0114
- p. 20
˜p, ˜q < (87.8)−1 = .0113 ,
Judge's note: The image confirms the printed text. The discrepancy is only in the last digit. Contrary to the reviewers, eq. (64) does not conflict with it. Because p̃ = p(1−p) (eq. 53), p̃ < .0113 gives p < ≈.01143, i.e. the printed .0114. The exact 1/87.8 would give ≈.0115. So .0113 looks like a deliberate conservative truncation inside a sufficient condition. Only the "=" sign is loose.
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Typo Eqs. (118)-(119) write n^(4)_SAW(ℓ) for 'self-avoiding closed loops', but Sec. VI uses n_SAP for closed self-avoiding polygons (eqs. (56)-(58)) and keeps it distinct from n_SAW.
- p. 37 eq. (118)
The abundance of self-avoiding closed loops of length ℓ behaves like [32] n(4) SAW(ℓ)
- p. 37 Eqs. (118)-(119)
The abundance of self- avoiding closed loops of length ℓ behaves like [32]
- p. 20 Eqs. (56)-(58)
n(3) SAP(ℓ) ≤P3(ℓ)(µ3)ℓ
Judge's note: On p.37, eq. (118) uses the SAW subscript for closed loops. On p.20, eqs. (57)-(58) use n_SAP, and the text around line 2617 distinguishes n_SAW from n_SAP. The two differ only by a polynomial factor, so μ4 and the argument are unaffected. This is a notational inconsistency only.
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Typo Eq. (88) prints 'p_single < 1/100 , q < 1/100' where the condition from the V-sum of eq. (87) is on q_single.
- p. 28 eq. (88)
q < 1/100
Judge's note: The page image of p.28 shows eq. (88) with a bare 'q < 1/100'. The last bracket of eq. (87) is 100 q_single(1+q_hook/(25 p_single q_single))^2, and it needs a value below 1, which requires q_single < 1/100. Eq. (89) and the surrounding text use q_single. This is a dropped subscript. Elsewhere, q is the syndrome-error probability.
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Typo 'can be replaced my unitary transformations' should read 'replaced by unitary transformations'.
- p. 34 Sec. X intro, right column
can be replaced my unitary transformations
Judge's note: The typo is visible in the right column of p.34.
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Typo 'But the classical computation not so local' is missing a verb; it should read 'is not so local'.
- p. 34 Sec. X intro, left column bottom
But the classical compu- tation not so local
Judge's note: The page image of p.34 (bottom of the left column) shows this wording.
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Typo Stray or missing punctuation in author initials: Ref. [20] 'X. .G. Wen', Ref. [6] 'E Knill', Ref. [24] 'P Pujol'.
- p. 38 Reference [20]
X. .G. Wen and Q. Niu
- p. 38 Ref [15]
“‘Reliable computation with cellular automata,”
- p. 38 Ref [20]
X. .G. Wen
- p. 38 Ref [6]
E Knill, R. Laflamme
Judge's note: All three are visible in the reference list on p.38. The doubled quotation mark in Ref. [15] is handled separately under C027.
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Typo The title in Ref. [15] opens with a doubled quote, '“‘Reliable computation with cellular automata,”', but closes with only one.
- p. 38 Reference [15]
“‘Reliable computation with cellular automata,”
Judge's note: The page image of p.38 shows the extra single quote after the opening double quote.
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Typo Ref. [41] gives the publisher as '(New York, Dekke, 1980)'; it is presumably Dekker.
- p. 39 Reference [41]
(New York, Dekke, 1980)
- p. 39 Reference [41]
New York, Dekke, 1980
- p. 39
(New York, Dekke, 1980), p. 549-575.
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: the publisher of Advances in Probability 6 is Marcel Dekker. The page image of p.39 shows 'Dekke'.
Second check: none (judge only)