A Variational Eigenvalue Solver on a Photonic Quantum Processor
Nature Communications, 2014 · paper · Computer science
Presents a variational method for finding eigenvalues that needs less coherent quantum evolution, using trial states and classical optimization. It is demonstrated on a small photonic quantum processor, computing the He–H+ molecule's ground-state energy to within chemical accuracy.
Checked: PDF from arxiv.org, fetched 4 Oct 2026. Page numbers are those of that PDF.
6 findings
Reference or label The main reference list has two entries labelled '[1]' (Thogersen & Olsen; Helgaker, Jorgensen & Olsen) and no [2], so 'atoms [1]. Many approximate methods [1]' cannot be resolved.
- p. 5 Reference list, first two entries
[1] T. Helgaker, P. Jorgensen, and J. Olsen, Molecular Elec-
- p. 1 Introduction, first paragraph
Many approximate methods [1] have been de-
- p. 1 Introduction, para 1
atoms [1]. Many approximate methods [1] have been de-
- p. 5 reference list
[1] T. Helgaker, P. Jorgensen, and J. Olsen
Judge's note: The p5 image shows '[1] L. Thogersen…' and '[1] T. Helgaker…' followed by [3]. The p1 image shows [1] cited twice in successive sentences. Helgaker matches the Appendix bibliography's [1] (p10), which points to a merge of the two bibliographies. The other duplicated labels raised in reviewer note 3 ([5], [4], [3]) are judged separately in C010, C011 and C012.
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Reference or label Main reference list: Ortiz et al. is printed as '[5]' in slot 23 (between [22] and [24]), duplicating '[5] G. H. Golub and H. A. van der Vorst'; [23] is missing, so the citations 'tors [5]' and 'ucts [5, 22]' on p2 are ambiguous.
- p. 6 Reference list, between [22] and [24]
[5] G. Ortiz, J. E. Gubernatis, E. Knill, and R. Laflamme,
- p. 5 Reference list
[5] G. H. Golub and H. A. van der Vorst, J. Comput. Appl.
- p. 2 Algorithm 1 text
product of an arbitrary number of simple Pauli opera- tors [5]
- p. 2 Main text, left column
tors [5], therefore with an n-qubit state we can efficiently
Judge's note: The p6 image shows '[5] G. Ortiz…' between [22] and [24]. The p5 image shows [5] Golub. The p2 claim about efficiently measuring Pauli tensor products fits Ortiz, but by number it resolves to Golub. The Appendix list has Ortiz as its [5] (p10), the source of the clash.
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Reference or label Fig. 3 caption: 'estimated directly from the state parameters {φ^j_i}', but the Appendix defines the state parameters as {θ_i} and {φ_i} as the chip phases, which are 'not equivalent'.
- p. 4 Fig. 3 caption
the state parameters {φj i}
- p. 9 Mapping from the state parameters to the chip phases
The set of phases {θi}, which uniquely identi-
- p. 3 Fig. 2a CPU box
{θ_i^j} → {φ_i^j}
- p. 9 Appendix, Mapping from the state parameters to the chip phases
is not equivalent to the phases which are written to the photonic circuit {φi}
Judge's note: p9 Appendix: 'The set of phases {θi}, which uniquely identifies the state |ψ⟩, is not equivalent to the phases which are written to the photonic circuit {φi}'. The Fig. 2a CPU box (p3 image) shows ⟨H⟩ → {θ^j_i} → {φ^j_i}. The Fig. 3 caption (p4 image) calls {φ^j_i} the state parameters, swapping the symbols.
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Reference or label Main reference list: Taube & Bartlett is printed as '[4]' in slot 28, duplicating '[4] L. Page, S. Brin…'; [28] is missing, so 'unitary coupled cluster ansatz [4]' and '“gold standard” [4]' resolve to the PageRank reference.
- p. 6 Reference list, between [27] and [29]
[4] A. G. Taube and R. J. Bartlett, Int. J. Quant. Chem.
- p. 5 Reference list
[4] L. Page, S. Brin, R. Motwani, and T. Winograd, (1999).
- p. 3 Unitary coupled cluster paragraph
cluster ansatz [4]
- p. 3 Main text, right column
superior results to even this “gold standard” [4]
Judge's note: Confirmed on the p6 image ([4] Taube between [27] and [29]), the p5 image ([4] Page et al.) and the p3 image (two UCC citations to [4]). The p1 'search engines [4]' correctly needs Page.
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Reference or label p4: 'The coefficients h^i_α(R) and h^{ij}_{αβ}(R) were determined using the PSI3 computational package [3] and tabulated in the Appendix', but the Appendix contains no table of coefficients.
- p. 4 Main text after Eq. 5
tabulated in the Appendix
- p. 7 Appendix, Second Quantized Hamiltonian (Eq. 11)
given explicitly by an FCI calculation in the PSI3 com- putational package [3] and can be written as
- p. 7 Second Quantized Hamiltonian
given explicitly by an FCI calculation in the PSI3 com-
Only the authors can say what is right here.
Judge's note: The Appendix (pp7-10; p7 image checked, pp8-10 text layer checked) only restates the symbolic form in Eq. 11 ('given explicitly by an FCI calculation in the PSI3 computational package [3] and can be written as'). No table or numeric coefficient values appear anywhere in the files, and no separate supplement is referenced. Only the authors can supply the data or correct the pointer.
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Reference or label Main reference list: Crawford et al. (PSI3) is printed as '[3]' in slot 32, duplicating '[3] Y. Saad, Numerical methods for large eigenvalue problems'; [32] is missing, so 'PSI3 computational package [3]' (p4) resolves to Saad.
- p. 6 Reference list, between [31] and [33]
[3] T. D. Crawford, C. D. Sherrill, E. F. Valeev, J. T. Fer-
- p. 4 Main text after Eq. 5
ing the PSI3 computational package [3]
- p. 5 Reference list
[3] Y. Saad, Numerical methods for large eigenvalue prob-
- p. 6 References, slot after [31]
[3] T. D. Crawford, C. D. Sherrill, E. F. Valeev
Judge's note: Confirmed on the p6 image ([3] Crawford between [31] and [33]), the p5 image ([3] Saad) and the p4 image (PSI3 [3]). The p1 'large eigenvalue problems [3]' correctly needs Saad.
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18 minor (typos and the like; most checked by the judge only)
Wording p5: 'More than 96% of the experimental data are within chemical accuracy' (and abstract 'to within chemical accuracy'): 'chemical accuracy' is never defined, and the claim is hard to square with the Fig. 4 inset scatter and the stated per-point SD of '37KJ/mol'.
- p. 5 Results, first lines
More than 96% of the experimental data are within chemical accuracy
- p. 1 Abstract
to within chem- ical accuracy
- p. 4 Fig. 4 magnified panel
Corrected Exp.
- p. 9 Statistical errors
37KJ/mol
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: the conventional chemical-accuracy threshold of about 1 kcal/mol (≈4.2 kJ/mol ≈ 0.004 MJ/mol). The paper gives no definition. In the Fig. 4 inset on the p4 image (gridlines 0.1 MJ/mol apart), several corrected points sit visibly off the curve; the point near 215 pm is at about -2.58 against a curve at about -2.81 MJ/mol. The p9 statistical SD is 37 kJ/mol, and E = -2.865±0.008 MJ/mol. The authors should state the threshold they used. Minor point: 'KJ/mol' (p9) should be 'kJ/mol'.
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Wording The Appendix reuses symbols: N is 'the number of single particle basis functions' (p7), then 'an N electron system' (Eq. 13) and O(N^k(M−N)^k) 'where M is the number of single particle orbitals' (p8), while M is 'the number of terms in the decomposition of the Hamiltonian' (p2, p9).
- p. 7 Second Quantized Hamiltonian
where N is the number of single particle basis functions used
- p. 7 Unitary Coupled Cluster Theory
N electron system
- p. 8 UCC term count
where M is the number of sin- gle particle orbitals
- p. 2 Algorithm 1 cost
where M is the number of terms in the decomposition of the Hamiltonian
Judge's note: Checked p7 (N as basis functions; Eqs. 9–10 use N for the number of qubits; Eq. 13 uses N for electrons) and the p8 text. M is redefined between the main text/Computational Scaling and the UCC paragraph. Each use can be inferred from context, so this is an ambiguity rather than an error.
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Wording "These errors can be arbitrarily reduced at a sublinear cost of measurement time (i.e. efficiently) since the magnitude of error is proportional to the square root of the count rate" conflicts with the stated cost of O(|h|^2/p^2) repetitions for precision p.
- p. 9 Experimental details, Statistical errors
These errors can be arbitrarily reduced at a sublinear cost of measurement time
- p. 2 Algorithm 1 cost
proach incurs a cost of O(|h|2/p2) repetitions
- p. 2 Main text, cost of Algorithm 1
incurs a cost of O(|h|2/p2) repetitions
Only the authors can say what is right here.
Judge's note: Page 9 image: the Statistical errors paragraph says "sublinear cost". Page 9 left column (and page 2) give a cost of O(|h|^2/p^2) repetitions for precision p, i.e. quadratic in 1/p. With shot noise, the relative error falls as 1/sqrt(N), so halving it takes four times as many counts. That cost is efficient (polynomial) but not sublinear. Only the error as a function of time decreases sublinearly. The wording misdescribes the scaling; the authors should say what they meant.
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Wording Algorithm 2 pseudocode: step 2 'Let n = 0' and step 3(a) 'Call Algorithm 1 with {θi} as input', but no step increments n, and the input is not the current iterate {θ^n_i}.
- p. 7 Quantum eigenvector preparation algorithm, steps 2–3
(a) Call Algorithm 1 with {θi} as input:
Judge's note: The p7 image confirms that steps 1–3(b) set n = 0, compute f({θ^n_i}) and determine {θ^{n+1}_i}, with no n ← n+1 step. The intent is clear, so this is a presentation gap.
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Wording Same setting (H is 2^n × 2^n) gives "O(log∗ n) queries" for Berry et al. but "O(k+log∗ N) queries" for Childs et al.; N is not defined in this section (and N on the same page means the number of electrons).
- p. 8 Application to k-sparse Hamiltonians
O(log∗n) queries
- p. 8 Application to k-sparse Hamiltonians
O(k+log∗N) queries
Judge's note: Page 8 image, right column: H defined as 2^n × 2^n, then O(log* n) and, two sentences later, O(k+log* N). N is not defined in this paragraph. In the left column N appears in O(N^k(M−N)^k) for a different quantity. Most likely just inconsistent notation, but the reader can't tell which size the second bound refers to.
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Wording Gradient descent is listed as a direct-search method: "other DS methods, such as the gradient descent, can perform better for smooth functions".
- p. 8 Classical optimization algorithm
other DS methods, such as the gradient descent, can per-
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: direct-search methods are derivative-free by the standard definition, and gradient descent uses gradients. Page 8 image: the paragraph introduces NM as "a simplex-based direct search (DS) method" and then calls gradient descent one of the "other DS methods". The paper itself never defines DS as derivative-free, so this is loose wording for the authors to fix. It does not change the reported result.
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Wording The growing ground-state error is attributed to "the first and second excited eigenstates of this Hamiltonian become degenerate"; that degeneracy alone does not explain harder ground-state minimization.
- p. 10 Systematic errors, last paragraph
the first and second excited eigenstates of this Hamiltonian be-come degenerate
- p. 10 Systematic errors, last paragraph
first and second excited eigenstates of this Hamiltonian be-
Only the authors can say what is right here.
Judge's note: Page 10 image, left column: the sentence is as quoted. Fig. 4 shows only ground-state energies. If two excited states become degenerate with each other, but not with the ground state, a variational minimisation toward the ground state should not get harder or produce mixtures. The argument as written would only work if the ground state were (near-)degenerate with an excited state. The HeH+ spectrum is not given in the files, so only the authors can say which states they meant.
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Typo Fig. 1 labels the last module 'quantum module n' but its output '⟨H_N⟩'; elsewhere n is the number of qubits and M the number of Hamiltonian terms.
- p. 2 Fig. 1 diagram
quantum module n
Judge's note: The p2 image shows 'quantum module n' feeding ⟨H_N⟩ and ⟨H_N⟩ in the adder. The text uses n for qubits ('n-qubit state') and M for the number of terms. This is a schematic label inconsistency and is cosmetic.
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Typo 'N-representability problem' (p2, twice) is written 'n-representability problem' in the Discussion (p5).
- p. 2 Algorithm 1 discussion
N-representability problem
- p. 5 Discussion
where the n- representability problem does not occur
- p. 5 Discussion
n- representability problem
- p. 5 Discussion
n- representability problem does not occur
Judge's note: Images: p2 has 'N-representability' twice; p5 right column has 'where the n-representability problem does not occur'. Lower-case n is the paper's symbol for the number of qubits. Cosmetic inconsistency.
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Typo Fig. 3(b) caption: 'state |ψ^j⟩ at each the optimization step j' has an extra word.
- p. 4 Fig. 3 caption (b)
at each the optimization step j
Judge's note: Seen on the p4 image in the Fig. 3 caption. It should read 'at each optimization step j'.
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Typo Ref. [6] (Nielsen & Chuang): 'edited by C. S. o. Information and t. N. Sciences' is a garbled editor field.
- p. 5 Reference [6]
edited by C. S. o. Information and t. N. Sciences
Judge's note: The p5 image shows the garbled text exactly as quoted. Read as names, these are not plausible editors; this is evidently a series title mangled by BibTeX into an editor field.
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Typo Misspellings: heading 'SUPPLEMETARY THEORY' (p7) and 'efficient algorthim' (p8).
- p. 7 Appendix heading
SUPPLEMETARY THEORY
- p. 8 k-sparse section
algorthim
Judge's note: Both are visible on the page images: the p7 heading and the p8 right column, first line.
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Typo Misspelling: "efficient algorthim for finding the locations and values" should read "algorithm".
- p. 8 Application to k-sparse Hamiltonians
efficient algorthim for finding the locations
- p. 8 k-sparse Hamiltonians paragraph
efficient algorthim for finding
- p. 8
efficient algorthim for finding the locations and values
Judge's note: Page 8 image, top of the right column: the misspelling "algorthim" is in the printed text.
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Typo Missing word: "has found that it possible to use a star decomposition" should read "that it is possible".
- p. 8 Application to k-sparse Hamiltonians
has found that it possible to use a star decomposition
Judge's note: Page 8 image, right column: the text reads "has found that it possible", with no "is".
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Typo Duplicated word: "must still respect the known complexity QMA-Complete complexity of this task".
- p. 9 Computational Scaling, last paragraph
respect the known complexity QMA-Complete complexity of this task
- p. 9 Computational Scaling, last paragraph
the known complexity QMA-Complete complexity of this task
- p. 9 Computational Scaling, final paragraph
known complexity QMA-Complete complexity of this task
Judge's note: Page 9 image, last paragraph of the left column: "complexity" appears twice.
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Typo Kilojoule is written with an uppercase K: "37KJ/mol" and "ϵ = 50KJ/mol" should read kJ/mol.
- p. 9 Statistical errors
37KJ/mol
- p. 9 Systematic errors
ϵ = 50KJ/mol
Judge's note: Page 9 image, right column: both instances print "KJ/mol". Elsewhere the paper writes MJ/mol correctly. Cosmetic only.
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Typo Number agreement: "The minima of the potential energy curve was determined" should read "The minimum ... was".
- p. 9 Statistical errors, curve fit paragraph
The minima of the potential energy curve was deter-
Judge's note: Page 9 image, right column: the sentence refers to a single equilibrium minimum, so "minima" should be "minimum".
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Typo "voltage-controlled phases shifters" should read "phase shifters".
- p. 10 Systematic errors, second effect
voltage-controlled phases shifters
Judge's note: Page 10 image, left column, Systematic errors paragraph: the printed text says "phases shifters".
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