Update kqd tutorial

[snow0369]

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✅ henryzou50
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[henryzou50]

Hi @snow0369, thanks for the great work on this! The extended-swap-test construction, the shifting technique, and the spectral-filter appendix are all well-motivated! Below is a summary of what I already fixed on the branch, plus some suggestions to address before merge.

What I already pushed (CI fixes, no action needed)

I fixed the CI blockers directly, so please pull the branch before making changes:

• Ran the repo tooling (tox -e fix): extracted the embedded PNG outputs to .avif files, ruff-formatted the code cells, and normalized notebook metadata
• Moved warnings.filterwarnings("ignore") below the imports in the Setup cell (ruff E402).
• Added the {/* cspell:ignore ... */} comment and deleted the stale images from the old tutorial version.
• Added notebook metadata for the usage estimate

Suggested changes before merge

Template conformance (comparing against the recently revised tutorials, e.g. multi-product-formula, nishimori-phase-transition)

• Requirements section: the Qiskit pin should be v2.3 or later, not v2.0 -- CommutativeOptimization doesn't exist before 2.3 (I confirmed it fails to import on 2.2.3). Also add the "with visualization support" clause -- draw("mpl") fails on a bare pip install qiskit without pylatexenc.
• Citations: the in-text citations were changed from linked ([\[1\]](#references)) to plain [1], please restore the linked form so they're clickable on the platform (your source notebook had this right).
• Job tags: use a single underscore-style tag per repo convention (e.g. TUT_KQD). Also drop "ManualQEM", which looks like a leftover from the development runs.
• Section order: the revised tutorials end with ... -> Next steps -> References; this notebook has References before Next steps.
• Headings: "Reference State", "Time Evolution", "Extended Swap Test Circuit" are bold Title-Case text used as headings, make them real sentence-case headings (#### Reference state, etc.). Also "Step 2: Optimize the problem..." -> "Optimize problem..." to match the standardized step titles, and "may vary" -> "might vary" in the usage estimate (style guide avoids "may").

Narrative vs. code suggestions

• Step 3 text says we measure X⊗H and Y⊗H, but the code (correctly) measures the shifted X⊗(H−T), Y⊗(H−T) per the shifting subsection. Please update the text to match.
• The printed "Estimated first row of H" in the small-scale Step 4 is actually the shifted H̃ row as it starts with 0.+0.j even though H₀₀ = ⟨ψ₀|H|ψ₀⟩ = 7, which could look like a bug to a eader. Relatedly, the small-scale section solves the shifted GEVP and adds τ back to the eigenvalue, while the large-scale section reconstructs H = H̃ + τS as the text promises. Both are equivalent, but please either make the small-scale cell reconstruct H like the large-scale one, or add a sentence explaining the shortcut.
• The ordering-comparison conclusion overstates its own table: "we clearly observe that even-odd edge-grouped ordering offers the lowest depth, two-qubit count, and infidelity", but in the table the two-qubit count is identical (33) for all three orderings, depth ties with even-then-odd (6), and the infidelities are 0.9999 vs 0.9998 (all ≈ 1). The real story the data supports: even/odd ordering collapses depth 15 → 6, and edge-grouping additionally reduces Trotter error. Please restate accordingly (the same applies to the earlier claim that edge-grouping "gives a shallower circuit after synthesis and transpilation").

Copyedits

• Present tense throughout: "we are going to build/present/implement" -> "we build", "here is", etc.
• There's a stray comma after \end{equation*} in the shifting subsection (renders as a floating comma), and the $T'$ prime notation is introduced once and never reused; the surrounding sentence is also a comma splice.
• "Other parameters such as Krylov dimension and the number of Trotter steps are determined." reads unfinished. We should state the chosen values and why.
• "Because of commutation relation" -> "Because all powers of H commute" (or similar); "noncommuting terms in neighbor" -> "next to each other"; "the number of 1 in the state" -> "the excitation number".
• Delete the leftover commented-out debug lines in the small-scale Step 4 (# pub_obs..., # assert...), and the unused retained_dim parameter of solve_thresholded_gevp.