Update multi-product-formula tutorial to conform to new template

[henryzou50]

Summary

Brings multi-product-formula.ipynb in line with new tutorial template and the conventions established by recent tutorial updates.

Both the small-scale simulator example and the large-scale hardware example now follow the standard four-step layout. Labels match content: Step 2 = transpile/optimize, Step 3 = execute, Step 4 = post-process (compute MPF coefficients and combine). Previously, Step 2 was doing execution and Step 3 was doing post-processing.

Other changes:

• Background section rewritten to focus on the Trotter-step (k) vs. circuit-depth trade-off rather than the PF order. r and k_j are introduced on first appearance, eta_n / chi exponents are made explicit for symmetric vs. non-symmetric PFs, and the "stability" paragraph is replaced with a derivation-based explanation of why t/k_min <~ 1 matters.
• Prerequisites point to the compilation-methods tutorial and SuzukiTrotter / LieTrotter API pages (the previous links were mismatched).
• Requirements trimmed (removed unused quimb / qiskit-quimb).
• Per-cell narrative restored for the LSE inspection cells, identity_factory / Néel MPS setup, and the dynamic-coefficients loop.
• Small-scale post-plot commentary corrected: no longer reads as if MPFs work better at t/k > 1, and the sampling-error formula sigma_MPF^2 = sum_j x_j^2 sigma_{k_j}^2 is given explicitly.
• Large-scale section corrected: the misleading "Steps 1-4 combined" header is dropped (that cell is only Step 1), the inline comment on the k=6 baseline is fixed (depth-comparable single-circuit run, not one that matches the MPF's effective Trotter error), and the results discussion is rewritten against the latest hardware run (dynamic MPF beats every individual product formula; exact-static MPF is worst because ||x||_1 = 5.63 amplifies hardware noise).
• TeNPy unit_cell_width UserWarning suppressed in the Setup cell. The default is correct for Chain lattices, which is what CouplingMap.from_line(...) produces here.
• cspell:ignore extended with the new technical / LaTeX terms.

Acknowledgments

Thanks to Boseong for the detailed review of the prior draft. The descriptions, Step-label fixes, and several of the Background corrections above incorporate that feedback directly.

[qiskit-bot]

One or more of the following people are relevant to this code:

• @jenglick
• @nathanearnestnoble

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[abbycross]

Which step 4 should this link to, since we have two headers with this same title?

Also - should the step names ever change, I suggest future-proofing it with this syntax on the line right before the relevant step 4:
<span id="step-4-or-whatever-anchor-name-you-want"></span>
and then you can use [Step 4](#step-4-or-whatever-anchor-name-you-want) to call it.

[henryzou50]

Oh this should link to step 4 of the small scale example. I just pushed a change for that and did your suggestion! That is good to know and thanks!

[abbycross]

Suggested change

"- **Trotter error.** The individual product formulas (grey markers) deviate from the exact curve more and more as time grows, with the largest deviation for $k=1$ — that circuit is the shallowest, but it is also already in the regime where $t/k \\gtrsim 1$, so the leading $1/k^{2}$ error term is large. The MPF combinations (colored markers) cancel several of these leading Trotter-error terms, so they track the exact curve much more closely than any single $k_j$ circuit. The remaining gap reflects the higher-order Trotter terms that the MPF does *not* cancel: an order-$2$, $r=3$ static MPF only kills the first two error orders, and at large $t/k_{\\min}$ the uncancelled tail eventually dominates — so MPF is not a free pass to use very shallow circuits at arbitrary times.\n",

"- **Trotter error.** The individual product formulas (grey markers) deviate from the exact curve more and more as time grows. The $k=1$ circuit has the largest deviation and is the shallowest, but it is also already in the regime where $t/k \\gtrsim 1$, so the leading $1/k^{2}$ error term is large. The MPF combinations (colored markers) cancel several of these leading Trotter-error terms, so they track the exact curve much more closely than any single $k_j$ circuit. The remaining gap reflects the higher-order Trotter terms that the MPF does *not* cancel: an order-$2$, $r=3$ static MPF only kills the first two error orders, and at large $t/k_{\\min}$ the uncanceled tail eventually dominates — so MPF does not necessarily invite using very shallow circuits at arbitrary times.\n",

[abbycross]

With translations in mind, I tried rewriting the last sentence to avoid colloquialisms, but perhaps you can come up with something better?

[henryzou50]

Good point, I went with: "so MPF does not guarantee that very shallow circuits remain accurate at arbitrary times." This should read a bit more naturally than "invite using" and should translate cleanly. Also fixed a stray double-backslash in \gtrsim that snuck in. Let me know if you'd prefer different phrasing!

[abbycross]

Sounds much better, thanks @henryzou50!

[abbycross]

I've finished a copyedit sweep ✅

[abbycross]

Thanks!

[nathanearnestnoble]

I think this tutorial looks good to publish! I am curious for us to evaluate how necessary the backend filtering is for performance and whether we should apply this principle to all of the large scale hardware demos (+ updating backend characterization).