Tight bound for the total time in digital-analog quantum computation

Mikel Garcia-de-Andoin; Mikel Sanz

REVIEW 4 major objections 4 minor 27 references

Tight bound for the total time in digital-analog quantum computation

T0 review · 4 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Digital-analog quantum computation has a tight worst-case time bound: any compatible two-body Hamiltonian can be implemented in total analog time at most T√3 times the 2-norm of the coupling ratios, and some three-coupling problems require

desk verdict The claimed √3 bound is fresh and plausible, but the all-n proof is asserted rather than shown; the paper is a promising draft, not a theorem. read the letter →

arxiv 2512.11619 v2 pith:QIH532TZ submitted 2025-12-12 quant-ph

Mikel Garcia-de-Andoin , Mikel Sanz This is my paper · ORCID

classification quant-ph MSC 81P6852B12

keywords digital-analogquantumcomputationDAQCtime-optimalcompilationHamiltoniansimulationconvexpolytopetwo-bodyHamiltonianstightboundTrotterdecomposition

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to settle the worst-case time cost of digital-analog quantum computation (DAQC), where a computation is compiled into single-qubit gates interleaved with evolutions under a fixed source Hamiltonian. It claims a tight upper bound: any compatible problem Hamiltonian can be implemented in total analog time at most T√3∥hP⊘hS∥2, and there exist problems—three non-zero couplings among three qubits—that reach this bound. This improves on earlier bounds that grew quadratically with the number of qubits, yielding a linear-in-qubits estimate for the worst case. If correct, resource estimates for DAQC simulations and comparisons with purely digital approaches become rigorous.

What carries the argument

The key object is the convex polytope M whose vertices are the columns of the sign matrix M, which encodes the sign flips produced by single-qubit Pauli gates sandwiching each analog block. Solving a DAQC problem b = T h^P ⊘ h^S as M t = b, t ≥ 0, is equivalent to writing b as a positive combination of those vertices; the optimal total time ∥t∥_1 is the scale factor needed to bring b onto the polytope's surface. The argument reduces the worst-case time to the minimal Euclidean distance from the origin to a facet of M, which is verified for small n to occur on the n=3 embedded tetrahedron. The recursive structure M(n+1) = [[L,L'],[M(n),M(n)]] carries the extension, with the new couplings para

What would settle it

Compute exactly, for the ZZ-Ising polytope with n=8 qubits, the minimum Euclidean distance from the origin to any facet (the columns of M are the vertices, so the closest facet gives the worst-case rescaling). If that distance, after normalizing the columns appropriately, is smaller than the corresponding distance for the embedded n=3 tetrahedron—i.e., if a problem b can be found with ∥t_opt∥_1 > T√3∥hP⊘hS∥2—then the claimed bound fails; the authors' own facet calculations stop at n=7.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1: for any problem Hamiltonian H_P compatible with a source Hamiltonian H_S, an optimal DAQC schedule exists with ∥t_opt∥_1 ≤ T√3 ∥h^P ⊘ h^S∥_2, where h^P and h^S are the vectors of the Hamiltonian couplings. The proof treats the columns of the sign matrix M as vertices of a convex polytope containing the origin; any problem vector b is a positive rescaling of a surface point of that polytope, and the optimal total time equals the rescaling factor. The paper identifies the worst facets as those belonging to the smallest non-trivial system: for ZZ Hamiltonians this is n=3, where the polytope is a regular tetrahedron and the closest facets correspond to four sign p

Load-bearing premise

The proof depends on an induction step that is asserted rather than fully demonstrated: the authors assume that for any number of qubits, the worst-case problem is the n=3 all-to-all problem embedded in the larger system, with all newly added couplings set to zero; the manuscript states this explicitly and gives only a sketch for n≥8.

Editorial extensions

If this is right

Any DAQC protocol with a compatible source Hamiltonian can be compiled with total analog time bounded by T√3 times the 2-norm of the coupling ratios, giving a direct resource estimate independent of the number of qubits except through that norm.
The bound is tight: some all-to-all problems with only three non-zero couplings require the full amount, so the constant √3 cannot be improved without restricting the problem class.
The result yields a two-sided characterization T∥hP⊘hS∥∞ ≤ ∥t_opt∥_1 ≤ T√3∥hP⊘hS∥2, with both endpoints achieved by explicit problems.
For all-to-all Hamiltonians, the 2-norm of the coupling ratios grows as the square root of the number of couplings, so the worst-case DAQC time scales linearly with the number of qubits, confirming the earlier conjecture's linear scaling though with a different constant and norm.
It also disproves a previously conjectured sharp bound based on the ∞-norm and n or n−1, providing a concrete counter-example.
The authors argue the result carries over to arbitrary two-body Hamiltonians and to DAQC protocols with fixed or arbitrary single-qubit rotations, since Pauli-based protocols give the worst case.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the recursive step holds for all n, the same geometric inradius argument should extend to other gate sets and non-Pauli sign-flipping operations, giving similar √3-type constants wherever the worst-case polytope is the smallest non-trivial one.
The polytope-inradius perspective suggests a practical way to generate hard DAQC instances: problems whose direction is perpendicular to the closest facets of M, which could be found for moderate system sizes by solving the facet-closest-point problem without full facet enumeration.
A direct device-level test is possible on small processors: compiling the identified worst-case problems (three equal-magnitude couplings of opposite sign on three qubits) should reproduce the √3 saturation; deviations would reveal gaps between the compilation model and hardware constraints such as finite gate speed or crosstalk.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the bound is new and the geometric angle is right, but the proof of Theorem 1 for arbitrary n is not there. The paper explicitly says so at two key points: "we assume that the worst problem corresponds to an n=3 all-to-all problem embedded in a larger system" and "by brute-forcing the resolution of this problem, we see..." That is an induction hypothesis plus a numerical check, not a proof. The reader's verdict is correct.

What is actually new: the bound ∥t∥1 ≤ √3 T ∥hP⊘hS∥2 is not in the earlier literature; the previous best was linear in n with an ∞-norm. The claim that the Baßler-Heinrich-Kliesch conjecture is not tight is also new, though the counter-example is never exhibited. The geometric picture (projecting onto the DAQC polytope and bounding the inradius) is a natural and likely correct way to attack the problem, and the n=3 case is computed exactly: the regular tetrahedron gives the √3 constant.

Where it falls apart: the step from n=3 to general n. For n=4–6 the paper parameterizes the new couplings and asserts that setting them to zero is worst. For n≥7 it says the topology reduces to n=6 and "we can extend the upper bound to arbitrary n." That is not a formal induction; it does not bound the facets introduced by the new qubit. The numerical exact facet calculation stops at n=7 for ZZ and n=4 for general two-body, so the tightness for larger n is unverified. There is also a minor slip in the decomposition for n=4: the α_i are not constrained to sum to 3α, so the decomposition does not exactly reproduce b. That is fixable, but it adds to the impression that the induction was never fully written down.

The citation pattern is fine. The main self-citation [22] is to a prior result that the sign-pattern matrix spans the space and the polytope contains the origin—that is a legitimate base to build on, not a circular argument.

Bottom line: this is a plausible conjecture with a suggestive partial proof and good small-n numerics. It deserves a serious referee, but not acceptance in its current form. The authors need to either prove the inradius bound for all n—probably via explicit facet bounds for the recursive polytope structure—or state the result as a conjecture and show the counter-example to BHK directly. As is, I would not cite it as a theorem. If you are working on DAQC resource estimates, it is worth reading for the geometric framing and the n=3 constant.

Referee Report

4 major / 4 minor

Summary. The paper studies digital-analog quantum computation (DAQC) and proposes a tight upper bound on the total analog evolution time required to simulate an arbitrary two-body Hamiltonian. After vectorizing Hamiltonians, the compilation problem is reduced to solving M t = b with t ≥ 0, where M is a sign-pattern matrix. The authors' main result, Theorem 1, states that for any compatible target and source Hamiltonian, ∥t_opt∥_1 ≤ T√3 ∥h^P ⊘ h^S∥_2, and that the bound is tight, realized by problems with three non-zero couplings. The proof is based on a geometric interpretation: the columns of M generate a polytope whose inradius would need to be at least 1/√3. The paper verifies the n=3 case exactly, gives numerical evidence for moderate n, and extends the claim to arbitrary two-body Hamiltonians. It also states that the previous Baßler-Heinrich-Kliesch bound is not optimal and that a counterexample is provided.

Significance. If Theorem 1 were rigorously established, the result would be a clean, parameter-free bound on DAQC time resources, scaling linearly with the number of qubits for all-to-all Hamiltonians and only polylogarithmically in the number of couplings in typical cases. The geometric reformulation in terms of the inradius of the DAQC sign-pattern polytope is appealing and could be a useful tool for future work. The paper also contains careful exact or numerical information for small system sizes. However, the central theorem for arbitrary n is not proven: the key inductive step is explicitly assumed, not derived, and no counterexample to the BHK conjecture is actually exhibited. The claimed tightness and the claimed improvement over the previous bound therefore remain unsupported. The numerical experiments are suggestive but cannot replace a proof of the general statement.

major comments (4)

[Section II.A, proof of Theorem 1] The induction proving the bound for all n is asserted, not demonstrated. The text says, 'we assume that the worst problem corresponds to an n=3 all-to-all problem embedded in a larger system, and assume that this holds up to n qubits,' and later 'we can extend the upper bound to arbitrary n.' This does not bound the new facets introduced when adding a qubit, nor does it justify that the n=3 configuration remains the closest facet to the origin. Exact facet information is reported only for n≤7 for ZZ Hamiltonians and n≤4 for arbitrary two-body Hamiltonians. Hence Theorem 1 for general n is not proven by the manuscript.
[Section II.A, paragraph beginning 'For any b̃'] The proof that a boundary point b̃ must have minimal 1-norm equal to 1 contains a scaling error. If M t = b̃ with ∥t∥_1 = ξ < 1, the proposed t' = (2−ξ)t satisfies ∥t'∥_1 = (2−ξ)ξ, not 1. The contradiction therefore does not follow as written. A correct argument could rescale by 1/ξ and use convexity, but that argument is not given. This is a load-bearing step in the geometric reduction.
[Introduction and Conclusion] The paper states that the BHK conjecture is 'actually not optimal' and that a counterexample is provided, but no explicit counterexample appears. The n=3 worst-case vector b = (−α,−α,−α) gives ∥t_opt∥_1 = T√3∥b∥_2 = 3Tα, which equals the BHK bound for odd n=3 and is smaller than the BHK bound for larger n. Thus this example does not violate Conjecture 1. No explicit problem is exhibited for which ∥t_opt∥_1 exceeds the BHK bound.
[Section II.A, arbitrary two-body Hamiltonians] The extension to arbitrary two-body Hamiltonians lacks a verified base case for n=2. The text states that n=2 is non-trivial and identifies candidate worst directions, but it does not compute the facets or the inradius for the 9-dimensional polytope. Since the subsequent argument again relies on the same assumed induction for ZZ Hamiltonians, the general theorem is not established beyond the explicitly checked small cases.

minor comments (4)

[Abstract] The phrase 'linear dependence with the number of couplings' is inaccurate: for equal-magnitude couplings the 2-norm grows as sqrt(N_couplings), not linearly in N_couplings. The scaling is linear in the number of qubits for all-to-all Hamiltonians.
[Section II] Typo: 'Firsly' should be 'Firstly'.
[Eq. (7), notation] The notation M ∈ M_{d,d'}(±1) is nonstandard; it would be clearer to state that M is a d×d' matrix with ±1 entries and full row rank.
[Section II.B and figures] The description of the numerical generation of 'green' problems is imprecise ('close to the axes' with 'a maximum of 6 nonzero elements'). Please specify the exact distribution to make the numerical results reproducible.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation; the all-n induction is asserted rather than proved, which is a rigor gap, not a circular reduction.

full rationale

The paper's central bound ||t_opt||_1 ≤ T√3||h^P ⊘ h^S||_2 is derived from a geometric inradius claim about the DAQC sign-pattern polytope. The n=3 case is computed exactly (regular tetrahedron). For larger n, the proof attempts an induction, but explicitly states 'we assume that the worst problem corresponds to an n=3 all-to-all problem embedded in a larger system, and assume that this holds up to n qubits' (Section II.A). This is an unproven assumption used as the inductive step, not a circular use of the theorem's conclusion: the theorem's upper bound and tightness would follow if the assumption were proved, but the paper does not prove it. This is a completeness/rigor gap, not a reduction by construction. The self-citation to Ref. [22] for the fact that the sign-pattern columns span R^d and the polytope contains the origin is independent support (prior work established universality, not the time bound), so it does not constitute circularity. Numerical checks are confirmatory and do not fit parameters into the bound. No fitted input is renamed as a prediction, and no result is forced by definition or by a self-citation chain. Thus no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The proof depends on the universality result from the authors' earlier work [22], on an unproven assertion about the inradius of the DAQC polytope, and on standard convex-geometry facts. No new physical entities are introduced, and there are no fitted free parameters.

assumptions (3)

domain assumption For any compatible H_P and H_S, the columns of the sign-pattern matrix M span R^d and the origin lies in the interior of their convex hull (universality of DAQC).
Used to assert that every b has a positive-coefficient representation; cited from Ref. [22], a previous paper by overlapping authors.
ad hoc to paper The Euclidean inradius of the DAQC polytope is at least 1/√3 for all n.
This is the crucial geometric assertion of the proof; the paper attempts to prove it by an inductive sketch, but the induction step is not rigorously demonstrated and the assumption is explicitly stated in Section II.A.
standard math The polyhedral gauge (minimal 1-norm of positive coefficients) of a boundary point is 1.
Standard convex geometry fact, but the paper's proof of this contains an algebraic error; the fact itself is true.

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Cite this review

Pith. "Pith review of Tight bound for the total time in digital-analog quantum computation." pith.science (2026). https://pith.science/paper/QIH532TZ

@misc{pith2026251211619,
  author       = {Pith},
  title        = {Pith review of: Tight bound for the total time in digital-analog quantum computation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QIH532TZ}},
  note         = {Machine review of arXiv:2512.11619}
}

read the original abstract

Digital-analog quantum computing (DAQC) is a universal computational paradigm that combines the evolution under an entangling Hamiltonian with the application of single-qubit gates. Since any unitary operation can be decomposed into a sequence of evolutions generated by two-body Hamiltonians, DAQC is inherently well-suited for realizing such operations. Suboptimal upper bounds for the total time required to perform these evolutions have been previously proposed. Here, we improve these limits by providing a tight bound for this crucial parameter, which shows a linear dependence with the number of couplings. This result enables a precise estimation of the time resources needed for quantum simulations and quantum algorithms implemented within the DAQC framework, facilitating a rigorous comparison with other approaches.

Figures

Figures reproduced from arXiv: 2512.11619 by the authors.

**Figure 2.** Minimum total analog block times for ZZ Hamiltonians for problems with norm ∥b∥2 = p n(n − 1)/2. Colored area shows the range between the obtained maximum and minimum total time for the different problem distributions, while the lines represent their mean value. The color code employed is blue for uniformly random problems, red for problems close to the axes and green to problems designed to be close to the upper… view at source ↗

**Figure 3.** Minimum total analog block times, in this case, for [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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Reference graph

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Reviewed August 3, 2026 · model on record in the stance chip above.

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ERDF A way of making Europe

or fixed rotation angles [6, 22, 23], as the protocols composed of Pauli gates are the worst of the three. This allows us to estimate the time resources needed to perform a simulation with DAQC, further allowing us to compare it with the time needed for its DQC counterpart. We provide an answer to an open question and confirm the previous intuition regard...

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David Elieser Deutsch, Adriano Barenco, and Artur Ek- ert, “Universality in quantum computation,” Proc. Roy. Soc. Lond. A Math.449, 669–677 (1995)

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Efstratios Manousakis, “A quantum-dot array as model for copper-oxide superconductors: A dedicated quantum simulator for the many-fermion problem,” Journal of Low Temperature Physics126, 1501–1513 (2002)

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D. Porras and J. I. Cirac, “Effective quantum spin sys- tems with trapped ions,” Phys. Rev. Lett.92, 207901 (2004)

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Universal quantum compu- tation and simulation using any entangling hamiltonian and local unitaries,

Jennifer L. Dodd, Michael A. Nielsen, Michael J. Brem- ner, and Robert T. Thew, “Universal quantum compu- tation and simulation using any entangling hamiltonian and local unitaries,” Phys. Rev. A65, 040301(R) (2002)

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2020