A quantum algorithm for modular flow

Ian T. Lim; Isaac H. Kim

A quantum algorithm for modular flow

T0 review · 4 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper gives a quantum algorithm that ε-approximates modular flow in O~(κ²|t| log(κ²/ε)) queries, with a matching lower bound.

desk verdict Modular flow via QSVT is a good idea, but the central lemma has a bogus error bound and the advertised complexity doesn't follow. read the letter →

arxiv 2508.16826 v2 pith:7VXZ7D6P submitted 2025-08-22 quant-ph

Ian T. Lim , Isaac H. Kim This is my paper

classification quant-ph

keywords modularflowHamiltonianquantumsingularvaluetransformQSVTentanglementvonNeumannentropychiralcentralchargeAdS/CFT

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

Entanglement entropy is a single number, but the modular Hamiltonian -log ρ and its flow ρ^{-it}Oρ^{it} carry more information, and physicists have proposed using modular flow to extract topological data and to describe observers inside holographic spacetimes. This paper supplies what was missing: an explicit quantum algorithm that implements modular flow with a provable cost. For any block-encoded state ρ whose smallest nonzero eigenvalue is 1/κ, Theorem 1 gives an ε-approximation to the flow in O~(κ²|t| log(κ²/ε)) queries to the encoding. The construction approximates -log ρ by a QSVT-compatible polynomial, then runs Hamiltonian simulation; a companion lower-bound argument shows the cost is near-optimal. If the algorithm works as claimed, modular flow becomes a computable primitive for condensed-matter and holographic experiments.

What carries the argument

The modular Hamiltonian polynomial: a degree-2N Chebyshev partial sum f_N(x) = -log 2 + Σ_{n=1}^N T_{2n}(x)(-1)^{n-1}/n, rescaled by 1/log(1/2κ) and multiplied by a rectangular polynomial that suppresses |x|≤1/2κ, making it bounded by 1 and of definite parity. This polynomial is admissible for the quantum singular value transform (QSVT), which applies it to the singular values of a block-encoded ρ, producing an ε-approximation to -log ρ; Hamiltonian simulation then applies ρ^{it}. The degree N=O(κ² log(κ²/ε)) of the log polynomial is the dominant cost.

What would settle it

Take a 2×2 state ρ = diag(1/κ, 1−1/κ), classically compute ρ^{-i}Oρ^i for a fixed traceless operator O, then run the algorithm with a block-encoding circuit and count the queries. If the query count exceeds C κ²|t| log(κ²/ε) for a fixed constant C, or the implemented operator differs from the exact flow by more than ε in spectral norm, Theorem 1 is false.

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Extended reading notes

Core claim

The paper's central claim is that modular flow is efficiently implementable from a block encoding of ρ. Theorem 1 states that for a block-encoded ρ with smallest nonzero eigenvalue 1/κ, one can construct Õ(t) with ||Õ(t)-ρ^{-it}Oρ^{it}||<ε using O~(κ²|t| log(κ²/ε)) queries to the encoding. The proof constructs an explicit polynomial approximating (1/2 log(1/2κ)) log(x²) on the spectrum outside a gap of width 1/κ, uses the quantum singular value transform to turn it into an approximate modular Hamiltonian, and feeds that into Hamiltonian simulation to produce ρ^{it}. A reduction from von Neumann entropy estimation to modular flow then gives a query lower bound, so the algorithm's scaling cann

Load-bearing premise

The complexity guarantee is measured only against an oracle that block-encodes ρ; if no such encoding can be made—likely for holographic states—the algorithm's cost bounds do not carry over.

Editorial extensions

If this is right

For any block-encoded ρ with smallest nonzero eigenvalue 1/κ, the modular flow of any operator can be ε-approximated at modular time t using O~(κ²|t| log(κ²/ε)) queries, making modular flow a concrete computational primitive rather than a formal object.
The reduction from von Neumann entropy estimation to modular flow implies any modular-flow algorithm must scale polynomially with the dimension n, so the κ² dependence in Theorem 1 cannot be improved to log n.
In a topologically ordered gapped ground state, running modular flow across a tripartite disk and measuring the entanglement entropy of one region reveals the chiral central charge c_- through the slope πc_-/3 of the entropy as a function of modular time.
Given oracle access to a block encoding of the reduced state and to operator insertions, modular flow plus quantum phase estimation can evaluate the correlation function proposed for probing behind the horizon in the holographic bulk-observer scenario.
When modular flow is applied to a purification |ψ⟩, the κ-dependence can be tamed: choosing κ=O(d²/δ²) gives error at most δ with cost linear in |t| and polynomial in the dimension d.

Reading between the lines

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

I infer that the Chebyshev expansion of log(x²) is the true workhorse: the same technique should apply to other operator functions with logarithmic singularities, giving κ²-type costs whenever the spectrum has a gap.
The paper's caveat that holographic states may lack efficient block encodings suggests the practical bottleneck is state preparation, not flow simulation; a testable next step is finding circuit families for which ρ^{it} is easy even though ρ itself is hard to block-encode.
If the purification construction is combined with existing ground-state preparation oracles for gapped systems, modular flow could become a practical probe of topological order at modest system sizes, since the cost no longer singles out tiny eigenvalues.
The polylog(n) entropy estimate for fixed κ indicates that the hardness of von Neumann entropy estimation resides in resolving tiny eigenvalues; for states with known condition number, entropy estimation may be much easier than the worst-case n² lower bound.

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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

John,

You should know before reading closely: the central theorem as stated is not supported. Lemma 2 claims the degree-2N Chebyshev partial sum for ½log(x²) is ε-close on [1/κ,1] with N = O(κ² log(κ²/ε)). The proof compares the tail to the Maclaurin tail of log(1+z) with z = -cos(2 arccos x). That comparison is invalid: the series is not a power series in that z. Its terms are cos(2n arccos x)/n. At x=1/κ, the tail is effectively -∑_{n>N} cos(2n/κ)/n. For small 2/κ, sums of cos(2n/κ) over intervals are O(κ), so the tail is bounded by about 2κ/(N+1). Plugging in their N leaves a residual ~1/(κ log(κ²/ε)), far above ε. So the error bound is false, and Theorem 1's query complexity is unsupported. The construction as written needs degree at least ~κ/ε, which kills the practical claim.

That's the bad news. The good news is the paper still has something. The task—implementing modular flow algorithmically—is genuinely new, and framing it with QSVT is natural. The review of QSVT is clear, and the applications (chiral central charge, AdS/CFT bulk observer, entropy estimation) are plausible directions. The reduction for the lower bound is at best a consistency check: they show their own algorithm has the n²-like scaling, but they don't prove a matching lower bound on modular flow. The abstract overstates this as a proven lower bound; it isn't. The QPE-based entropy estimation also ignores the cost of high-precision phase estimation; the variance argument needs more care.

On the citation pattern and assumptions: the block-encoding oracle for ρ is a real limitation, but they acknowledge it. The external results are standard. No signs of fitted parameters or invented entities.

So who is this for? Someone working on quantum algorithms for entanglement-related quantities will find the framing useful and the flaw instructive. I would not cite it as a reliable result until the polynomial approximation is fixed or replaced. The authors may be able to salvage the idea by using a different polynomial, but the current write-up needs a major revision.

If this lands on my desk, I'd send it to a referee—the question is important enough—but with a clear request to check the approximation lemma and resubmit after fixing it.

Best.

Referee Report

4 major / 4 minor

Summary. The paper proposes a QSVT-based algorithm for implementing modular flow ρ^{-it}Oρ^{it} for a block-encoded density matrix ρ. The central construction approximates -log ρ by a truncated Chebyshev series, then feeds the resulting approximate modular Hamiltonian into a QSVT Hamiltonian-simulation routine. The authors claim a query complexity of O~(κ²|t| log(κ²/ε)), where 1/κ is the smallest nonzero eigenvalue of ρ. They also discuss applications to chiral central charge extraction, to the Jafferis–Lamprou/Gao–Lamprou holographic observer proposal, and to von Neumann entropy estimation, and they claim a query-complexity lower bound for modular flow based on the known entropy-estimation lower bound. The paper is clearly written and situates the algorithm within standard QSVT tools, but the main technical lemma and the lower-bound argument have serious defects.

Significance. If correct, the paper would provide the first explicit provable algorithm for modular flow and would connect a well-studied physics quantity to the QSVT toolbox. The applications, especially to chiral central charge and to holographic bulk-observer probes, are potentially interesting. The paper also has the virtue of being constructive and explicit. However, the central polynomial approximation lemma is demonstrably incorrect, and the claimed lower bound is not a lower bound. The significance is therefore conditional: the conceptual idea is natural, but the results as stated are not supported.

major comments (4)

[§3.1, Lemma 2] The error bound in Lemma 2 is false. The proof identifies the tail of ∑ T_{2n}(x)(-1)^{n-1}/n with the Maclaurin tail of log(1+z) at z=cos(2 arccos x), but the actual tail at x=cos θ is ∑_{n>N} (-1)^{n-1} cos(2nθ)/n = Re ∑_{n>N} (-1)^{n-1} e^{2inθ}/n, which is not a power series in cos(2θ). At x=1 the tail is the alternating harmonic tail ∑_{n>N} (-1)^{n-1}/n ~ 1/(2N), not exponentially small. At x=1/κ, the tail is -∑_{n>N} cos(2n/κ)/n, and Abel summation with |∑_{n=1}^M cos(2n/κ)| ≤ 1/sin(1/κ) gives |tail| ≤ 2κ/(N+1). Thus for N=O(κ² log(κ²/ε)) the error at x=1/κ is ~1/(κ log(κ²/ε)), which is far larger than ε for large κ and small ε. Consequently Eq. (12), the degree bound Eq. (13), and the query complexity of Theorem 1 are unsupported.
[§4.3, 'query complexity lower bound'] This section does not prove a lower bound on modular flow. The argument says that the von Neumann entropy lower bound Ω(n²/ε) forces any modular flow implementation to scale polynomially in n. But a reduction from entropy estimation to modular flow only shows that the particular entropy-estimation algorithm built on modular flow must respect the entropy lower bound; it does not constrain the complexity of an arbitrary modular-flow oracle. The inequality in Eq. (47) compares an upper bound on the proposed entropy algorithm with a known lower bound; that is a consistency check, not a lower bound on modular flow. The sentence 'There can be no modular flow implementation scaling like log n' is a non sequitur. A genuine lower bound would require showing that any algorithm implementing modular flow with q queries can be converted into an entropy estimator whose complexity violates the known bo
[§4.3, Eq. (39)–(41)] The QPE-based entropy estimation omits the phase-precision cost. The entropy is recovered from the mean of phases θ_n = π log λ_n / log κ. To achieve additive error ε in S(ρ), the phases must be estimated to precision O(ε/log κ), and QPE then requires O(log κ/ε) controlled-U applications per sample (or a comparable number of ancilla qubits). This factor is absent from the sample count in Eq. (40) and from the total query count in Eq. (41). In addition, Eq. (33) defines QPE(ρ,U) as ∑ e^{iθ_k}⟨k|ρ|k⟩ = Tr(ρU), while Eq. (38) uses QPE(V)_ρ = ∑ θ_n λ_n; these are different quantities, and QPE does not directly output Tr(ρU) for a general U. The section therefore does not provide a valid estimate of von Neumann entropy with the stated complexity.
[§4.2, Eq. (33)–(36)] The claimed computation of the correlation function W(s,t) by QPE is not justified. For a mixed state ρ and a unitary U that does not commute with ρ, QPE on ρ gives a distribution over U-eigenstates with weights ⟨k|ρ|k⟩ in the eigenbasis of U; it does not return Tr(ρU). The operators ilde U = ρ^{-is}U_1ρ^{is}U_2 are not generally diagonal in the eigenbasis of ρ, so the identity in Eq. (35) does not follow. A controlled-Hadamard test on a purification would be a standard alternative, but no such procedure is described. Thus the holographic application, as written, is not established.

minor comments (4)

[§2.1] The definition of a block encoding is imprecise: 'acts on H_R⊗H_A as |0⟩⟨0|⊗O' should be 'there exists a unitary U whose top-left block equals O/α', with α≥∥O∥.
[Eq. (33) and Eq. (38)] The notation QPE(ρ,U) is used inconsistently: Eq. (33) writes it as a trace with e^{iθ_k}, while Eq. (38) writes it as a phase average with θ_n. These are different quantities and should be distinguished.
[Appendix A and Conclusions] Typos: 'Cheybshev' should be 'Chebyshev'; 'langauge' should be 'language'.
[Appendix B and §3.2] The signs around log(1/2κ) are confusing. The text says P^{MH} uses 1/(2 log(1/2κ)), which is negative for κ>1/2, while Appendix B says f^{log}_{κ',ε} is bounded by log(2κ). Please normalize consistently to avoid ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central claim derives from QSVT and an external entropy lower bound; self-citations appear only in applications.

full rationale

The paper's central derivation is self-contained. The modular flow algorithm approximates -log ρ by constructing a Chebyshev polynomial for log(x^2) (Lemmas 1-2) and then applies QSVT-based Hamiltonian simulation. No parameter is fitted to the modular flow itself, and the target quantity ρ^{-it}Oρ^{it} is not used to define the logarithmic approximation. The query-complexity bound follows from the polynomial degree and the known Hamiltonian simulation scaling; it is not obtained by assuming the conclusion. The lower-bound argument in Section 4.3 reduces von Neumann entropy estimation to modular flow and then invokes an external lower bound from [22]; it does not assume the modular-flow query complexity, so the reduction is a legitimate complexity-theoretic implication rather than a circular step. Self-citations [9,10,13] occur only in the applications discussion (chiral central charge, modular commutator) and are not load-bearing for Theorem 1 or Theorem 2. The skeptical concern about Lemma 2's error bound is a mathematical-correctness issue, not a circularity issue; even if the bound were wrong, that would invalidate the proof but not constitute the derivation reducing to its own input. Overall, no circular step can be exhibited from the manuscript's own equations.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The algorithm introduces no new physical entities. It relies on standard QSVT machinery and a domain assumption of block-encoded state access. The main hand-chosen parameters are analytical, not data-fitted. The presence of an unverified model assumption in the lower-bound argument is a concern.

free parameters (2)

rectangular function parameter ε' = min(2ε/(5 log κ), log(1/2κ)/N(κ,ε))
Chosen by hand in Eq. (17) to ensure the modular Hamiltonian polynomial is bounded by 1 and has the desired approximation error. It is an analytical choice, not fitted to data, but it is a free auxiliary parameter in the construction.
truncation point κ in Section 3.4 = (d/(|t|ε))^{2/3}
Chosen to balance two error terms in the purification bound. It is a hand-picked algorithmic parameter, not an observable constant.

assumptions (4)

standard math QSVT can implement any polynomial of the singular values of a block-encoded operator with degree equal to the number of encoding queries.
The paper relies on the QSVT framework from [20,23] as a black box, quoting its polynomial transformation property in Section 2.1.
standard math Hamiltonian simulation via QSVT has query complexity Θ(α|t| + log(1/ε)/log(e + log(1/ε)/(α|t|))) for a Hamiltonian with norm bound α.
Invoked in Section 3.3 using the result of [23]. This is an external theorem the paper depends on.
domain assumption The algorithm has oracle access to a block encoding of ρ.
Stated at the start of Section 3.3 and discussed in the Conclusions. It is essential to the query complexity; without it the algorithm cannot be run.
domain assumption The known query lower bound Ω(n² log(1/δ)/ε) for von Neumann entropy estimation holds in the same oracle model.
Used in Section 4.3 to infer a lower bound on modular flow. This assumption is not justified in the paper; the cited [22,27] lower bounds may be in a copy model rather than a block-encoding query model.

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Cite this review

Pith. "Pith review of A quantum algorithm for modular flow." pith.science (2026). https://pith.science/paper/7VXZ7D6P

@misc{pith2026250816826,
  author       = {Pith},
  title        = {Pith review of: A quantum algorithm for modular flow},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7VXZ7D6P}},
  note         = {Machine review of arXiv:2508.16826}
}

read the original abstract

Entanglement is a defining property of quantum systems. For a subsystem of a larger quantum system, one can formally define an operator known as the modular Hamiltonian, which is closely linked to the entanglement properties of that subsystem, and a corresponding operator flow called the modular flow. Algorithms for estimating the von Neumann entropy, the best-known entanglement measure, are well-established, but no equivalent procedures have been previously described for the modular flow. In this work, we briefly review the quantum singular value transform (QSVT) framework for developing quantum algorithms, and then discuss the implementation of modular flow within this framework. We conclude by describing select applications of our modular flow algorithm, such as extracting the chiral central charge of a topologically ordered system and simulating the experience of the bulk observer in holography. We also prove a query complexity lower bound for modular flow, which shows that our method cannot be improved further substantially.

Figures

Figures reproduced from arXiv: 2508.16826 by the authors.

**Figure 1.** Partition of a disk used in Ref. [11]. 4.2 Modular flow in AdS/CFT In [16], Jafferis and Lamprou proposed a framework for “describing the experience of bulk observers in AdS/CFT.” Roughly, in the AdS/CFT correspondence, properties of a certain gravitational theory in d + 1 dimensions (the bulk) are shown to correspond in a precise way to properties of a quantum theory in d dimensions (the boundary) and vice versa. T… view at source ↗

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Reviewed August 5, 2026 · model on record in the stance chip above.