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A quantum algorithm for Khovanov homology

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Pith's one-line read A quantum algorithm could compute Khovanov homology, a knot invariant richer than the Jones polynomial.

desk verdict A conditional quantum algorithm for Khovanov homology with a solid boundary-operator construction and clean hardness reductions; the efficiency claim hinges on an unproven thermalization step, so it deserves peer review but not acceptance as a proven speedup. read the letter →

arxiv 2501.12378 v2 pith:INSUBGJY submitted 2025-01-21 math.GT math.QAquant-ph

Alexander Schmidhuber , Michele Reilly , Paolo Zanardi , Seth Lloyd , Aaron Lauda This is my paper · ORCID
classification math.GTmath.QAquant-ph MSC 57K1868Q1281P6805C50 PACS 03.67.Ac
keywords quantumalgorithmKhovanovhomologyBettinumbersHodgeLaplacianspectralgapJonespolynomialunknottingproblemGibbssampling
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

This paper proposes the first quantum algorithm for Khovanov homology, a topological invariant of knots that refines the Jones polynomial and is known to detect the unknot. The algorithm outputs an additive approximation to the graded Betti numbers of Khovanov homology, and would run in polynomial time if two conditions hold: the Hodge Laplacian of the Khovanov complex has an inverse-polynomial spectral gap and can be thermally prepared in polynomial time. The paper proves that increasingly accurate additive approximations to these Betti numbers are DQC1-hard, BQP-hard, and #P-hard, and it provides numerical and analytic evidence for the spectral-gap condition. A new pre-thermalization subroutine using low-temperature Gibbs states and a SWAP test lets the algorithm succeed even when the Betti numbers are tiny compared with the chain-space dimension.

What carries the argument

The load-bearing object is the Hodge Laplacian $\Delta_{(i,j)}$ of the Khovanov chain complex, whose kernel is isomorphic to the Khovanov homology group $\mathrm{Kh}^{i,j}(K)$; the algorithm exponentiates the Hermitian operator $B = d + d^\dagger$ via quantum signal processing, since $B^2 = \Delta$. The boundary operator is implemented with Jordan–Wigner creation operators that automatically generate the alternating signs, acting on a resolution register and an enhanced-state register for loop labelings. To handle small Betti numbers, the algorithm replaces uniform sampling with a Gibbs-state preparation step followed by quantum phase estimation and a SWAP test, so that a smaller kernel makes the estimate easier rather than harder. For the spectral-gap condition, the paper identifies the degree-zero Khovanov Laplacian $\Delta_{(0,q)}$ with the signless Laplacian $Q = D + A$ of a weighted graph $G_{n,k}(K)$ built from the all-zero resolution, and applies known bounds on the smallest eigenvalue of $Q$ to control the gap.

What would settle it

Compute the exact spectral gap of the Khovanov Hodge Laplacian for a sequence of knots (e.g., 3-strand pretzels with growing twist lengths) in all bidegrees; if the minimum gap decays faster than any inverse polynomial in crossing number, the numerical evidence underlying the algorithm's efficiency is refuted. Alternatively, exhibit a knot for which preparing the low-temperature Gibbs state provably requires exponential time.

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Extended reading notes

Core claim

The central claim is that the ranks $\beta_{i,j}(K) = \dim \mathrm{Kh}^{i,j}(K)$ of Khovanov homology can be approximated on a quantum computer by encoding the Khovanov chain complex of a knot diagram into qubits, expressing the boundary operator as a Jordan–Wigner-style fermionic operator, and treating the Hodge Laplacian $\Delta = d^\dagger d + d d^\dagger$ as a Hamiltonian whose kernel is exactly $\mathrm{Kh}^{i,j}(K)$. The algorithm prepares a low-temperature Gibbs state of $\Delta$ (pre-thermalization), projects onto the kernel via quantum phase estimation, and estimates the kernel dimension with a SWAP test; this replaces the uniform-sampling step of earlier quantum homology algorithms and works when Betti numbers are small. The efficiency statement is conditional: the runtime is $O(m^4 \beta^4 t_{\mathrm{therm}} / \delta^{1/2})$, and the paper does not prove that $t_{\mathrm{therm}}$ (the thermalization time) is polynomial, identifying it as the key open question. As corollaries the authors prove that estimating the Betti numbers is DQC1-hard, BQP-hard, and #P-hard at increasing accuracy, and they establish analytic lower bounds on the spectral gap in homological degree zero by showing the Laplacian there coincides with the signless Laplacian of an associated weighted graph.

Load-bearing premise

The algorithm is only efficient if the Khovanov Hodge Laplacian can be cooled to a Gibbs state with inverse-polynomial overlap with its kernel in polynomial time, and the paper does not prove this can be done for any knot.

Editorial extensions

If this is right

  • If the algorithm is efficient, Khovanov homology joins the short list of concrete topological invariants with a proposed exponential quantum speedup, and the unknotting problem becomes a plausible target because Khovanov homology detects the unknot.
  • The algorithm approximates the Jones polynomial at arbitrary values of the variable, extending known quantum algorithms that only work at roots of unity, although not always efficiently.
  • The hardness results imply that efficient thermalization cannot hold for every knot unless #P collapses into BQP, so the pre-thermalization step must be analyzed as an average-case or restricted-instance subroutine.
  • The analytic gap bounds guarantee that at least for twisted unknots in homological degree zero the gap is inverse-polynomial, supporting the spectral-gap hypothesis where enumeration alone cannot reach.
  • The preprocessing simplification of long twist sequences (replacing them by homotopy-equivalent short complexes) can raise the spectral gap above 1, suggesting a classical preprocessing step that improves the quantum runtime.

Reading between the lines

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

  • The density-of-states numerics suggest a concrete testable threshold: if the exponent in $N(dE) \sim e^{\alpha E}$ continues to grow with crossing number, the required Gibbs temperature will eventually become exponentially small, and the algorithm's efficiency would degrade even for average knots.
  • The signless-Laplacian correspondence in degree zero is likely to extend to other extremal homological degrees by symmetry, and possibly to Lee homology, offering a graph-theoretic route to gap bounds for filtered Khovanov invariants.
  • A natural extension is to run the algorithm's preparation subroutine on unknot diagrams with many twists: if cooling to the two-dimensional kernel is efficient, it yields a quantum certificate of unknottedness that avoids the full Betti-number readout.
  • One could test the thermalization assumption directly on small knots by simulating the proposed Lindblad evolution classically, checking whether the convergence time scales polynomially in crossing number before any quantum hardware is needed.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes a quantum algorithm that additively approximates the bigraded Betti numbers of Khovanov homology of a knot. The algorithm encodes the Khovanov complex using resolution, resolved-knot, loop-enumeration, and enhanced-state registers; implements the boundary operator as a sparse Jordan-Wigner-type Hamiltonian; and estimates Betti numbers by preparing a low-temperature Gibbs state of the Hodge Laplacian, projecting onto the kernel by quantum phase estimation, and applying SWAP tests. The runtime bound in Eq. (4.20) is polynomial provided two conditions hold: the Hodge Laplacian has an inverse-polynomial spectral gap, and a thermalizing Lindbladian prepares a Gibbs state with inverse-polynomial kernel overlap in polynomial time. The paper also proves DQC1-, BQP-, and #P-hardness for increasingly accurate additive approximations to the Betti numbers, develops an analysis of spectral gaps via homological perturbation theory, introduces graph-theoretic encodings of homological-degree-zero Khovanov Laplacians as signless Laplacians, and reports numerical gap data for knots and links up to 10--11 crossings.

Significance. If the two efficiency conditions are eventually established, this would be the first quantum algorithm for a non-simplicial homology theory, and it would give a concrete route toward the unknotting problem because Khovanov homology detects the unknot. The paper has several genuine strengths: the reversible encoding of the Khovanov complex is explicit; the boundary operator is shown to be sparse and efficiently exponentiating; the hardness theorems are clean corollaries of known Jones-polynomial results; and the connection between homological-degree-zero Khovanov Laplacians and signless Laplacians of associated graphs is new and potentially useful. The authors are also unusually candid about what is not proved: Section 9.3 explicitly identifies efficient thermalization as a key open question, and the spectral-gap evidence, while substantial, is analytic only in homological degree zero for twisted unknots and numerical elsewhere. As it stands, the central contribution is a well-motivated conditional reduction together with supporting evidence, rather than a complete efficient algorithm with all subroutines specified.

major comments (3)
  1. [4.5, 4.6, 9.3] The central efficiency claim is dominated by the undefined quantity t_therm. In §4.5 the algorithm assumes 'the ability to implement arbitrary sparse thermalizing Lindbladians', and the performance analysis in §4.6, Eq. (4.20), multiplies the runtime by t_therm without giving any Lindbladian, quantum Metropolis step, or spectral-filtering construction for the specific operator Δ=B². Section 9.3 then concedes that efficient thermalization is a key open question. The cited Gibbs-sampling results [37-42] do not automatically apply, because their mixing-time and detailed-balance hypotheses have not been verified for this nonlocal sparse Laplacian. Please either supply a concrete pre-thermalization subroutine with stated assumptions, or reformulate the main result explicitly as a conditional reduction whose theorem statement separates the unconditional encodings and hardness results from the unproven thermalization assumption.
  2. [6.3, Proposition 18] The proof of Proposition 18 is incomplete. The statement that the adjoint F1† is a chain map is justified only by the remark that the source is a one-term complex, but for an inclusion F1 : C → D with C one-term and D two-term one must separately verify both d_D F1 = 0 (the chain-map condition) and F1† d_D = 0 (the adjoint chain-map condition). The second condition is not automatic and generally fails when the target differential acts nontrivially on the complement of the image of F1. Since Proposition 18 is the basis for the claim that Reidemeister-one moves never increase the spectral gap, please write out the explicit differentials and verify both identities, or revise the proposition and its consequences.
  3. [7, 8, Conjecture 20] The spectral-gap assumption that enters Eq. (4.20) is not established in the required generality. Section 8 proves lower bounds only in homological degree zero, and the sharp identification of the minimal gap for twisted unknots remains Conjecture 20, verified numerically only up to n=10. The numerical evidence in Section 7 covers all knots through 10 crossings but only 8 of 552 eleven-crossing knots, and the Lorentzian/cubic fits involve free parameters that are not predictive beyond the fitted range. Please state clearly, in the abstract and in the main theorems, that the inverse-polynomial global gap is a conjecture, and distinguish unconditional results (encoding, Hamiltonian simulation, hardness reductions) from conditional algorithmic claims.
minor comments (4)
  1. [4.3] The enhanced-state register is first described as a 2m-qutrit register but the relevant degrees of freedom are only the ℓ(r) loop labels; please clarify the physical encoding and state which registers are uncomputed before the boundary operator is applied.
  2. [7] The sentence that the twisted-unknot data 'perfectly follows a Lorentzian fit' overstates the evidence; please report the number of data points, fit parameter uncertainties, and residuals rather than relying on a visual statement.
  3. [References] Several references contain malformed identifiers or incomplete author lists, for example the arXiv identifiers '2405.2404.05998' in [42] and [70], and the citation formatting of [40] and [41]; these should be corrected before publication.
  4. [Throughout] There are inconsistencies in spelling ('Kauffmann' vs 'Kauffman') and a few garbled displayed equations (for example, the bracket notation around Eq. (2.12)); a careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 1.0 of 10

No material circularity: the algorithm's correctness chain is self-contained, efficiency rests on two explicitly flagged assumptions (Gibbs-state thermalization time and spectral gap), and the hardness and spectral-gap results reduce to external mathematics (Jones hardness, Hodge theory, Desai–Rao) rather than to the paper's own fitted values.

full rationale

This paper's three central claims — a conditional quantum algorithm for additively approximating Khovanov Betti numbers, hardness theorems for increasingly accurate approximations, and partial spectral-gap evidence — do not reduce to their own inputs. The algorithm's correctness chain is self-contained and built on external mathematics: the |r⟩|s⟩ encoding reproduces the Khovanov chain complex by direct transcription of the Section 2 definitions; B = ϑ + ϑ† satisfies Δ = B² (Section 4.4); Hodge theory (Theorem 9, citing the external Friedman result [79]) identifies ker Δ with Kh^{i,j}(K); and the SWAP test extracts β from Tr(ρ²) = 1/β with no self-reference. Efficiency is conditional on two assumptions the paper explicitly declines to prove: the Gibbs-preparation time t_therm dominates eq. (4.20), and the paper states in Section 9.3 that characterizing efficient thermalization 'is a key open question' and in Section 10 that it does 'not prove that these two conditions are satisfied in general.' No fitted parameter is renamed as a prediction: the numerical fits of Section 7 are presented, with explicit hedging, as evidence for inverse-polynomial gap scaling, while the analytic lower bounds (Corollaries 37 and 41) derive from the external Desai–Rao theorem [89] applied to graphs whose signless Laplacian is shown by direct computation (Theorem 28) to equal Δ_{0,q}. The hardness theorems (Theorems 7–8) are one-way reductions from external results [36, 5] via the graded Euler characteristic (eqs. 5.1–5.2); the direction (Betti numbers determine the Kauffman bracket) is Khovanov's external categorification theorem, not a renamed tautology. Self-citations exist ([22] base algorithm being generalized, [28] on clique-homology hardness, [69, 71] QPE tools, [56] background) but none is load-bearing: the Khovanov-specific correctness argument is re-derived in Sections 3, 4, and 9 rather than imported, and [22] is a published, peer-reviewed result whose correctness is independent of this paper. The genuine weaknesses — t_therm is not instantiated by any concrete Lindbladian, and the inverse-polynomial gap is proven only in homological degree zero with the rest resting on numerics — are completeness/correctness risks that the paper itself flags, not circularity.

Assumptions & free parameters 2 free parameters · 7 assumptions · 1 invented entities

The paper's efficiency claims rest on unproven conditions (efficient thermalization, inverse-polynomial spectral gap) and on numerical fits; the graph-theoretic bounds cover only homological degree zero and specific knots.

free parameters (2)
  • Lorentzian fit parameters for twisted unknot spectral gap = a≈10.3732, b≈-1.98306, c≈0.974362
    Fitted in Figure 6 to the gap of Δ(0,3-n)(TU_n); used as numerical evidence that the gap decays polynomially, not exponentially.
  • Density-of-states growth exponents for Gibbs cooling = ~0.19 (7 crossings), ~0.24 (8), ~0.27 (9), ~0.36 (10)
    Fitted from eigenvalue distributions in Figure 9; used to argue that cooling to inverse temperatures β=0.19,...,0.36 suffices for overlap with the kernel.
assumptions (7)
  • standard math Standard Hamiltonian simulation (Low-Chuang) and quantum phase estimation are reliable and applicable to sparse matrices.
    Used in Section 4.6 to bound the cost of implementing e^{-iBt} and phase estimation; these are accepted results in quantum computing.
  • domain assumption Khovanov homology is a well-defined knot invariant that categorifies the Jones polynomial and detects the unknot (Kronheimer-Mrowka).
    Foundation of the problem; cited from [12], [34]. The paper does not reprove these facts.
  • domain assumption The Desai-Rao bound (Theorem 24) correctly bounds the smallest eigenvalue of signless Laplacians of weighted graphs.
    Used in Section 8.1 to translate graph quantities Ψ into spectral gap bounds; taken from [89].
  • ad hoc to paper Efficient thermalization of the Khovanov Hodge Laplacian to a Gibbs state at temperature O(gap) with inverse-polynomial kernel overlap is achievable for the knots considered.
    Necessary for the algorithm's efficiency (Section 4.5, 9.3); the paper explicitly leaves it as a key open question and provides no construction or proof.
  • ad hoc to paper The spectral gap of the Khovanov Hodge Laplacian is at least inverse-polynomial in the crossing number for the relevant instances.
    Required for phase estimation to resolve the kernel (Section 4.6). Supported by numerical fits (Section 7) and analytic bounds only in homological degree zero for specific knots (Section 8); not proven in general.
  • ad hoc to paper Conjecture 20: the smallest spectral gap for the twisted unknot TU_n occurs in bidegree (0, 3-n).
    Used to extend the analytic lower bound Corollary 41 to the global minimum gap of TU_n; verified numerically only up to n=10.
  • domain assumption The Betti numbers of Khovanov homology are not exponentially large for knots up to 11 crossings (empirical observation).
    Motivates the pre-thermalization procedure (Section 7) and is required for the SWAP test to be efficient; based on database checks rather than a theorem.
invented entities (1)
  • 0-homology knot graphs G_{n,k}(K)
    purpose: Represent the Khovanov Laplacian in homological degree zero as a signless Laplacian of a weighted graph, enabling spectral gap bounds via extremal graph theory.
    A new mathematical construction introduced in Section 8.2.2; it is a definition used internally to derive bounds, with no falsifiable prediction outside the paper.

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Cite this review

Pith. "Pith review of A quantum algorithm for Khovanov homology." pith.science (2026). https://pith.science/paper/INSUBGJY

@misc{pith2026250112378,
  author       = {Pith},
  title        = {Pith review of: A quantum algorithm for Khovanov homology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/INSUBGJY}},
  note         = {Machine review of arXiv:2501.12378}
}
abstract

Khovanov homology is a topological knot invariant that categorifies the Jones polynomial, recognizes the unknot, and is conjectured to appear as an observable in $4D$ supersymmetric Yang--Mills theory. Despite its rich mathematical and physical significance, the computational complexity of Khovanov homology remains largely unknown. To address this challenge, this work initiates the study of efficient quantum algorithms for Khovanov homology. We provide simple proofs that increasingly accurate additive approximations to the ranks of Khovanov homology are DQC1-hard, BQP-hard, and #P-hard, respectively. For the first two approximation regimes, we propose a novel quantum algorithm. Our algorithm is efficient provided the corresponding Hodge Laplacian thermalizes in polynomial time and has a sufficiently large spectral gap, for which we give numerical and analytical evidence. Our approach introduces a pre-thermalization procedure that allows our quantum algorithm to succeed even if the Betti numbers of Khovanov homology are much smaller than the dimensions of the corresponding chain spaces, overcoming a limitation of prior quantum homology algorithms. We introduce novel connections between Khovanov homology and graph theory to derive analytic lower bounds on the spectral gap.

Figures

Figures reproduced from arXiv: 2501.12378 by the authors.

Figure 1
Figure 1. Three different planar diagrams for a trefoil knot. [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Left: Implementing the recursive algorithm that produces the Kauffmann bracket poly [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. An illustration of the Khovanov complex associated with the Hopf link. Each resolution [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: A trefoil knot with 𝑚 = 3 and 2𝑚 = 6 edge labels. This knot is encoded by |𝐾⟩. • In section 4.5 we describe the quantum algorithm. To our knowledge, this is the first quantum algorithm for computing Betti numbers of a concrete non-simplicial complex. Our quantum algori…
Figure 5
Figure 5. Figure 5: Two examples of resolutions |𝑟⟩ of the trefoil knot from [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]
Figure 6
Figure 6. Figure 6: A graph of the spectral gap of the twisted unknot [PITH_FULL_IMAGE:figures/full_fig_p039_6.png]
Figure 7
Figure 7. Figure 7: Numerical computations of the spectral gap of the Hodge Laplacian as a function of [PITH_FULL_IMAGE:figures/full_fig_p041_7.png]
Figure 8
Figure 8. Figure 8: Numerical computations of the spectral gap of the Hodge Laplacian as a function of [PITH_FULL_IMAGE:figures/full_fig_p041_8.png]
Figure 9
Figure 9. Figure 9: Example of the density of states for Khovanov homology. The figure displays the number [PITH_FULL_IMAGE:figures/full_fig_p043_9.png]
Figure 10
Figure 10. Figure 10: An illustration of the chain spaces of the knot 6 [PITH_FULL_IMAGE:figures/full_fig_p048_10.png]

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Pith tools

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