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Tail-Resolved Schmidt-Rank Width of Weighted Phase Networks: Noncommuting Activation Limits, Gauge-Stable Perturbations, and Approximate Tree Complexity

Byoungwoo Lee

Published Aug 9, 2026• 0 citations
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Abstract

This work develops a finite-resolution counterpart of Schmidt-rank width for weighted phase networks and uses it to distinguish algebraic Schmidt-rank activation from spectrally resolved entanglement complexity. For a bipartition $A|A^c$ with Schmidt coefficients $s_1 \ge s_2 \ge \cdots$, define the discarded Schmidt-tail mass $$\tau_r^A(\psi)=\left(\sum_{j>r}s_j^2\right)^{1/2},$$ and the corresponding $\epsilon$-resolved Schmidt rank $$R_\epsilon^A(\psi)=\min\left\{r:\tau_r^A(\psi)\le\epsilon\right\}.$$ The global tail-resolved Schmidt-rank width is then $$\chi_{\mathrm{res}}^{(\epsilon)}(\psi)=\min_T\max_{e\in E(T)}\log_2R_\epsilon^{A_e}(\psi),$$ where $T$ ranges over subcubic decomposition trees and $A_e|A_e^c$ is the bipartition induced by deleting the tree edge $e$. At zero resolution, $\epsilon=0$, this quantity reduces to ordinary Schmidt-rank width. For graph states, the flat cut Schmidt spectrum gives an exact calibration: $$\chi_{\mathrm{res}}^{(\epsilon)}(|G\rangle)=\log_2\left\lceil(1-\epsilon^2)\,2^{\operatorname{rw}(G)}\right\rceil,\qquad0\le\epsilon<1,$$ where $\operatorname{rw}(G)$ is the graph rank-width. Consequently, graph families with linear rank-width retain linear tail-resolved Schmidt-rank width at every fixed finite resolution $\epsilon<1$. The paper next establishes a gauge-stable perturbation theory. After optimization over cut-local diagonal phases, Schmidt truncation tails satisfy the Lipschitz estimate $$\left|\tau_r^A(\psi)-\tau_r^A(\phi)\right|\led_{\mathrm{diag}}^A(\psi,\phi),$$ which yields a global resolution-shift sandwich for $\chi_{\mathrm{res}}^{(\epsilon)}$. For balanced pairwise weighted phase networks with crossing phase errors $\eta_e$, $$d_{\mathrm{diag}}^A\le\frac14\left\|\eta_{E(A,A^c)}\right\|_2.$$ If a fixed coupling template $W$ is implemented with coupling error $\Delta W$ and timing error $\Delta t$, then $$\eta=t\,\Delta W+\Delta t\,W+\Delta t\,\Delta W,$$ giving an explicit certificate that converts finite calibration and timing uncertainty into a guaranteed resolved-width bound. The tail-resolved width also controls approximate subcubic tree-tensor-network complexity. If $D_{\mathrm{TTN}}^{(\epsilon)}$ denotes the minimum possible maximal bond dimension of a subcubic TTN approximating the state within Hilbert-space norm error $\epsilon$, then $$2^{\chi_{\mathrm{res}}^{(\epsilon)}}\leD_{\mathrm{TTN}}^{(\epsilon)}\le2^{\chi_{\mathrm{res}}^{\left(\epsilon/\sqrt{2n-3}\right)}}.$$ The central application concerns the Clifford point of the complete weighted phase network. Consider $$|\Psi_V(\lambda)\rangle=\exp\left[i\sum_e\left(\pi+\lambda V_e\right)n_u n_v\right]|+\rangle^{\otimes n}.$$ For every fixed resolution $$0<\epsilon<\frac{1}{\sqrt{2}},$$ the tail-resolved width remains equal to $1$ for all sufficiently small $|\lambda|$, independently of the perturbation direction $V$. In contrast, at exact zero resolution an arbitrarily small generic nonuniform perturbation activates the maximal complete-support Schmidt-rank width $$\left\lceil\frac{n}{3}\right\rceil,$$ whereas a uniform perturbation activates only $$\log_2\left(\left\lceil\frac{n}{3}\right\rceil+1\right).$$ Thus the zero-resolution and zero-perturbation limits do not commute. For an open, dense, full-measure class of nonuniform perturbation directions, $$\lim_{\epsilon\downarrow0}\lim_{\lambda\to0}\chi_{\mathrm{res}}^{(\epsilon)}=1,$$ while $$\lim_{\lambda\to0,\ \lambda\neq0}\lim_{\epsilon\downarrow0}\chi_{\mathrm{res}}^{(\epsilon)}=\left\lceil\frac{n}{3}\right\rceil.$$ This establishes a sharp distinction between algebraic Schmidt activation and finite-resolution activation: infinitesimal symmetry breaking can create maximal exact Schmidt-rank width while remaining invisible at any fixed finite spectral resolution sufficiently close to the Clifford point. The results connect weighted phase and fixed $ZZ$/Ising networks, graph width parameters, Schmidt-spectrum stability, and approximate tensor-network complexity. The global phase-error certificate is intentionally conservative: it does not by itself establish thermodynamic robustness against a size-independent coherent error on every edge.

DOI
10.5281/zenodo.21862358

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This work develops a finite-resolution counterpart of Schmidt-rank width for weighted phase networks and uses it to distinguish algebraic Schmidt-rank activation from spectrally resolved entanglement complexity. For a bipartition A∣AcA|A^cA∣Ac with Schmidt coefficients s1≥s2≥⋯s_1 \ge s_2 \ge \cdotss1​≥s2​≥⋯, define the discarded Schmidt-tail mass τrA(ψ)=(∑j>rsj2)1/2,\tau_r^A(\psi)=\left(\sum_{j>r}s_j^2\right)^{1/2},τrA​(ψ)=(∑j>r​sj2​)1/2, and the corresponding ϵ\epsilonϵ-resolved Schmidt rank RϵA(ψ)=min⁡{r:τrA(ψ)≤ϵ}.R_\epsilon^A(\psi)=\min\left\{r:\tau_r^A(\psi)\le\epsilon\right\}.RϵA​(ψ)=min{r:τrA​(ψ)≤ϵ}. The global tail-resolved Schmidt-rank width is then χres(ϵ)(ψ)=min⁡Tmax⁡e∈E(T)log⁡2RϵAe(ψ),\chi_{\mathrm{res}}^{(\epsilon)}(\psi)=\min_T\max_{e\in E(T)}\log_2R_\epsilon^{A_e}(\psi),χres(ϵ)​(ψ)=minT​maxe∈E(T)​log2​RϵAe​​(ψ), where TTT ranges over subcubic decomposition trees and Ae∣AecA_e|A_e^cAe​∣Aec​ is the bipartition induced by deleting the tree edge eee. At zero resolution, ϵ=0\epsilon=0ϵ=0, this quantity reduces to ordinary Schmidt-rank width. For graph states, the flat cut Schmidt spectrum gives an exact calibration: χres(ϵ)(∣G⟩)=log⁡2⌈(1−ϵ2) 2rw⁡(G)⌉,0≤ϵ<1,\chi_{\mathrm{res}}^{(\epsilon)}(|G\rangle)=\log_2\left\lceil(1-\epsilon^2)\,2^{\operatorname{rw}(G)}\right\rceil,\qquad0\le\epsilon<1,χres(ϵ)​(∣G⟩)=log2​⌈(1−ϵ2)2rw(G)⌉,0≤ϵ<1, where rw⁡(G)\operatorname{rw}(G)rw(G) is the graph rank-width. Consequently, graph families with linear rank-width retain linear tail-resolved Schmidt-rank width at every fixed finite resolution ϵ<1\epsilon<1ϵ<1. The paper next establishes a gauge-stable perturbation theory. After optimization over cut-local diagonal phases, Schmidt truncation tails satisfy the Lipschitz estimate ∣τrA(ψ)−τrA(ϕ)∣\leddiagA(ψ,ϕ),\left|\tau_r^A(\psi)-\tau_r^A(\phi)\right|\led_{\mathrm{diag}}^A(\psi,\phi),​τrA​(ψ)−τrA​(ϕ)​\leddiagA​(ψ,ϕ), which yields a global resolution-shift sandwich for χres(ϵ)\chi_{\mathrm{res}}^{(\epsilon)}χres(ϵ)​. For balanced pairwise weighted phase networks with crossing phase errors ηe\eta_eηe​, ddiagA≤14∥ηE(A,Ac)∥2.d_{\mathrm{diag}}^A\le\frac14\left\|\eta_{E(A,A^c)}\right\|_2.ddiagA​≤41​​ηE(A,Ac)​​2​. If a fixed coupling template WWW is implemented with coupling error ΔW\Delta WΔW and timing error Δt\Delta tΔt, then η=t ΔW+Δt W+Δt ΔW,\eta=t\,\Delta W+\Delta t\,W+\Delta t\,\Delta W,η=tΔW+ΔtW+ΔtΔW, giving an explicit certificate that converts finite calibration and timing uncertainty into a guaranteed resolved-width bound. The tail-resolved width also controls approximate subcubic tree-tensor-network complexity. If DTTN(ϵ)D_{\mathrm{TTN}}^{(\epsilon)}DTTN(ϵ)​ denotes the minimum possible maximal bond dimension of a subcubic TTN approximating the state within Hilbert-space norm error ϵ\epsilonϵ, then 2χres(ϵ)\leDTTN(ϵ)≤2χres(ϵ/2n−3).2^{\chi_{\mathrm{res}}^{(\epsilon)}}\leD_{\mathrm{TTN}}^{(\epsilon)}\le2^{\chi_{\mathrm{res}}^{\left(\epsilon/\sqrt{2n-3}\right)}}.2χres(ϵ)​\leDTTN(ϵ)​≤2χres(ϵ/2n−3​)​. The central application concerns the Clifford point of the complete weighted phase network. Consider ∣ΨV(λ)⟩=exp⁡[i∑e(π+λVe)nunv]∣+⟩⊗n.|\Psi_V(\lambda)\rangle=\exp\left[i\sum_e\left(\pi+\lambda V_e\right)n_u n_v\right]|+\rangle^{\otimes n}.∣ΨV​(λ)⟩=exp[i∑e​(π+λVe​)nu​nv​]∣+⟩⊗n. For every fixed resolution 0<ϵ<12,0<\epsilon<\frac{1}{\sqrt{2}},0<ϵ<2​1​, the tail-resolved width remains equal to 111 for all sufficiently small ∣λ∣|\lambda|∣λ∣, independently of the perturbation direction VVV. In contrast, at exact zero resolution an arbitrarily small generic nonuniform perturbation activates the maximal complete-support Schmidt-rank width ⌈n3⌉,\left\lceil\frac{n}{3}\right\rceil,⌈3n​⌉, whereas a uniform perturbation activates only log⁡2(⌈n3⌉+1).\log_2\left(\left\lceil\frac{n}{3}\right\rceil+1\right).log2​(⌈3n​⌉+1). Thus the zero-resolution and zero-perturbation limits do not commute. For an open, dense, full-measure class of nonuniform perturbation directions, lim⁡ϵ↓0lim⁡λ→0χres(ϵ)=1,\lim_{\epsilon\downarrow0}\lim_{\lambda\to0}\chi_{\mathrm{res}}^{(\epsilon)}=1,limϵ↓0​limλ→0​χres(ϵ)​=1, while lim⁡λ→0, λ≠0lim⁡ϵ↓0χres(ϵ)=⌈n3⌉.\lim_{\lambda\to0,\ \lambda\neq0}\lim_{\epsilon\downarrow0}\chi_{\mathrm{res}}^{(\epsilon)}=\left\lceil\frac{n}{3}\right\rceil.limλ→0, λ=0​limϵ↓0​χres(ϵ)​=⌈3n​⌉. This establishes a sharp distinction between algebraic Schmidt activation and finite-resolution activation: infinitesimal symmetry breaking can create maximal exact Schmidt-rank width while remaining invisible at any fixed finite spectral resolution sufficiently close to the Clifford point. The results connect weighted phase and fixed ZZZZZZ/Ising networks, graph width parameters, Schmidt-spectrum stability, and approximate tensor-network complexity. The global phase-error certificate is intentionally conservative: it does not by itself establish thermodynamic robustness against a size-independent coherent error on every edge.
This work studies the global multipartite entanglement width generated by fixed pairwise weighted-phase networks acting on full-support product inputs. It develops the global-width consequences of the fixed-cut weighted-phase theory established in the preceding study, Schmidt-Rank Capacity and Spectral Robustness in Fixed Weighted Phase Networks; the corresponding public research record is available at DOI 10.5281/zenodo.21858429. For a graph G=(V,E)G=(V,E)G=(V,E) with a fixed real edge-weight template W=(We)e∈EW=(W_e)_{e\in E}W=(We​)e∈E​, we consider the commuting weighted-phase evolution UW(t)=exp⁡(it∑e={u,v}∈EWenunv),U_W(t) = \exp\left( i t \sum_{e=\{u,v\}\in E} W_e n_u n_v \right),UW​(t)=exp(it∑e={u,v}∈E​We​nu​nv​), with a single scalar evolution parameter ttt. For every bipartition A∣AcA\vert{}A^cA∣Ac, let νG(A)\nu_G(A)νG​(A) be the maximum matching number of the crossing graph G[A,Ac]G[A,A^c]G[A,Ac]. The main result globalizes the fixed-cut matching law. There exists an open, dense, full-Lebesgue-measure class of fixed templates WWW such that, outside one locally finite exceptional set of evolution times, the same scalar ttt simultaneously saturates the matching capacity of every nontrivial bipartition: log⁡2SR⁡A∣Ac(∣ΨW(t)⟩)=νG(A)\log_2 \operatorname{SR}_{A\vert{}A^c}\bigl(\vert{}\Psi_W(t)\rangle\bigr) = \nu_G(A)log2​SRA∣Ac​(∣ΨW​(t)⟩)=νG​(A) for all nontrivial cuts A∣AcA\vert{}A^cA∣Ac. Consequently, χwd(∣ΨW(t)⟩)=mmw⁡(G)\chi_{\mathrm{wd}}\bigl(\vert{}\Psi_W(t)\rangle\bigr) = \operatorname{mmw}(G)χwd​(∣ΨW​(t)⟩)=mmw(G) for generic fixed weighted-phase networks, where χwd\chi_{\mathrm{wd}}χwd​ is Schmidt-rank width and mmw⁡(G)\operatorname{mmw}(G)mmw(G) is the maximum matching width of the interaction graph. This provides a continuous weighted counterpart to the standard graph-state relation between Schmidt-rank width and rank-width. For complete support KnK_nKn​, mmw⁡(Kn)=⌈n3⌉,\operatorname{mmw}(K_n) = \left\lceil\frac{n}{3}\right\rceil,mmw(Kn​)=⌈3n​⌉, so almost every nonuniform fixed template attains χwd=⌈n3⌉,\chi_{\mathrm{wd}} = \left\lceil\frac{n}{3}\right\rceil,χwd​=⌈3n​⌉, which is the largest Schmidt-rank width permitted for any nnn-qubit pure state. Uniform complete-graph templates behave very differently. Their exact width is χwdunif=log⁡2min⁡{⌈n3⌉+1, ord⁡(eiθ)},\chi_{\mathrm{wd}}^{\mathrm{unif}} = \log_2 \min\left\{ \left\lceil\frac{n}{3}\right\rceil+1,\, \operatorname{ord}(e^{i\theta}) \right\},χwdunif​=log2​min{⌈3n​⌉+1,ord(eiθ)}, with ord⁡(eiθ)=∞\operatorname{ord}(e^{i\theta})=\inftyord(eiθ)=∞ allowed. Thus the same complete interaction support and the same one-dimensional run-time control exhibit a sharp template hierarchy: Clifford⟶χwd=1,\text{Clifford} \quad \longrightarrow \quad \chi_{\mathrm{wd}}=1,Clifford⟶χwd​=1, generic uniform⟶χwd=Θ(log⁡n),\text{generic uniform} \quad \longrightarrow \quad \chi_{\mathrm{wd}}=\Theta(\log n),generic uniform⟶χwd​=Θ(logn), generic nonuniform⟶χwd=Θ(n).\text{generic nonuniform} \quad \longrightarrow \quad \chi_{\mathrm{wd}}=\Theta(n).generic nonuniform⟶χwd​=Θ(n). This separation is not restricted to dense complete graphs. Since rw⁡(G)≤mmw⁡(G),\operatorname{rw}(G) \le \operatorname{mmw}(G),rw(G)≤mmw(G), graph families with linear rank-width transfer directly to fixed weighted-phase families with linear generic Schmidt-rank width. In particular, known deterministic bounded-degree high-rank-width constructions yield sparse O(n)O(n)O(n)-coupler networks with generic Schmidt-rank width Θ(n)\Theta(n)Θ(n). The width separation has an exact tensor-network consequence. For generic uniform complete templates, an explicit subcubic tree tensor network exists with optimal maximal bond dimension Dunif=⌈n3⌉+1=Θ(n),D_{\mathrm{unif}} = \left\lceil\frac{n}{3}\right\rceil+1 = \Theta(n),Dunif​=⌈3n​⌉+1=Θ(n), whereas generic nonuniform complete templates require Dgen=2⌈n/3⌉=2Θ(n).D_{\mathrm{gen}} = 2^{\left\lceil n/3\right\rceil} = 2^{\Theta(n)}.Dgen​=2⌈n/3⌉=2Θ(n). Thus fixed template diversity, rather than run-time control dimension alone, can change exact global tree-tensor complexity from polynomial to exponential while the interaction support and the number of run-time control parameters remain unchanged. The computational interpretation is deliberately limited. Superlogarithmic Schmidt-rank width lies outside the logarithmic-width hypothesis underlying the standard low-width exact-TTN simulation guarantee, and linear width is a particularly strong realization of this separation. However, large or linear Schmidt-rank width is neither asserted to be necessary nor sufficient for universal measurement-based quantum computation, hard classical simulation in general, fault tolerance, or quantum advantage. Version v0.2r2 is the frozen theorem-core release. It preserves the mathematical results of v0.2r1 while finalizing the claim boundary concerning tensor-network simulation, measurement-based quantum computation, and computational advantage.
High-dimensional bipartite entanglement depends on how probability is distributed across the Schmidt spectrum, whereas a single reference-state fidelity resolves only one spectral scale. We develop multiscale Schmidt-spectrum bounds that connect geometric measures with quantitative Schmidt-number witnesses. For any partition of a pure-state Schmidt spectrum, several nested Vidal tails determine block masses. We derive the sharp upper boundary of the associated normalized nuclear-norm coordinate and show that equality holds if and only if the spectrum is uniform within each block. Refining the partition gives a monotone hierarchy of tighter bounds whenever the added tail data distinguish unequal block means. We also solve a relaxed weighted multiscale optimization globally: one scalar parameter specifies its unique full-support optimizer and explicit value on the nontrivial branch. Using the established single-tail fidelity--resource curve as a baseline, we obtain exact-fidelity equality refinements, convex-roof lower bounds for mixed states, and quantitative calibrations of Schmidt-number witnesses. A higher-tail relation further bounds convex-roof extended negativity in terms of Vidal tails and identifies the pure-state equality spectra. Leakage-aware and joint-confidence formulations state how these bounds can be used with incomplete data. Multistep tails require block-resolved or independently certified spectral information; they are not determined by one projector expectation.
This work studies bipartite Schmidt rank in continuously weighted diagonal phase networks, with particular emphasis on the distinction between interaction support, parameter freedom, and higher-order algebraic dependencies. For a fixed bipartition and a full-support product input, the Schmidt rank is reduced exactly to the complex rank of a cross-cut phase kernel. In the independently weighted pairwise qubit sector, the main result gives an exact generic matching law: rgen=2ν(G×),r_{\mathrm{gen}} = 2^{\nu(G_\times)},rgen​=2ν(G×​), where ν(G×)\nu(G_\times)ν(G×​) is the maximum matching number of the crossing bipartite interaction graph. The symbolic rank, maximum attainable rank, and almost-everywhere physical rank coincide. The paper then shows that the interaction support alone does not determine the generic Schmidt rank once the parameter family is constrained. For a uniformly weighted complete bipartite network Kp,qK_{p,q}Kp,q​, the exact rank is rank⁡Kθ=min⁡{p+1, q+1, ord⁡(eiθ)},\operatorname{rank} K_\theta = \min\left\{ p+1,\, q+1,\, \operatorname{ord}(e^{i\theta}) \right\},rankKθ​=min{p+1,q+1,ord(eiθ)}, with infinite multiplicative order interpreted in the natural way. The K2,2K_{2,2}K2,2​ example therefore exhibits a rank stratification (4→3→24 \to 3 \to 24→3→2) when passing from independently weighted phases to a uniform one-parameter family and then to the Clifford point. For genuine higher-order phase interactions, the cross-cut kernel is factorized through a union-coefficient matrix. This reveals a different obstruction: even with independent hyperedge weights, repeated algebraic use of the same phase parameters can force the symbolic rank below the structural matching rank. The paper proves that no such obstruction occurs with at most two crossing hyperedges and gives an explicit three-hyperedge example with rstr=4,rsym=3.r_{\mathrm{str}}=4, \qquad r_{\mathrm{sym}}=3.rstr​=4,rsym​=3. Exact finite enumeration further certifies, within the stated simple crossing-hypergraph model, that five total cut vertices are the smallest size at which this structural-versus-symbolic rank mismatch occurs. The results distinguish three mechanisms controlling cross-cut Schmidt-rank capacity: pairwise interaction combinatorics, external parameter tying, and intrinsic higher-order union-algebraic dependencies. The manuscript is accompanied by exact verification code and reproducibility checks for the finite classification statements. Version: v0.2r3
Exponentially small spectral gaps are the central obstacle to adiabatic quantum annealing, yet whether such bottlenecks demand correspondingly complex entanglement remains unclear. We show that gap scaling and entanglement complexity need not be linked. Conventional first-order transitions combine exponentially small gaps with bounded O(1)O(1)O(1) bipartite Schmidt rank, even though the critical states retain global multipartite correlations, while continuous transitions combine polynomially closing gaps with logarithmically growing entanglement. We show that each pairing can also be inverted: in a known first-order transition in a staggered-field antiferromagnetic Ising chain the gap closes polynomially despite bounded Schmidt rank, while we construct a mirror-twin manifold model whose gap closes exponentially even as its entanglement entropy grows logarithmically with system size. We derive bounds connecting Schmidt rank to Hamming-space geometry, and identify manifold density and connectivity as the ingredients that determine which combination occurs. These results show that entanglement complexity alone cannot characterize annealing difficulty, and point toward a broader search for quantum resources that capture the geometry of state evolution in Hilbert space.
Graph states provide a bridge between graph theory and multipartite quantumentanglement, with cut-rank, rank-width, and related parameters playing a centralrole in both structural graph theory and the characterization of stabilizer states.Building on this interplay, we introduce a new scalar invariant—the S M NazmuzSakib entanglement rank entropy—defined as the Shannon entropy of the cut-rankdistribution over vertex subsets of fixed cardinality. For a simple graph G = (V, E)of order n, and k ∈ {1, . . . , n − 1}, we consider the random variable ρG(X) whereX ⊆ V is a uniformly random k-subset, and we define the Sakib entanglementrank entropy H(k)ρ (G) as the Shannon entropy (in bits) of its distribution.Forgraph states, where ρG(X) coincides with the Schmidt rank exponent across thebipartition (X, V \ X), H(k)ρ (G) is precisely the Shannon entropy of the bipartiteentanglement (in ebits) seen by random size-k bipartitions.We formally define the S M Nazmuz Sakib entanglement rank entropy, provethat it is invariant under local complementation and hence a local-Clifford invariantfor graph states, and study a balanced version Hbalρ (G) = H(n/2)ρ(G) (for even n).We then perform a data-based analysis on explicit graph families (paths, cycles,stars, complete graphs, and 2 × m grid graphs) of size up to n = 12, computingHbalρ (G), the average cut-rank, and the full cut-rank distributions. The resultingdataset shows that, while complete and star graphs have vanishing Sakib entangle-ment rank entropy, more “distributed” structures such as paths, cycles, and gridsexhibit increasing entropy with system size, despite similar ranges of cut-rank.Motivated by these observations, we formulate the S M Nazmuz Sakib entan-glement heterogeneity hypothesis, proposing that, at fixed rank-width and averagecut-rank, a larger Sakib entanglement rank entropy correlates with higher classicalsimulation complexity of the corresponding graph states.We outline how pub-licly available graph-state orbit datasets and average cut-rank techniques can becombined with the proposed invariant to further test this hypothesis.
We study the computational complexity of estimating the normalized trace 2−nTr[f(A)]2^{-n}\mathrm{Tr}[f(A)]2−nTr[f(A)] for a log-local Hamiltonian AAA acting on nnn qubits. This problem arises naturally in the DQC1 model, yet its complexity is only understood for a limited class of functions f(x)f(x)f(x). We show that if f(x)f(x)f(x) is a continuous function with approximate degree Ω(poly(n))Ω(\mathrm{poly}(n))Ω(poly(n)), then estimating 2−nTr[f(A)]2^{-n}\mathrm{Tr}[f(A)]2−nTr[f(A)] up to constant additive error is DQC1-complete, under a technical condition on the polynomial approximation error of f(x)f(x)f(x). This condition holds for a broad class of functions, including exponentials, trigonometric functions, logarithms, and inverse-type functions. We further prove that when AAA is sparse, the classical query complexity of this problem is exponential in the approximate degree. Together, these results identify the approximate degree as the key parameter governing the complexity of normalized trace estimation: it characterizes both the quantum complexity (via efficient DQC1 algorithms) and the classical hardness, yielding an exponential quantum-classical separation. Our proof develops a unified framework that cleanly combines circuit-to-Hamiltonian constructions, periodic Jacobi operators, and tools from polynomial approximation theory, including the Chebyshev equioscillation theorem.