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arXiv:2608.20113

Counterexamples to the fractional coloring conjecture for triply efficient shadow tomography

Jędrzej Stempin, Santiago Llorens, Felix Huber

quant-phSubmitted Aug 20, 2026• Announced Aug 21, 2026• 0 citations
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Abstract

Fractional graph colorings are useful for the Shadow tomography of Pauli observables. In practice, it is desirable that any experimentally interesting set of Pauli operators has a small fractional chromatic number $χ_{f}$ for its anticommutation graph. Conjecture 13 in King, Gosset, Kothari, and Babbush [PRX Quantum 6, 010336 (2025)] states that if $B_ε(\varrho)$ is the set of Pauli observables having expectation value magnitude at least $ε$ in some given quantum state $\varrho$, then the fractional chromatic number of the anticommutation graph $G$ induced by $B_ε(\varrho)$ is $O(ε^{-2})$. In other words, it asserts that there exists a constant $C$ such that $χ_{f} \cdot ε^2 \leq C$ on all states and graphs. If the conjecture were true, it would imply that there exists a triply efficient Pauli shadow tomography algorithm for {\it any} subset $S$ of Pauli observables, provided that there is also an efficient fractional coloring algorithm for the set $B_ε$. Here we show that the conjecture is false by constructing a family of states and observables for which no finite $C$ satisfying the bound exists. We also give a more general construction relying on the commutation index or $β$ number of a graph. The key ingredient in the proofs can be seen as an instance of the amplification trick, where fractional chromatic numbers, $β$ numbers, and expectation values are amplified through lexicographic graph products.

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14 pages. Found with GPT Sol 5.6. Comments welcome!

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Fractional graph colorings are useful for the Shadow tomography of Pauli observables. In practice, it is desirable that any experimentally interesting set of Pauli operators has a small fractional chromatic number χfχ_{f}χf​ for its anticommutation graph. Conjecture 13 in King, Gosset, Kothari, and Babbush [PRX Quantum 6, 010336 (2025)] states that if Bε(ϱ)B_ε(\varrho)Bε​(ϱ) is the set of Pauli observables having expectation value magnitude at least εεε in some given quantum state ϱ\varrhoϱ, then the fractional chromatic number of the anticommutation graph GGG induced by Bε(ϱ)B_ε(\varrho)Bε​(ϱ) is O(ε−2)O(ε^{-2})O(ε−2). In other words, it asserts that there exists a constant CCC such that χf⋅ε2≤Cχ_{f} \cdot ε^2 \leq Cχf​⋅ε2≤C on all states and graphs. If the conjecture were true, it would imply that there exists a triply efficient Pauli shadow tomography algorithm for {\it any} subset SSS of Pauli observables, provided that there is also an efficient fractional coloring algorithm for the set BεB_εBε​. Here we show that the conjecture is false by constructing a family of states and observables for which no finite CCC satisfying the bound exists. We also give a more general construction relying on the commutation index or βββ number of a graph. The key ingredient in the proofs can be seen as an instance of the amplification trick, where fractional chromatic numbers, βββ numbers, and expectation values are amplified through lexicographic graph products.
Godsil, Roberson, Šámal and Severini conjectured that the quantum chromatic number of the categorical product of two graphs equals the minimum of the quantum chromatic numbers of the factors. We disprove this conjecture: we construct explicit finite graphs G,HG,HG,H with χ(G×H)≤1538<1539=min⁡(χq(G),χq(H)). χ(G\times H) \leq 1538 < 1539 = \min(χ_q(G),χ_q(H)).χ(G×H)≤1538<1539=min(χq​(G),χq​(H)).The graphs are obtained from Zhu's counterexample to Hedetniemi's conjecture by using a base graph for which the Lovász theta number of the complement, and not only the fractional chromatic number, is large. The lower bound for the first factor is the theta bound. For the second factor we adapt Zhu's argument to projections that do not commute: the step that fixes the colors of a clique is replaced by identities between operators. Both lower bounds hold for colorings by projections in an arbitrary nonzero unital C∗C^*C∗-algebra. Hence the conjecture also fails for the spatial, approximate, commuting-operator and C∗C^*C∗-algebraic variants of the quantum chromatic number. We also give smaller counterexamples certified by exact integer data. The graph constructions, the certificates and the counterexample statements in the projective formulation are formalized in Lean~4.
Given copies of a quantum state ρ, a shadow tomography protocol aims to learn all expectation values from a fixed set of observables, to within a given precision ε. We say that a shadow tomography protocol is triply efficient if it is sample- and time-efficient, and only employs measurements that entangle a constant number of copies of ρ at a time. The classical shadows protocol based on random single-copy measurements is triply efficient for the set of local Pauli observables. This and other protocols based on random singlecopy Clifford measurements can be understood as arising from fractional colorings of a graph G that encodes the commutation structure of the set of observables. Here we describe a framework for two-copy shadow tomography that uses an initial round of Bell measurements to reduce to a fractional coloring problem in an induced subgraph of G with bounded clique number. This coloring problem can be addressed using techniques from graph theory known as chi-boundedness. Using this framework we give the first triply efficient shadow tomography scheme for the set of local fermionic observables, which arise in a broad class of interacting fermionic systems in physics and chemistry. We also give a triply efficient scheme for the set of all n-qubit Pauli observables. Our protocols for these tasks use two-copy measurements, which is necessary: sample- efficient schemes are provably impossible using only single-copy measurements. Finally, we give a shadow tomography protocol that compresses an n-qubit quantum state into a poly(n )-sized classical representation, from which one can extract the expected value of any of the 4n Pauli observables in poly(n ) time, up to a small constant error.
We give a shadow tomography algorithm that uses O(log⁡(m)log⁡(m/δ)/ε2)O(\log(m) \log(m/δ) / \varepsilon^2)O(log(m)log(m/δ)/ε2) copies of the input state. Specifically, given a quantum state ρρρ and observables O1,…,OmO_1,\ldots,O_mO1​,…,Om​, the algorithm outputs estimates of tr(Oiρ)\mathrm{tr}(O_iρ)tr(Oi​ρ) for all i∈[m]i\in[m]i∈[m] that are all accurate to within ±ε\pm\varepsilon±ε, except with probability at most δδδ. The key idea is to work in the average-case setting: for a random state drawn from a known prior, a pretty good measurement on O(log⁡(m)/ε2)O(\log(m)/\varepsilon^2)O(log(m)/ε2) copies gives a constant-error guarantee on all but a small fraction of the prior. Boosting this guarantee gives an average-case shadow tomography algorithm, and the minimax principle then (non-constructively) implies the existence of a measurement with the same worst-case guarantee. We also identify an obstruction to further improvement with this approach. While pretty good measurements amplify success over the randomness of the prior, they do not amplify success over the randomness of measurement. This issue appears to be inherent, and gives rise to the remaining log⁡(m)\log(m)log(m) factor separating the complexity of the quantum problem from its classical counterpart.
Shadow tomography addresses the task of efficiently predicting many expectation values of an unknown quantum state from randomized measurements on comparatively few copies. Existing analyses promise large scaling advantages, but the optimal strategies realizing these guarantees are not always known, and the required measurements are potentially challenging to implement on current hardware. We address this gap for Pauli observables by computing optimal sample-complexity parameters and constructing optimal measurement strategies under realistic resource constraints. We focus on memoryless protocols, where each copy is measured only once, and on measurements with bounded interaction range. Our approach reduces the problem to the analysis of graph parameters of the frustration graph encoding the Pauli anticommutation relations. We provide efficient numerical methods for the general case and analytically prove that Clifford measurements are optimal in many situations. This includes all perfect graphs, all single-qubit, all two-qubit measurement scenarios, and more. Applied to Hamiltonian energy estimation, our framework yields constructive strategies and improved variance bounds for molecular benchmarks.
Shadow tomography protocols have recently emerged as powerful tools for efficient quantum state learning, aiming to reconstruct expectation values of observables with considerably fewer resources than traditional quantum state tomography. For the particular case of estimating Pauli observables, entangling two-copy measurement schemes can offer an exponential improvement in sample complexity over any single-copy strategy conceivable [1, Huang, Kueng, Preskill, PRL (2021)]. A recent refinement of these ideas by King et al. [2, King, Gosset, Kothari, Babbush, SODA (2025)] does not only achieve polynomial sample complexity, but also maintains reasonable computational demands and utilizes joint measurements on only a small constant number of state copies. This 'triple efficiency' is achievable for any subset ofnnn-qubit Pauli observables, whereas single-copy strategies can only be efficient if the Pauli observables in question have advantageous structure. In this work, we complement existing theoretical performance guarantees with the empirical evaluation of triply efficient shadow tomography using classical, noise-free simulations. Our findings indicate that the empirical sample complexity aligns closely with theoretical predictions for stabilizer states and, notably, demonstrates slightly improved scaling for random Gibbs states compared to established theoretical bounds. In addition, we improve a central subroutine in the triply-efficient shadow protocol by leveraging insights from a refined quantum and quantum-inspired convex optimization algorithm [3, Henze et al. arXiv:2502.15426 (2025)] [Henze et al. arXiv:2502.15426 (2025)]. To summarize, our empirical sample complexity studies of triply efficient shadow tomography not only confirm existing theoretical scaling behavior, but also showcase that the actual constants involved are comparatively benign. Hence, this protocol has the potential to also be very sample-efficient in practice.