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arxiv: 2606.07425 · v1 · pith:N5IWSZMQnew · submitted 2026-06-05 · 🪐 quant-ph · cs.CC· math.CO

Tomography of quantum states with bounded extent

Srinivasan Arunachalam , Arkopal Dutt This is my paper

Pith reviewed 2026-06-27 21:35 UTC · model grok-4.3

classification 🪐 quant-ph cs.CCmath.CO
keywords quantum tomographybounded extentstabilizer statesagnostic learningquantum state learningregularity lemmalinear combinations
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The pith

A weak agnostic learner for a family of quantum states can be boosted to a tomography algorithm for states with bounded extent relative to that family.

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

The paper establishes a general framework showing that if a class C of n-qubit states is succinctly representable and has a weak agnostic learner, then tomography is possible for any unknown state that decomposes as a linear combination of states from C with bounded l1-norm of coefficients. This reduction is black-box and turns the learner directly into the tomography protocol. As a result, for the class of stabilizer states, efficient tomography is achieved for states with small stabilizer extent. The work also gives an efficient way to extract the C-explainable part of an arbitrary state via a weak regularity lemma. This demonstrates that learnability of a base class extends to its low-complexity spans.

Core claim

If C is succinctly representable and admits a weak agnostic learner, a black-box reduction produces a tomography protocol for states with bounded extent with respect to C. For stabilizer states this yields a protocol running in time poly(n, (ξ/ε)^log(ξ/ε)) for extent ξ and accuracy ε, which improves to poly(n, ξ, 1/ε) under the algorithmic polynomial Freiman-Ruzsa conjecture. The same techniques give an algorithmic decomposition of arbitrary states that efficiently learns the component explainable by C.

What carries the argument

The black-box reduction that boosts a weak agnostic learner for C into a tomography algorithm for bounded-extent combinations of states from C.

If this is right

  • For stabilizer states, tomography up to trace distance ε for extent ξ states runs in poly(n, (ξ/ε)^log(ξ/ε)) time.
  • Under the polynomial Freiman-Ruzsa conjecture the time improves to poly(n, ξ, 1/ε).
  • Arbitrary states admit an efficient decomposition that recovers the part spanned by C.
  • Agnostic learning of C implies learnability of the low l1-norm span of C.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The framework may extend to other structured classes beyond stabilizers if they satisfy the two conditions.
  • The conceptual link suggests that classical learning theory ideas like boosting and regularity lemmas transfer to quantum state learning.
  • Testing whether an unknown state has bounded extent with respect to C could be a natural follow-up question.

Load-bearing premise

The family C must be succinctly representable and admit a weak agnostic learner.

What would settle it

A specific family C that is succinctly representable and has a weak agnostic learner, yet for which no efficient tomography protocol exists for states of bounded extent with respect to C.

Figures

Figures reproduced from arXiv: 2606.07425 by Arkopal Dutt, Srinivasan Arunachalam.

Figure 1
Figure 1. Figure 1: Circuit for preparing the state |βe⟩. implements A−1Φ † up to some subnormalization. Let this normalization be ν > 0 to be determined and let us evaluate A−1Φ † − νW: A −1Φ † − ν(⟨0 a | ⊗ Im ⊗ ⟨0 n |)(UA−1 ⊗ In)(Ia ⊗ U † Φ )(Ia ⊗ H⊗m ⊗ In)(|0 a ⟩ |0 m⟩ ⊗ In) =A −1Φ † − ν(Im ⊗ ⟨0 n |) h(⟨0 a | ⊗ Im)UA−1 (|0 a ⟩ ⊗ Im) i ⊗ In  U † Φ (|+⟩ m ⊗ In) =A −1Φ † − ν(Im ⊗ ⟨0 n |)  1 α (A −1 + E) ⊗ In  U † Φ (|+⟩ m… view at source ↗
read the original abstract

We give a general framework for tomography of states that have bounded-extent with respect to a structured class of states. Let $\textsf{C}$ be a family of $n$-qubit states such that: $(i)$ $\textsf{C}$ is succinctly representable and $(ii)$ there is a weak agnostic learner of $\textsf{C}$. We give a tomography protocol for an unknown state $|\psi\rangle$ that is promised to admit a decomposition of the form $|\psi\rangle = \sum_i c_i |\phi_i\rangle$, where $|\phi_i\rangle \in \textsf{C}$ with bounded $\ell_1$-norm of the coefficients (which we call extent). Our main contribution is to show that a weak agnostic learner for $\textsf{C}$ can be boosted into a tomography algorithm for states with bounded extent with respect to $\textsf{C}$. Our reduction is black-box and applies broadly across model classes. As an application, when $\textsf{C}$ is the class of stabilizer states, we obtain tomography algorithms for states with stabilizer extent $\xi$ up to trace distance $\varepsilon$, in time $\textsf{poly}(n,(\xi/\varepsilon)^{\log(\xi/\varepsilon)})$, which is improvable to $ \textsf{poly}(n,\xi,1/\varepsilon)$ assuming the algorithmic polynomial Freiman-Ruzsa conjecture in the high-doubling regime. When the unknown state $|\psi\rangle$ is arbitrary, we give an algorithmic decomposition result in the spirit of a weak regularity lemma for quantum states with respect to $\textsf{C}$ and show that the structure in $|\psi\rangle$ that is explainable by $\textsf{C}$ can be efficiently learned. Our main conceptual message is that agnostic learning of a structured base class automatically yields learnability of its low-complexity linear span.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper claims a general black-box reduction showing that any weak agnostic learner for a succinctly representable class C of n-qubit states yields a tomography algorithm for unknown states |ψ⟩ that admit a decomposition |ψ⟩ = ∑ c_i |φ_i⟩ with |φ_i⟩ ∈ C and bounded ℓ1-norm of the coefficients (the extent). It applies the framework to obtain tomography for states of bounded stabilizer extent, gives runtime bounds (poly(n, (ξ/ε)^log(ξ/ε)) or poly(n, ξ, 1/ε) under a Freiman-Ruzsa conjecture), and proves an algorithmic weak regularity lemma that efficiently learns the C-explainable structure in arbitrary states.

Significance. If the reduction holds, the result provides a broadly applicable link between agnostic learning of a base class and tomography of its bounded-extent linear combinations, which is a conceptual contribution to quantum state learning. The stabilizer application yields concrete, improvable runtimes for a natural class, and the regularity lemma extends the technique beyond the bounded-extent promise. The black-box nature and explicit preconditions on C are strengths.

minor comments (3)
  1. The abstract introduces the extent via the decomposition but does not define the ℓ1-norm bound explicitly until the body; adding a parenthetical definition in the abstract would improve readability for readers outside the subfield.
  2. The runtime expression poly(n,(ξ/ε)^log(ξ/ε)) is stated without an explicit dependence on the weak-learner parameters; a sentence clarifying how the learner's advantage and sample complexity propagate into the final bound would help.
  3. The Freiman-Ruzsa conjecture is invoked only for the improved runtime; a brief remark on whether the conjecture is needed for correctness or only for polynomiality would clarify the result's unconditional status.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the paper, including the conceptual contribution of the black-box reduction, the concrete runtime bounds for stabilizer extent, and the algorithmic weak regularity lemma. The recommendation for minor revision is noted. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

Black-box reduction from stated assumptions; no circularity

full rationale

The paper's central claim is a black-box reduction: given that C is succinctly representable and admits a weak agnostic learner (explicitly listed as the two preconditions in the abstract), any such learner can be boosted to a tomography algorithm for states with bounded ℓ1 extent w.r.t. C. No quantity in the target result is defined in terms of a fitted parameter of the output, no self-citation is invoked as a load-bearing uniqueness theorem, and the derivation does not reduce to renaming or self-definition. The result is therefore self-contained against the stated external inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard linear-algebraic definitions of quantum states and trace distance together with the domain assumption that the input class C admits a weak agnostic learner; no free parameters or new entities are introduced.

axioms (2)
  • standard math Quantum states are vectors in a 2^n-dimensional complex Hilbert space and trace distance is the standard metric on density operators.
    Invoked when defining the decomposition |ψ⟩ = ∑ c_i |φ_i⟩ and the tomography guarantee to trace distance ε.
  • domain assumption There exists a weak agnostic learner for the class C.
    Explicitly required in the statement of the general framework (first sentence of abstract).

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