Tomography of quantum states with bounded extent

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 $|ψ\rangle$ that is promised to admit a decomposition of the form $|ψ\rangle = \sum_i c_i |φ_i\rangle$, where $|φ_i\rangle \in \textsf{C}$ 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 $ξ$ up to trace distance $\varepsilon$, in time $\textsf{poly}(n,(ξ/\varepsilon)^{\log(ξ/\varepsilon)})$, which is improvable to $ \textsf{poly}(n,ξ,1/\varepsilon)$ assuming the algorithmic polynomial Freiman-Ruzsa conjecture in the high-doubling regime. When the unknown state $|ψ\rangle$ 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 $|ψ\rangle$ 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.

• The paper establishes a framework that boosts weak agnostic learners to perform efficient tomography on quantum states with bounded extent.
• It decomposes a quantum state into a linear combination of structured states with a bounded ℓ1-norm, ensuring control over the residual error.
• Applications to stabilizer and matrix product states demonstrate the protocol's efficiency through polynomial/quasipolynomial time complexities and optimal sample bounds.

Tomography of Quantum States with Bounded Extent: An Expert Summary

Problem Setting and Motivation

The paper "Tomography of quantum states with bounded extent" (2606.07425) addresses a central challenge in quantum learning theory: under what structural assumptions can quantum states be learned efficiently using only copies of the state? Specifically, attention is focused on quantum states that exhibit "bounded extent" with respect to a structured model class $\mathcal{C}$, meaning the target state can be decomposed into a linear combination of a small number of states from $\mathcal{C}$, with bounded $\ell_1$-norm of the coefficients. Key examples of such classes include stabilizer states and matrix product states, which are foundational for both quantum simulation and quantum error correction.

Previous work exploited low-complexity structure for simulation and partial learning, often with demanding assumptions such as access to state-preparation unitaries or tolerating only constant-error approximations. It was unclear if efficient learning (in trace distance) was possible for all low-extent states using only copies—a limitation that has significant implications for both the theoretical boundary between learnability and simulability, and the efficient characterization of many-body quantum systems.

Main Contributions and Theoretical Results

The paper establishes a formal connection between weak agnostic learning and efficient tomography for quantum states with bounded extent. The pivotal insight is that if an efficient weak agnostic learner exists for a base class $\mathcal{C}$—that is, an algorithm that finds a state in $\mathcal{C}$ with non-trivial overlap with the target—then this learner can be "boosted" to perform tomography on states in the linear span of $\mathcal{C}$, assuming the coefficients in the decomposition are bounded in $\ell_1$.

The main results are:

1. Algorithmic Decomposition Theorem: For any unknown $n$-qubit state $\ket{\psi}$, and any model class $\mathcal{C}$ with a succinct representation and weak agnostic learner, there is a protocol that outputs a list of $\mathcal{C}$0-states $\mathcal{C}$1 and coefficients $\mathcal{C}$2, such that

$\mathcal{C}$3

with low $\mathcal{C}$4-fidelity in the residual $\mathcal{C}$5 and norm $\mathcal{C}$6 bounded by the extent.

1. Boosting Weak to Strong Agnostic Learning: The decomposition and residual-control techniques provide a generic boosting framework for quantum state learning. Specifically, any weak agnostic learner for $\mathcal{C}$7 (achieving overlap $\mathcal{C}$8 with the best-in-class state) can be transformed into a strong agnostic learner (achieving fidelity within $\mathcal{C}$9 of optimum).
2. Tomography Protocols for States with Stabilizer or MPS Extent: Applying the framework to stabilizer states, the authors achieve:
   • Unconditional tomography up to trace distance $\ell_1$0 in quasipolynomial time in $\ell_1$1 and $\ell_1$2 (the extent).
   • Conditional (on algorithmic polynomial Freiman–Ruzsa conjecture) tomography in polynomial time for $\ell_1$3. For matrix product states, the time and sample complexity scale polynomially in $\ell_1$4, the MPS bond dimension $\ell_1$5, $\ell_1$6, and $\ell_1$7.
3. Optimal Sample Complexity Lower Bound: Any constant-error tomography protocol for stabilizer-extent $\ell_1$8 states requires at least $\ell_1$9 copies, establishing near-optimality of the proposed algorithms in terms of sample complexity.

Algorithmic and Technical Insights

The protocol is built upon an iterative, multistage process reminiscent of boosting and regularity lemmas:

• Structure Learning: Iteratively, the residual state is projected (with regularization, via a 'ridge projector') orthogonally to the span of the previously identified dictionary states from $\mathcal{C}$0, and a weak agnostic learner is run on this residual. Each step appends a new state to the dictionary, efficiently improving the structured component and ensuring numerical stability (in stark contrast to naive projections which may be ill-conditioned).
• Parameter Learning: Once the structured dictionary is fixed, coefficient estimation is reduced to estimating overlaps and Gram matrices via quantum block-encoding techniques and quantum singular value transformation (QSVT).
• Complexity Control: The number of boosting iterations is $\mathcal{C}$1, and the sample/time costs are dominated by the complexity of the weak agnostic learner and the regularized inversion of the Gram matrix.
• Agnostic Boosting: The process is formalized as agnostic boosting: starting from a learner that can only weakly distinguish structure, the protocol efficiently amplifies performance to strong tomography—mirroring (and generalizing) classical boosting in machine learning.

Numerical and Provable Performance

• For stabilizer states, using the recent agnostic learning protocols ("Stabilizer Bootstrapping: A Recipe for Efficient Agnostic Tomography and Magic Estimation" [chen2024stabilizer]), the time and sample complexity for tomography are $\mathcal{C}$2 unconditionally, and $\mathcal{C}$3 conditionally (algorithmic PFR).
• For MPS with bond dimension $\mathcal{C}$4, the complexity is polynomial in $\mathcal{C}$5.
• The sample complexity lower bound, $\mathcal{C}$6, demonstrates that the protocol is optimal up to polynomial factors for realistic physical scaling regimes.

Impact and Implications

This work decisively clarifies the relationship between structure, simulability, and learnability in the quantum tomography landscape. The black-box reduction from weak learners to tomography subroutines has broad implications for quantum algorithm design: advances in even specialized agnostic learning algorithms for structured classes (e.g., Clifford circuits, product states, MPS) can be immediately leveraged for robust, efficient tomography of states in their linear spans.

The protocol is not limited to the stabilizer formalism and applies to any model class for which an agnostic learner is given, including graph states and sum-of-MPS modalities. Moreover, the approach is robust across different access models and is optimal with respect to sample complexity in the stabilizer extent setting.

Future Directions

Several open directions arise from this work:

• Improving the quasipolynomial dependence on the extent in the stabilizer protocol, possibly via progress on the algorithmic PFR conjecture.
• Tightening the dependence on error and constants.
• Extending the full boosting and tomography pipeline to mixed states, continuous-variable classes (e.g., Gaussian states), or to the learning of structured unitaries.
• Exploring the implications of agnostic learning for higher rungs in the circuit complexity or state-complexity hierarchy.

Conclusion

The results of "Tomography of quantum states with bounded extent" (2606.07425) establish a powerful and generic framework that transforms any weak agnostic learner for a structured quantum model class into an efficient tomography protocol for states with bounded extent over that class. This provides both general theory and practical tools for quantum information science, sharply delimiting the frontier of learnability for physically relevant quantum states, and offering pathways for deep algorithmic advances in quantum learning and simulation.