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arXiv:2609.27128

Efficient Learning of Clifford-Scrambled Product States

Tobias Haug

quant-phSubmitted Sep 22, 2026• Announced Sep 24, 2026• 0 citations
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Abstract

Clifford circuits acting on product magic states provide a compact ansatz exhibiting extensive magic, volume-law entanglement, and even classically hard sampling under standard complexity assumptions. Here, we efficiently recover its hidden product subsystems from two-copy Bell sampling. Quadratic relations between Bell samples determine the irreducible blocks after removing Pauli stabilizers, and binary linear algebra constructs a Clifford disentangler. For logarithmic-size blocks, approximate learning of the full state is efficient whenever the state remains inverse-polynomially separated from acquiring additional Pauli stabilizers. We efficiently learn $n$-qubit states prepared by random $T$-doped Clifford circuits with $T$-gate density below one via $O(n^2)$ Bell samples, and disentangle hidden product blocks in Clifford-augmented matrix product states. For unitary learning, we exactly learn $T$-depth-one circuits using $O(n^2)$ queries. Finally, we rule out pseudorandom states and unitaries with a product bipartition hidden by Clifford circuits.

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Clifford circuits acting on product magic states provide a compact ansatz exhibiting extensive magic, volume-law entanglement, and even classically hard sampling under standard complexity assumptions. Here, we efficiently recover its hidden product subsystems from two-copy Bell sampling. Quadratic relations between Bell samples determine the irreducible blocks after removing Pauli stabilizers, and binary linear algebra constructs a Clifford disentangler. For logarithmic-size blocks, approximate learning of the full state is efficient whenever the state remains inverse-polynomially separated from acquiring additional Pauli stabilizers. We efficiently learn nnn-qubit states prepared by random TTT-doped Clifford circuits with TTT-gate density below one via O(n2)O(n^2)O(n2) Bell samples, and disentangle hidden product blocks in Clifford-augmented matrix product states. For unitary learning, we exactly learn TTT-depth-one circuits using O(n2)O(n^2)O(n2) queries. Finally, we rule out pseudorandom states and unitaries with a product bipartition hidden by Clifford circuits.
Efficient learning of Clifford disentanglers and typical ttt-doped unitaries with exponentially more TTT gates
Highly entangled and highly non-stabilizer quantum states need not be hard to learn. We give efficient algorithms for testing and recovering hidden tensor-product structure in unknown pure state vectors of the form ∣ψ⟩=UC⨂i∣ψi⟩\lvertψ\rangle = U_C \bigotimes_i \lvertψ_i\rangle∣ψ⟩=UC​⨂i​∣ψi​⟩, where UCU_CUC​ is an arbitrary unknown Clifford unitary. Although the Clifford can thoroughly scramble the visible product structure, we prove that the Bell distribution retains a characteristic family of quadratic symmetries. By simultaneously block-diagonalising these symmetries, our algorithms linearize the problem and manage to recover both a disentangling Clifford and the hidden partitions with polynomial sample and computational complexity. This may be viewed as an extension of the abelian StateHSP paradigm in which classical post-processing exposes genuinely quadratic structure. Applied to Choi states, the method yields efficient proper learning algorithms for typical ttt-doped Clifford unitaries in regimes containing exponentially more TTT gates than previously accessible: the required condition fails only for an exponentially small fraction of circuits when t∼nt\sim nt∼n, and continues to hold for a constant fraction even when t=2nt=2nt=2n. Our framework also provides tools for compressing structured many-body Hamiltonians and suggests benchmarking protocols for encoded logical product states in the early fault-tolerant regime.
Full tomography of a generic many-qubit quantum state requires exponentially many copies, while suitable structural constraints can make reconstruction sample-efficient. Existing approaches exploit, for example, limited entanglement structure, low magic, or constrained state-preparation circuits. Here we consider a class of mixed states that can simultaneously exhibit extensive entanglement and extensive magic. Specifically, we introduce Clifford-encoded block-product (CEBP) states, obtained by applying an unknown global Clifford unitary to a tensor product of arbitrary mixed states supported on unknown blocks of bounded size. We show that, for a fixed block size, CEBP states can be reconstructed using polynomially many copies by exploiting Clifford-preserved Pauli correlations to recover the latent structure and reduce the remaining problem to local tomography. Our result demonstrates that sample-efficient tomography can arise from bounded complexity in a latent frame even when the physical state is highly entangled, highly magical, and mixed, and suggests complexity modulo structured transformations as a broader organizing principle for quantum-state learnability.
Determining the quantum-classical boundary between quantum circuits that can be efficiently simulated classically and those that cannot remains a fundamental question. One approach to classical simulation is to represent the output of a quantum circuit as a Clifford-augmented matrix product state (CAMPS), which, via a disentangling algorithm, decomposes the wave function into Clifford and MPS components, and from which Pauli expectation values can be computed in time polynomial in the MPS bond dimension. In this work, we develop an optimization-free disentangling (OFD) algorithm for Clifford circuits either doped with T gates or, equivalently, preceded by multiqubit gates of the form α I + β P . We give a simple and easily computed algebraic criterion that characterizes the individual quantum circuits for which OFD generates an efficient CAMPS—the bond dimension is exponential in the null space of a Galois binary field GF(2) matrix induced by a tableau of the twisted Pauli strings P . This significantly increases the number of circuits with rigorous polynomial-time classical simulations. We also give evidence that the typical N qubit random Clifford circuit doped with N uniformly distributed T gates of polylogarithmic depth or greater has a CAMPS with polynomial bond dimension. In addition, we compare OFD against disentangling by optimization. We further explore the representability of CAMPS for random Clifford circuits doped with more than N T gates. We also propose algorithms for sampling, probability, and amplitude estimation of bitstrings, and evaluation of entanglement Rényi entropy from CAMPS, which, though still having exponential complexity, are more efficient than standard MPS simulations. This work establishes a versatile framework for understanding classical simulatability of Clifford + T circuits and explores the interplay between quantum entanglement and quantum magic in quantum systems.
We show that states obtained from deep random Clifford circuits doped with non-Clifford phase gates (including T gates and sqrt[T] gates) can be disentangled completely, provided the number of non-Clifford gates is smaller or approximately equal to the number of qubits. This implies that Pauli expectation values of such states can be efficiently simulated classically, despite them exhibiting both extensive entanglement and extensive nonstabilizerness. We prove this result analytically using a quantum error correction formulation, demonstrate its applicability numerically, and discuss consequences for the disentanglability of states generated through Hamiltonian dynamics. We show that this result implies a novel representation of approximate state designs that can also facilitate their efficient generation, and we propose a novel quantum circuit compression scheme for Clifford circuits doped with non-Clifford phase gates.
Scrambling of quantum information is an important feature at the root of randomization and benchmarking protocols, the onset of quantum chaos, and black-hole physics. Unscrambling this information is possible given perfect knowledge of the scrambler [arXiv:1710.03363]. We show that one can retrieve the scrambled information even without any previous knowledge of the scrambler, by a learning algorithm that allows the building of an efficient decoder. Remarkably, the decoder is classical in the sense that it can be efficiently represented on a classical computer as a Clifford operator. It is striking that a classical decoder can retrieve with fidelity one all the information scrambled by a random unitary that cannot be efficiently simulated on a classical computer, as long as there is no full-fledged quantum chaos. This result shows that one can learn the salient properties of quantum unitaries in a classical form and sheds a new light on the meaning of quantum chaos. Furthermore, we obtain results concerning the algebraic structure of ttt-doped Clifford circuits, i.e., Clifford circuits containing ttt non-Clifford gates, their gate complexity, and learnability that are of independent interest. In particular, we show that a ttt-doped Clifford circuit Ut{U}_{t}Ut​ can be decomposed into two Clifford circuits ${U}_{0},{U}_{0}^{\ensuremath{'}}$ that sandwich a local unitary operator ut{u}_{t}ut​, i.e., ${U}_{t}={U}_{0}{u}_{t}{U}_{0}^{\ensuremath{'}}$. The local unitary operator ut{u}_{t}ut​ contains ttt non-Clifford gates and acts nontrivially on at most ttt qubits. As simple corollaries, the gate complexity of the ttt-doped Clifford circuit Ut{U}_{t}Ut​ is O(n2+t3)O({n}^{2}+{t}^{3})O(n2+t3), and it admits a efficient process tomography using poly(n,2t)poly(n,{2}^{t})poly(n,2t) resources.
Random Clifford operators have numerous applications in quantum computing, including randomized benchmarking, classical shadows, and quantum authentication. However, sampling and implementing uniformly random nnn-qubit Clifford incur near-quadratic complexity due to the size of Clifford group. We introduce a cryptographic way to overcome these barriers: trapdoored Clifford operator distributions whose samples are computationally indistinguishable from uniformly random Cliffords, yet implementing them can be much faster given the trapdoor. We construct a distribution of trapdoored Clifford operators whose elements can be sampled and implemented in near-linear time under a variant of the learning parity with noise assumption. Our constructions allow fast tableau action on Pauli labels for classical simulation, and also can be optimized to admit polylogarithmic-depth implementation. Along the way, we construct trapdoored matrices over finite fields that support efficient multiplication by both a matrix and its inverse, resolving an open question left by Vaikuntanathan and Zamir [SODA'26]. We use these constructions to obtain faster protocols based on random Cliffords. We also explore their applications to the worst-case to average-case reductions for matrix and Clifford problems including the iterated matrix multiplication and Clifford circuit synthesis. In particular, we show the hardness of batching Clifford circuits: synthesizing circuits that apply the same Clifford to multiple registers is at least as hard as worst-case matrix multiplication, even when synthesis succeeds on a small constant fraction of random Cliffords. This extends to approximate implementations by general quantum circuits.