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arXiv:2602.20299

Entanglement Barriers from Computational Complexity: Matrix-Product-State Approach to Satisfiability

Tim Pokart, Frank Pollmann, Jan Carl Budich

quant-phcond-mat.stat-mechcond-mat.str-elphysics.comp-phSubmitted Feb 23, 2026• Announced Feb 25, 2026• 0 citations• Revised Mar 6, 2026 (v2)
v1 · Feb 23, 2026v2 · Mar 6, 2026
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Abstract

We approach the 3-SAT satisfiability problem with the quantum-inspired method of imaginary time propagation (ITP) applied to matrix product states (MPS) on a classical computer. This ansatz is fundamentally limited by a quantum entanglement barrier that emerges in imaginary time, reflecting the exponential hardness expected for this NP-complete problem. Strikingly, we argue based on careful analysis of the structure imprinted onto the MPS by the 3-SAT instances that this barrier arises from classical computational complexity. To reveal this connection, we elucidate with stochastic models the specific relationship between the classical hardness of the $\sharp$P $\supseteq$ NP-complete counting problem $\sharp$3-SAT and the entanglement properties of the quantum state. Our findings illuminate the limitations of this quantum-inspired approach and demonstrate how purely classical computational complexity can manifest in quantum entanglement. Furthermore, we present estimates of the non-stabilizerness required by the protocol, finding a similar resource barrier. Specifically, the necessary amount of non-Clifford operations scales superlinearly in system size, thus implying extensive resource requirements of ITP on different architectures such as Clifford circuits or gate-based quantum computers.

Comments
17 pages, 12 figures

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•Sep 24, 2026•0 citations
We approach the 3-SAT satisfiability problem with the quantum-inspired method of imaginary time propagation (ITP) applied to matrix product states (MPS) on a classical computer. This ansatz is fundamentally limited by a quantum entanglement barrier that emerges in imaginary time, reflecting the exponential hardness expected for this NP-complete problem. Strikingly, we argue based on careful analysis of the structure imprinted onto the MPS by the 3-SAT instances that this barrier arises from classical computational complexity. To reveal this connection, we elucidate with stochastic models the specific relationship between the classical hardness of the ♯\sharp♯P ⊇\supseteq⊇ NP-complete counting problem ♯\sharp♯3-SAT and the entanglement properties of the quantum state. Our findings illuminate the limitations of this quantum-inspired approach and demonstrate how purely classical computational complexity can manifest in quantum entanglement. Furthermore, we present estimates of the non-stabilizerness required by the protocol, finding a similar resource barrier. Specifically, the necessary amount of non-Clifford operations scales superlinearly in system size, thus implying extensive resource requirements of ITP on different architectures such as Clifford circuits or gate-based quantum computers.
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.
Satisfiability (SAT) is a central problem in computer science, and advances in SAT-solving algorithms have a far-reaching impact across many fields. Recent works have proposed quantum SAT solvers based on Grover's algorithm, a quantum search technique. However, Grover-based approaches face a key limitation: they typically require prior knowledge of the number of satisfying assignments of the target Boolean formula. This information is unavailable in most practical settings. Quantum counting can be used to estimate this quantity, but it incurs a computational overhead that is several orders of magnitude higher than Grover search. In this paper, we propose a novel quantum SAT solver based on entanglement and equivalence checking. Our method does not assume prior knowledge of the number of solutions and is computationally more efficient than quantum counting. Although the worst case time complexity is inevitably exponential, we prove that the expected time complexity of our approach is only constant time O(1) over random Boolean functions. Experimental results also support our theoretical claim.
The states accepted by a quantum circuit are known as the witnesses for the quantum circuit's satisfiability. The assumption BQP does not equal QMA implies that no efficient algorithm exists for constructing a witness for a quantum circuit from the circuit's classical description. However, a similar complexity-theoretic lower bound on the computational hardness of cloning a witness is not known. In this note, we derive a conjecture about cloning algorithms for maximally entangled states over hidden subspaces which would imply that no efficient algorithm exists for cloning witnesses (assuming BQP does not contain NP). The conjecture and result follow from connections between quantum computation and representation theory; specifically, the relationship between quantum state complexity and the complexity of computing Kronecker coefficients.
The advent of quantum computers promises exponential speed ups in the execution of various computational tasks. While their capabilities are hindered by quantum decoherence, they can be exactly simulated on classical hardware at the cost of an exponential scaling in terms of number of qubits. To circumvent this, quantum states can be represented as matrix product states (MPS), a product of tensors separated by so-called bond dimensions. Limiting bond dimensions growth approximates the state, but also limits its ability to represent entanglement. Methods based on this representation have been the most popular tool at simulating large quantum systems. But how to trust resulting approximate quantum states for such intractable systems sizes ? I propose here a method for inferring the fidelity of an approximate quantum state without direct comparison to its exact counterpart, and use it to design an ``entanglement-aware'' (EA) algorithm for both pure and mixed states. As opposed to state of the art methods which limit bond dimensions up to an arbitrary maximum value, this algorithm receives as input a fidelity, and adapts dynamically its bond dimensions to both local entanglement and noise such that the final quantum state fidelity at least reaches the input fidelity. I show that this algorithm far surpasses standard fixed bond dimension truncation schemes. In particular, a noiseless random circuit of 300 qubits and depth 75 simulated using MPS methods takes one week of computation time, while EA-MPS only needs 2 hours to reach similar quantum state fidelity.
Identifying the boundary between classical and quantum computation is a central challenge in quantum information. In multi-qubit systems, entanglement and magic are the key resources underlying genuinely quantum behaviour. While entanglement is well understood, magic - essential for universal quantum computation - remains relatively poorly characterised. Here we show that determining membership in the stabilizer polytope, which defines the free states of magic-state resource theory, requires super-exponential time $\class{exp} ( n^2)$ in the number of qubits nnn, even approximately. We reduce the problem to solving a 333-\class{SAT} instance on n2n^2n2 variables and, by invoking the exponential time hypothesis, the result follows. As a consequence, both quantifying and certifying magic are fundamentally intractable: any magic monotone for general states must be super-exponentially hard to compute, and deciding whether an operator is a valid magic witness is equally difficult. As a corollary, we establish the robustness of magic as computationally optimal among monotones. This barrier extends even to classically simulable regimes: deciding whether a state lies in the convex hull of states generated by a logarithmic number of non-Clifford gates is also super-exponentially hard. Together, these results reveal intrinsic computational limits on assessing classical simulability, distilling pathological magic states, and ultimately probing and exploiting magic as a quantum resource.
The study of classical simulability is a key part of quantum information theory, providing new tools for fundamental research and the development of quantum computing. An increasingly successful framework for the study of simulability is resource theory. The best-known and best-understood quantum resource is entanglement, but in recent years nonstabilizerness, or magic, has also been gathering attention from the community. It is not yet well understood to what extent different resources play off each other in making systems hard to classically simulate. It has recently been shown that Clifford-augmented matrix product states (CAMPS) allow for efficient circuit simulation in some cases where both entanglement and magic are present. Another emerging method which has shown the ability to simulate systems with both resources is Pauli propagation (PP). Furthermore, the interplay between entanglement and magic at the level of physical phenomena is also poorly understood, in particular in the context of real-time dynamics. In this work, I implement an analytical algorithm proposed by Fux et al. (2025) which allows CAMPS to efficiently simulate essentially the first N non-Clifford Pauli exponential gates in an N-qubit quantum circuit, reproducing results obtained by greedy-search entanglement cooling in the paper. I then develop CAMPS-PP, a method for expectation value dynamics which evolves the state as a CAMPS and the observable in the Heisenberg picture using PP. I show that this method is significantly faster than pure CAMPS for slightly deep random Pauli rotation circuits at the tested system sizes, up to N=100, with improvements of up to two orders of magnitude for N=20. Finally, in order to investigate the physical relevance of magic I apply CAMPS-PP to an XXZ Heisenberg Floquet circuit with Clifford and non-Clifford angles, finding that CAMPS-PP slightly extends the simulable non-Clifford gate density and significantly reduces memory requirements compared to PP.