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.
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.
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(n2) 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(n2) queries. Finally, we rule out pseudorandom states and unitaries with a product bipartition hidden by Clifford circuits.
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 study the classical simulability of open quantum dynamics using random Clifford circuits doped with non-Clifford phase rotations and subject to local noise. We unravel the dynamics into stochastic quantum trajectories simulated with Clifford-augmented matrix product states, and introduce a simulation cost that quantifies the classical resources required. Optimizing this cost over stochastic unravelings, we identify noise-induced classical phases: extended parameter regions in which the dynamics can be fully disentangled by Clifford operations at arbitrary circuit depth. Their existence depends on both the noise model \emph{and} the unraveling. Using a geometric representation of quantum channels, we analytically determine optimal unravelings for a broad class of noise models, with numerical simulations confirming the predicted phase boundaries. We further relate the optimal cost to the unraveling-independent nonstabilizerness of the channel and show that, together with trajectory-resolved entanglement and nonstabilizerness, it classifies distinct dynamical regimes. Finally, we show that these classical phases disappear in the averaged density-matrix description, where no unraveling freedom remains. Our results show that the emergence of classicality in noisy random circuits depends on the measurement scheme adopted to probe it, and paves the way to further studies on the classical simulability of driven-dissipative dynamics.
We introduce a family of quantum circuits that possess standard indicators of classical simulation hardness including high entanglement entropy, magic, and non-Gaussianity, yet admit efficient classical simulation via matrix product states (MPS). Our construction uses logical circuits of high-rate Calderbank-Shor-Steane (CSS) codes with enhanced symmetries. Using code automorphisms and transversal diagonal gates from higher levels of the Clifford hierarchy, we realize nonlocal logical Clifford and non-Clifford gates, showing how error-correcting codes can compile complex logical circuits into simple physical operations. Simulation efficiency rests on two properties: (i) diagonal transversal gates do not increase bond dimension, and (ii) permutations are tracked classically via on-the-fly relabeling, avoiding costly SWAP networks. Unlike Clifford or matchgate simulation, our method accepts a broad class of initial states, including dense entangled, magic, and non-Gaussian inputs, provided the encoded state retains an efficient MPS representation. We also release an exact phase-polynomial backend for monomial subfamilies, whose cost is set by higher-degree phase terms rather than entanglement growth. We demonstrate the method on an infinite polar CSS code family, showing bond dimension stays bounded by the encoding cost regardless of circuit depth. These results show that for some circuit families, standard resource measures are individually insufficient to indicate simulation hardness. As a near-term application, we use the compiled MPS as a classical reference for direct fidelity estimation of a quantum device running nontrivial logical circuits. Pauli sampling on the encoded reference, with a Clifford pushback through the known encoder, provides the ideal expectation values, so the logical output fidelity can be estimated from local Pauli readout alone, without costly state tomography.
Tensor network methods leverage the limited entanglement of quantum states to efficiently simulate many-body systems. Alternatively, Clifford circuits provide a framework for handling highly entangled stabilizer states, which have low magic and are thus also classically tractable. Clifford tensor networks combine the benefits of both approaches, exploiting Clifford circuits to reduce the classical complexity of the tensor network description of states, with promising effects on simulation approaches. We study the disentangling power of Clifford transformations acting on tensor networks, with a particular emphasis on entanglement cooling strategies. We identify regimes where exact or heuristic Clifford disentanglers are effective, explain the link between the two approaches, and characterize their breakdown as non-Clifford resources accumulate. Additionally, we prove that, beyond stabilizer settings, no Clifford operation can universally disentangle even a single qubit from an arbitrary non-Clifford rotation. Our results clarify both the capabilities and fundamental limitations of Clifford-based simulation methods.
We develop classical simulation algorithms for adaptive quantum circuits that produce states with low levels of ``magic'' (i.e., non-stabilizerness). These algorithms are particularly well-suited to circuits with high rates of Pauli measurements, such as those encountered in quantum error correction and monitored quantum circuits. The measurements serve to limit the buildup of magic induced by non-Clifford operations arising from generic noise processes or unitary gates, respectively. Our algorithms also allow a systematic truncation procedure to achieve approximate simulation. To benchmark our approach, we study the dynamics of all-to-all monitored quantum circuits with a sub-extensive rate of T-gates per unit of circuit depth, where we can simulate previously inaccessible system sizes and depths. We characterize measurement-induced phase transitions in the output wavefunction, including in the entanglement, purification, and magic. We outline the utility of our algorithms to simulate dynamics with low magic and high entanglement, complementary to the leading matrix-product state approaches.