Adding noise can make a random quantum circuit look faster without making its unitary dynamics more random. This distinction is especially relevant in modular processors, where local gates randomize each core and scarce inter-core communication must spread that randomness across the full device. In this paper, we study this problem with a reduced second-moment transfer-matrix theory for Pauli second moments in distributed random circuits affected by the amplitude-damping, depolarizing, and dephasing noise channels. The key step is to resolve the noisy spectrum into two branches: a radial branch, describing dissipative loss of non-identity Pauli weight, and an angular branch, describing Haar-like mixing within the surviving nontrivial sector. This separation gives a simple weak-noise criterion: noise is useful for angular randomization only when it suppresses the longitudinal Bloch component more strongly than the transverse plane. Among the three channels considered, this selects amplitude damping as the only locally favorable case, while depolarizing noise is neutral and dephasing is dominated by radial loss. For multicore architectures, we derive a universal first-order law for radial leakage and track the angular branch numerically across different channels, topologies, and core partitions. The results reveal narrow windows of genuine noise-assisted Haar mixing, most clearly for amplitude damping, but rule out a generic speed-up by noise. The framework therefore distinguishes useful noisy randomization from mere dissipation.
Adding noise can make a random quantum circuit look faster without making its unitary dynamics more random. This distinction is especially relevant in modular processors, where local gates randomize each core and scarce inter-core communication must spread that randomness across the full device. In this paper, we study this problem with a reduced second-moment transfer-matrix theory for Pauli second moments in distributed random circuits affected by the amplitude-damping, depolarizing, and dephasing noise channels. The key step is to resolve the noisy spectrum into two branches: a radial branch, describing dissipative loss of non-identity Pauli weight, and an angular branch, describing Haar-like mixing within the surviving nontrivial sector. This separation gives a simple weak-noise criterion: noise is useful for angular randomization only when it suppresses the longitudinal Bloch component more strongly than the transverse plane. Among the three channels considered, this selects amplitude damping as the only locally favorable case, while depolarizing noise is neutral and dephasing is dominated by radial loss. For multicore architectures, we derive a universal first-order law for radial leakage and track the angular branch numerically across different channels, topologies, and core partitions. The results reveal narrow windows of genuine noise-assisted Haar mixing, most clearly for amplitude damping, but rule out a generic speed-up by noise. The framework therefore distinguishes useful noisy randomization from mere dissipation.
As quantum devices grow in size while remaining affected by noise, a key challenge is to determine when and how they can surpass classical computation on practical tasks. This requires the development of highly efficient classical simulation algorithms. Pauli backpropagation algorithms are among the most promising, having shown efficiency in simulating certain classes of parameterized quantum circuits, strong candidates for near‑term quantum advantage, under random circuit assumptions and depolarizing noise. However, their efficiency had not been demonstrated for more realistic nonunital noise models, such as amplitude damping, which better capture the behaviour of current hardware. We close this gap by adapting Pauli backpropagation to most single‑qubit noise channels, including nonunital noise, and proving that it remains efficient under these conditions. Our algorithm is parallelizable, memory‑efficient, and requires no assumptions on the locality of the observable, the geometry of the hardware, or the depth of the circuit. While hardware noise can strongly affect computational power in highly structured, fault‑tolerant circuits, our findings indicate little difference for most near‑term variational architectures. This establishes Pauli backpropagation as an efficient method for simulating noisy quantum devices under realistic noise assumptions.While these advances improve our ability to reproduce noisy expectation values of observables, many practical applications, including combinatorial optimization, require direct access to the bitstrings output by quantum circuits, as these encode candidate solutions. Classical replication of such noisy samples has so far been limited beyond the random circuit setting. To bridge this gap, we first focus on problems whose objective functions depend only on two‑body correlations, such as Max‑Cut, and apply Gaussian randomized rounding to the circuit's two‑qubit marginals. This produces classical samples whose expected cost is provably close to that of the noisy quantum device. Large‑scale simulations and experiments on IBMQ hardware confirm that the rounded samples reproduce the full energy distribution, with similar behaviour under various noise models. Coupled with Pauli backpropagation for efficiently estimating noisy marginals, this yields an end‑to‑end classical sampler operating in polynomial time for a broad class of variational circuits. Our results therefore show that sufficiently noisy quantum circuits for combinatorial optimization can be sampled from classically, clarifying where near‑term hardware may or may not outperform classical algorithms.Finally , we go beyond one and two‑body marginals and present an algorithm that samples valid bitstrings directly from a maximum‑entropy distribution matching the given expectation values of the quantum circuit. This method does not rely on the locality or sparsity of the problem's Hamiltonian and performs well across a variety of quantum circuits, including deep circuits on many qubits. When combined with Pauli propagation or error mitigation, it provides a practical pipeline for obtaining samples from quantum circuits in the near term, where full error correction might not yet be available.
We present a polynomial-time classical algorithm for estimating expectation values of arbitrary observables on typical quantum circuits under any incoherent local noise, including non-unital or dephasing. Although previous research demonstrated that some carefully designed quantum circuits affected by non-unital noise cannot be efficiently simulated, we show that this does not apply to average-case circuits, as these can be efficiently simulated using Pauli-path methods. Specifically, we prove that, with high probability over the circuit gates’ choice, Pauli propagation algorithms with tailored truncation strategies achieve an inversely polynomially small simulation error. This result holds for arbitrary circuit topologies and for any local noise, under the assumption that the distribution of each circuit layer is invariant under single-qubit random gates. Under the same minimal assumptions, we also prove that most noisy circuits can be truncated to an effective logarithmic depth for the task of estimating expectation values of observables, thus generalizing prior results to a significantly broader class of circuit ensembles. We further numerically validate our algorithm with simulations on a 6 × 6 lattice of qubits under the effects of amplitude damping and dephasing noise, as well as real-time dynamics on an 11 × 11 lattice of qubits affected by amplitude damping.
We systematically characterize quantum entanglement in all sixteen combinations of two-qubit Bell states across three operational regimes: pure superposition, noiseless mixing, and asymmetric mixing of one noisy and one pure Bell state. For pure superpositions, we calculate the von Neumann entropy and show that entanglement depends critically on the relative phase phi between superposed states. For mixed states, we use concurrence C under three physically motivated decoherence channels. The Depolarizing (DP) and Phase Damping (PD) channels are modeled as global, collective operations on the two-qubit Hilbert space, while Amplitude Damping (AD) is treated as independent local decay on each qubit, reflecting the asymmetric, energy-dissipating nature of spontaneous emission. A central and counterintuitive result is that increasing noise can raise concurrence in specific parameter regimes across all three channels. We further map the boundary between mathematical entanglement (C > 0) and operational quantum non-locality using the Horodecki criterion, finding this boundary to be strongly channel-dependent. Under Phase Damping, the Bell-local entanglement regime vanishes entirely for identical and same-bases mixtures, meaning any surviving entanglement guarantees a CHSH violation. Under Depolarizing noise, a large Bell-local region requires mixing probability r > 1/sqrt(2) to recover non-locality at maximum noise. For Amplitude Damping, certain mixtures exhibit a non-monotonic Horodecki parameter M(T): concurrence decreases monotonically with noise, yet the capacity for CHSH violation is lost at intermediate noise levels and restored at high noise, as maximal damping repurifies the noisy branch. These results provide a unified analytical reference for entanglement and non-locality management across Bell-state combinations and decoherence channels relevant to NISQ architectures.
Expectation values of measurement operators, interpreted as measurement probabilities, arise frequently throughout quantum algorithms. When quantum states are randomly distributed, their expectation values are also randomly distributed. In this work, with the goal of understanding non-unitary dynamics, we generalize previous derivations for distributions of expectation values [Campos et al., Phys. Lett. A 377, 1854 (2013)] to the case of sets of measurement operators and random mixed quantum states within variable sized environments. Using combinatorics approaches, we derive expressions for their moments. We proceed to construct empirical distributions of simulated Haar random brickwork quantum circuits with local depolarizing noise, and compare their form to a proposed effective global-depolarizing-like model with variable effective noise scales and environment dimensions. The fitted effective distributions reproduce peak behaviour across circuit depths, noise scales, and system sizes, while deviations in the distribution tails arise from local noise effects. The fit effective model parameters are also shown to vary smoothly and consistently with circuit depth and noise scale. Finally, sets of non-symmetric measurement operators are shown to exhibit distinct multi-modal distributions relative to uni-modal distributions for symmetric measurement operators, opening up questions about their simulability.
Random unitary circuits have become a model system to investigate information scrambling in quantum systems. In the literature, mostly random circuits with Haar-distributed gate operations have been considered. In this work, we investigate operator spreading in random unitary circuits in which the elementary gate operations are drawn from general unitary-invariant ensembles, which include the well-studied Haar-distributed random unitary circuits as a special case. Similar to the Haar-distributed case, the long-time behavior of operator spreading with the more general unitary-invariant gate distribution is governed by drift-diffusion equations characterized by the butterfly velocity v B and a diffusion constant D . Differences with the Haar-random case are (i) that it takes a finite time τ b until ensemble-averaged Pauli-string weights take a “binary” form, in which they depend only on whether Pauli operators inside the support of the Pauli strong are equal to the identity matrix, and (ii) that the operator spreading is characterized by a finite “domain-wall width” n DW separating regions with a random-matrix-like Pauli-string distribution. To illustrate these findings, we perform explicit calculations for random unitary circuits distributed according to the Poisson kernel, which interpolates between the trivial and Haar-distributed circuits.
For random quantum circuits on n qubits of depth Θ ( log n ) with depolarizing noise, the task of sampling from the output state can be efficiently performed classically using a Pauli path method \cite{Aharonov2023_paulipaths} . This paper aims to study the performance of this method beyond random circuits. We first consider the classical simulation of local observables in circuits composed of Clifford and T gates – going beyond the average case analysis, we derive sufficient conditions for simulatability in terms of the noise rate and the fraction of gates that are T gates, and show that if noise is introduced at a faster rate than T gates, the simulation becomes classically easy. As an application of this result, we study 2D QAOA circuits that attempt to find low-energy states of classical Ising models on general graphs. There, our results shows that for hard instances of the problem, which correspond to Ising model's graph being geometrically non-local, a QAOA algorithm mapped to a geometrically local circuit architecture using SWAP gates does not have any asymptotic advantage over classical algorithms if depolarized at a constant rate. Finally, we illustrate instances where the Pauli path method fails to give the correct result, and also initiate a study of the trade-off between fragility to noise and classical complexity of simulating a given quantum circuit.