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arXiv:2609.31830

Probing the classical complexity of quantum dynamics experiments

Thomas Schuster, Andreas Elben

quant-phcond-mat.str-elcs.CChep-thSubmitted Sep 25, 2026• Announced Sep 29, 2026• 0 citations
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Abstract

A confluence of recent works has shown that many quantum circuits and dynamics are efficiently simulable by classical algorithms that track local information, even when conventional complexity measures such as the entanglement and magic are high. Here, we introduce a novel measure of complexity, the reactivity, to capture this new method of classical attack. Unlike conventional complexity measures, the reactivity does not capture a property of a quantum state or operator in isolation, but rather a quantum experiment as a whole. We provide numerical and rigorous evidence that quantum experiments with low reactivity are simple by a host of measures: they are efficient to classically simulate, learn, and fast-forward. This motivates the search for quantum experiments with high reactivity, which may evade these simplistic features. To this end, we introduce easily implementable experimental protocols---dubbed Pauli path spectroscopy---that allow one to efficiently measure the reactivity of any quantum experiment of interest. Our protocols are applicable even when the experiment itself is beyond the reach of classical simulation.

Comments
6+29 pages, 12 figures

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A confluence of recent works has shown that many quantum circuits and dynamics are efficiently simulable by classical algorithms that track local information, even when conventional complexity measures such as the entanglement and magic are high. Here, we introduce a novel measure of complexity, the reactivity, to capture this new method of classical attack. Unlike conventional complexity measures, the reactivity does not capture a property of a quantum state or operator in isolation, but rather a quantum experiment as a whole. We provide numerical and rigorous evidence that quantum experiments with low reactivity are simple by a host of measures: they are efficient to classically simulate, learn, and fast-forward. This motivates the search for quantum experiments with high reactivity, which may evade these simplistic features. To this end, we introduce easily implementable experimental protocols---dubbed Pauli path spectroscopy---that allow one to efficiently measure the reactivity of any quantum experiment of interest. Our protocols are applicable even when the experiment itself is beyond the reach of classical simulation.
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.
Claims of quantum advantage rest on the classical hardness of simulating quantum circuits. Magic, operator scrambling, anticoncentration, and non-Gaussianity for fermionic circuits are standard diagnostics of complex quantum dynamics. For pure states, some of these have been rigorously connected to classical simulability. Whether these diagnostics reliably track the limits of efficient classical simulation under noise remains unclear. Here, we show that in noisy Clifford+TTT and nearest-neighbour matchgate+SWAP circuits, dynamical diagnostics and classical simulability can separate in both directions. In particular, the diagnostics can remain nontrivial after classical simulation becomes efficient, or become trivial before known efficient classical algorithms apply. We trace this mismatch to the different statistical properties they probe: magic and scrambling depend on fourth-order statistics of the Pauli spectrum, whereas the simulation algorithms depend primarily on second-order moments, which local noise suppresses at different rates. Thus, dynamical diagnostics measured on a noisy quantum device do not by themselves constitute evidence for classical hardness.
In this this paper we present an inexpensive protocol to perform runtime and memory estimation for large-scale experiments with Pauli Path simulators (PPS). Additionally, we propose a conceptually simple solution for studying whether PPS can be used as a scientific discovery tool, rather than reproducing existing answers. We start by analyzing the dynamics of the Pauli coefficients tracked in the Heisenberg picture. In addition to surprisingly generic convergence features of the Pauli coefficient distributions, we find certain regularities that allow for extrapolation of memory and runtime requirements for smaller and smaller coefficient truncation parameter δδδ. We then introduce a framework for understanding convergence in the absence of rigorous error guarantees on PPS. Combined with runtime analysis, we propose bifurcating quantum simulation problems broadly into two classes, based on whether there is apparent convergence of expectation values as a function of δδδ. This serves as a way for practitioners to understand where their problem falls on the frontier of classical simulability. In the case without apparent convergence, PPS may still serve useful as a Monte Carlo-like estimate. Applied to IBM's utility-scale experiments, we show parameter regimes where both behaviors are realized. Some of our key findings challenge conventional intuition: reducing δδδ does not always improve accuracy, and deeper quantum circuits may actually be easier to simulate than shallower ones. The BlueQubit SDK implementing these methods has been released publicly, offering researchers a comprehensive toolkit for evaluating this frontier classical simulation approach. These results establish practical guidelines for when PPS can serve as a reliable verification tool versus when it should be used as a complementary estimate alongside quantum experiments.
Overcoming experimental errors is a central challenge in quantum science. This challenge is particularly acute in large and complex quantum circuits, where quantum information spreads into exponentially many paths, foiling any direct attempt to understand errors' impact. In this work, we present a novel approach to understanding and mitigating experimental errors, based on a surprisingly compact representation of a quantum circuit's information dynamics, the reactivity function. The shape of a circuit's reactivity function governs its noise response, and its amenability to different error mitigation methods. From this viewpoint, we introduce two complementary such methods. The first, Pauli-path zero-noise extrapolation, fits a parameterized model to a circuit's reactivity function and uses it to accurately extrapolate to zero noise. The second, tunable error cancellation, filters a circuit's reactivity function to cancel noise on all information up to a threshold locality. Crucially, both methods feature a tuning parameter that allows one to systematically improve their accuracy until convergence. Across three circuit families from published experiments, our methods overcome conventional zero-noise extrapolation biases as large as 30%; at ∼\sim∼10−210^{-2}10−2 root-mean-square error, they reduce sampling cost relative to probabilistic error cancellation by up to a factor of 10310^3103.
The classical simulation of highly entangling quantum dynamics is conjectured to be generically hard. Thus, recently discovered measurement-induced transitions between highly entangling and low-entanglement dynamics are phase transitions in classical simulability. Here, we study simulability transitions beyond entanglement: noting that some highly entangling dynamics (e.g., integrable systems or Clifford circuits) are easy to classically simulate, thus requiring “magic”—a subtle form of quantum resource—to achieve computational hardness, we ask how the dynamics of magic competes with measurements. We study the resulting “dynamical magic transitions” focusing on random monitored Clifford circuits doped by T gates (injecting magic). We identify dynamical “stabilizer purification”—the collapse of a superposition of stabilizer states by measurements—as the mechanism driving this transition. We find cases where transitions in magic and entanglement coincide, but also others with a magic and simulability transition in a highly (volume-law) entangled phase. In establishing our results, we use Pauli-based computation, a scheme distilling the quantum essence of the dynamics to a magic state register subject to mutually commuting measurements. We link stabilizer purification to “magic fragmentation” wherein these measurements separate into disjoint, O ( 1 ) -weight blocks, and relate this to the spread of magic in the original circuit becoming arrested. Published by the American Physical Society 2024
The interface between the quantum and the classical is an intriguing and, at times, hotly contested subject of ongoing research. The quantum regime is characterized by interference, made possible by the superposition principle, while such phenomena are absent in macroscopic, everyday experience. Here, we investigate the link of this absence (or, as we will argue, unobservability) to computational complexity. We show how the assumption that quantum systems cannot solve NP-complete problems efficiently implies that certain formally valid quantum measurements on finite-dimensional systems are unperformable. We study several consequences of this restriction. First, Pauli matrices in an inconveniently transformed basis are a simple example of unobservables. Furthermore, some quantum states are not connected by any physically realizable time evolution. Finally there are quantum states whose coherence cannot be observed, i.e. superpositions of pure quantum states which are indistinguishable from mixtures. We discuss the connection of this phenomenon to the presence of superselection sectors. Our results suggest that the apparent classicality of macroscopic systems may be partly due to limitations on measurements and time evolutions imposed by computational complexity.