Entanglement is the defining characteristic of quantum mechanics. Bipartite entanglement is characterized by the von Neumann entropy. Entanglement is not just described by a number, however; it is also characterized by its level of complexity. The complexity of entanglement is at the root of the onset of quantum chaos, universal distribution of entanglement spectrum statistics, hardness of a disentangling algorithm and of the quantum machine learning of an unknown random circuit, and universal temporal entanglement fluctuations. In this paper, we numerically show how a crossover from a simple pattern of entanglement to a universal, complex pattern can be driven by doping a random Clifford circuit with T gates. This work shows that quantum complexity and complex entanglement stem from the conjunction of entanglement and non-stabilizer resources, also known as magic.
Entanglement is the defining characteristic of quantum mechanics. Bipartite entanglement is characterized by the von Neumann entropy. Entanglement is not just described by a number, however; it is also characterized by its level of complexity. The complexity of entanglement is at the root of the onset of quantum chaos, universal distribution of entanglement spectrum statistics, hardness of a disentangling algorithm and of the quantum machine learning of an unknown random circuit, and universal temporal entanglement fluctuations. In this paper, we numerically show how a crossover from a simple pattern of entanglement to a universal, complex pattern can be driven by doping a random Clifford circuit with T gates. This work shows that quantum complexity and complex entanglement stem from the conjunction of entanglement and non-stabilizer resources, also known as magic.
Abstract Entanglement entropy has long served as a central diagnostic of complexity and information scrambling in quantum many-body systems, and its behavior in random quantum circuits is by now well characterized. However, as a single scalar, entanglement entropy captures only the total uncertainty of the reduced density matrix spectrum while discarding the detailed structure of that spectrum — including the shape of the eigenvalue distribution, level repulsion signatures, and the transition from integrable to chaotic behavior. In this work, we propose spectral entropy — the Shannon entropy of the eigenvalue distribution of the reduced density matrix — as a structured diagnostic that goes beyond standard entanglement entropy. Using numerical simulations, we compute the spectral entropy of subsystem reduced states in Haar-random and Clifford circuits as a function of circuit depth, and compare its behavior against von Neumann entanglement entropy, Rényi-2 entropy, and entanglement spectrum level-spacing statistics. We expect to find three key results: (i) spectral entropy exhibits a growth timescale distinct from that of entanglement entropy, potentially reflecting spectral restructuring more sensitively; (ii) in the depth regime where entanglement entropy has already saturated, spectral entropy may continue to evolve, revealing late-time complexification that standard entanglement measures cannot resolve; (iii) spectral entropy may display a distinctive response to measurement-induced or noise-induced disentanglement that differs qualitatively from entanglement entropy. This work positions spectral entropy as a complementary — not replacement — tool to entanglement entropy, offering a finer spectral lens on information scrambling, thermalization, and complexity growth in random quantum circuits.
Universal features of chaotic quantum dynamics underlie our understanding of thermalization in closed quantum systems and the complexity of quantum computations. Reversible automaton circuits, comprised of classical logic gates, have emerged as a tractable means to study such dynamics. Despite generating no entanglement in the computational basis, these circuits nevertheless capture many features expected from fully quantum evolutions. In this work, we demonstrate that the differences between automaton dynamics and fully quantum dynamics are revealed by the operator entanglement spectrum, much like the entanglement spectrum of a quantum state distinguishes between the dynamics of states under Clifford and Haar random circuits. While the operator entanglement spectrum under random unitary dynamics is governed by the eigenvalue statistics of random Gaussian matrices, we show evidence that under random automaton dynamics it is described by the statistics of Bernoulli random matrices, whose entries are random variables taking values 0 or 1 . We study the crossover between automaton and generic unitary operator dynamics as the automaton circuit is doped with gates that introduce superpositions, namely Hadamard or R x = e − i π 4 X gates. We find that a constant number of superposition-generating gates is sufficient to drive the operator dynamics to the random-circuit universality class, similar to earlier results on Clifford circuits doped with T gates. This establishes the operator entanglement spectrum as a useful tool for probing the chaoticity and universality class of quantum dynamics.
We study the emergence of complexity in deep random N -qubit T -gate-doped Clifford circuits, as reflected in their spectral properties and in magic generation, characterized by the stabilizer Rényi entropy distribution and the nonstabilizing power of the circuit. For pure (undoped) Clifford circuits, a unique periodic orbit structure in the space of Pauli strings implies peculiar spectral correlations and level statistics with large degeneracies. T -gate doping induces an exponentially fast transition to chaotic behavior, described by random matrix theory. We compare these complexity indicators with magic generation properties of the Clifford + T ensemble, and determine the distribution of magic, as well as the average nonstabilizing power of the quantum circuit ensemble. In the dilute limit, N T ≪ N , magic generation is governed by single-qubit behavior. Magic is generated in approximate quanta, increases approximately linearly with the number of T gates, N T , and displays a discrete distribution for small N T . At N T ≈ N , the distribution becomes quasicontinuous, and for N T ≫ N , it converges to that of Haar-random unitaries, and averages to a finite magic density, m 2 , lim N → ∞ 〈 m 2 〉 Haar = 1 . This is in contrast to the spectral transition, where O ( 1 ) T gates suffice to remove spectral degeneracies and to induce a transition to chaotic behavior in the thermodynamic limit. Magic is therefore a more sensitive indicator of complexity.
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
We investigate the emergence of quantum complexity and chaos in doped Clifford circuits acting on qudits of odd prime dimension d . Using doped Clifford Weingarten calculus and a replica tensor network formalism, we derive exact results and perform large-scale simulations in regimes challenging for tensor network and Pauli-based methods. We begin by analyzing generalized stabilizer entropies, computable magic monotones in many-qudit systems, and identify a dynamical phase transition in the doping rate, marking the breakdown of classical simulability and the onset of Haar-random behavior. The critical behavior is governed by the qudit dimension and the magic content of the non-Clifford gate. Using the qudit T -gate as a benchmark, we show that higher-dimensional qudits converge faster to Haar-typical stabilizer entropies. For qutrits ( d = 3 ), analytical predictions match numerics on brickwork circuits, showing that locality plays a limited role in magic spreading. We also examine anticoncentration and entanglement growth, showing that O ( log N ) non-Clifford gates suffice for approximating Haar expectation values to precision ε , and relate antiflatness measures to stabilizer entropies in qutrit systems. Finally, we analyze out-of-time-order correlators and show that a finite density of non-Clifford gates is needed to induce chaos, with a sharp transition fixed by the local dimension, twice that of the magic transition. Altogether, these results establish a unified framework for diagnosing complexity in doped Clifford circuits and deepen our understanding of resource theories in multiqudit systems.
Entanglement and non-stabilizerness are complementary resources underlying the complexity of quantum states, yet extracting either quantity from large quantum circuits generally requires exponentially scaling resources. In diagrammatic approaches to quantum computation, such as the ZX-calculus, both quantities likewise typically require expensive tensor contractions that are not native to graphical rewriting. Here we show that both entanglement and magic can be efficiently estimated directly from ZX-diagrams. Flow enables efficient extraction of a normal form separating a potentially extensively entangled graph-state backbone from non-Clifford Pauli gadgets. This structure yields additive upper and lower bounds on bipartite entanglement entropy, which we further tighten through a preprocessing procedure that removes or merges redundant non-Clifford contributions. The same normal form gives an upper bound on logarithmic stabilizer extent. We benchmark these methods in random unitary and monitored circuits, and a circuit combining Trotterized Hamiltonian evolution with Clifford layers, finding that the resulting bounds remain informative even at large qubit numbers, circuit depths, and internal spider counts. Our results provide scalable probes of entanglement and non-stabilizerness of ZX-diagrams and establish a bridge between ZX-calculus and quantum many-body resource characterization, with applications across quantum computing, quantum information, and condensed-matter physics.