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arXiv:2606.30396

Provable Quantum Advantage for Dynamical Phase Transition

Jue Xu, Xiao Yuan, Qi Zhao

quant-phSubmitted Jun 29, 2026• Announced Jun 30, 2026• 0 citations
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Abstract

The universal scaling of critical behavior in phase transitions is a cornerstone of physics. Dynamical quantum phase transitions (DQPTs) are their nonequilibrium analogues: abrupt nonanalyticities that emerge as a quantum system evolves in time. Yet the hardness and cost of detecting this phenomenon remain largely unexplored. We prove that estimating DQPT to a certain precision is intractable even for quantum computers, whereas deciding a subsystem variant of DQPT is as hard as simulating generic quantum circuits, implying a provable exponential quantum advantage. Furthermore, to search for critical times of local DQPTs, we show a quadratically faster quantum algorithm that estimates observables of Hamiltonian dynamics at multiple time points with Heisenberg-limited precision and sublinear scaling in the number of time points. Moreover, through encoding classical evolution into quantum dynamics, our framework enables broader quantum speedups for detecting anomalous phenomena in classical systems.

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The universal scaling of critical behavior in phase transitions is a cornerstone of physics. Dynamical quantum phase transitions (DQPTs) are their nonequilibrium analogues: abrupt nonanalyticities that emerge as a quantum system evolves in time. Yet the hardness and cost of detecting this phenomenon remain largely unexplored. We prove that estimating DQPT to a certain precision is intractable even for quantum computers, whereas deciding a subsystem variant of DQPT is as hard as simulating generic quantum circuits, implying a provable exponential quantum advantage. Furthermore, to search for critical times of local DQPTs, we show a quadratically faster quantum algorithm that estimates observables of Hamiltonian dynamics at multiple time points with Heisenberg-limited precision and sublinear scaling in the number of time points. Moreover, through encoding classical evolution into quantum dynamics, our framework enables broader quantum speedups for detecting anomalous phenomena in classical systems.
Dynamical quantum phase transitions (DQPTs) at critical times appear as nonanalyticities during nonequilibrium quantum real-time evolution. Although there is evidence for a close relationship between DQPTs and equilibrium phase transitions, a major challenge is still to connect to fundamental concepts such as scaling and universality. In this work, renormalization group transformations in complex parameter space are formulated for quantum quenches in Ising models showing that the DQPTs are critical points associated with unstable fixed points of equilibrium Ising models. Therefore, these DQPTs obey scaling and universality. On the basis of numerical simulations, signatures of these DQPTs in the dynamical buildup of spin correlations are found with an associated power-law scaling determined solely by the fixed point's universality class. An outlook is given on how to explore this dynamical scaling experimentally in systems of trapped ions.
Dynamical quantum phase transitions (DQPTs) are non-equilibrium transitions characterized by the orthogonality between an initial quantum state and its time-evolved counterpart following a sudden quench. Recently, studies of this phenomenon have been extended beyond closed quantum systems to include environmental interactions, often modeled through non-Hermitian effects. However, because non-Hermitian descriptions neglect both quantum jump processes and interaction effects, the ultimate fate of interacting quantum systems under full open-system quantum dynamics remains an open question. In this paper, by incorporating both interactions and full Liouvillian dynamics, we prove that DQPTs in open quantum systems remain robust when subject to either particle loss or particle gain alone, but are generically smeared out when both processes coexist, as a result of many-body particle backflow. Furthermore, we uncover a non-perturbative dynamical effect: even a weak admixture of gain (loss) into a system with loss (gain) can dramatically reshape the long-time behavior of DQPT dynamics, leading to substantial deviations over time. These phenomena--including the universal smearing of DQPTs and the emergence of large dynamical deviations in the long-time limit--arise intrinsically from non-equilibrium many-body effects in open quantum systems. Our findings are general and substantiated by both analytical arguments and numerical simulations.
Considerable theoretical and experimental efforts have been devoted to the quench dynamics, in particular, the dynamical quantum phase transition (DQPT) and the steady-state transition. These developments have motivated us to study the quench dynamics of the topological systems, from which we find the connection between these two transitions, that is, the DQPT, accompanied by a nonanalytic behavior as a function of time, always merges into a steady-state transition signaled by the nonanalyticity of observables in the steady limit. As the characteristic time of the DQPT diverges, it exhibits universal scaling behavior, which is related to the scaling behavior at the corresponding steady-state transition.
The dynamics in the vicinity of quantum phase transitions entails a variety of important and universal phenomena, such as critical slowing down or the Kibble-Zurek mechanism, which are associated with the low-energy and long-time properties at criticality. Here, the author reveals that the opposite regime of transient nonequilibrium time evolution at criticality can similarly host universal temporal behavior. By strongly perturbing a quantum critical state through quenching the associated order parameter, the system experiences a dynamical quantum phase transition with physical quantities becoming nonanalytic at critical times. These dynamical quantum phase transitions are caused by superextensive energy fluctuations induced by the particular protocol, which turn out to have profound consequences for quantum speed limits and potential restricted thermalization despite the nonintegrability of the system.
Phase transitions have recently been formulated in the time domain of quantum many-body systems, a phenomenon dubbed dynamical quantum phase transitions (DPTs), whose phenomenology is often divided in two types. One refers to distinct phases according to long-time averaged order parameters, while the other is focused on the nonanalytical behavior emerging in the rate function of the Loschmidt echo. Here we show that such DPTs can be found in systems with few degrees of freedom; i.e., they can take place without resorting to the traditional thermodynamic limit. We illustrate this by showing the existence of the two types of DPTs in a quantum Rabi model---a system involving a spin-12\frac{1}{2}21​ and a bosonic mode. The dynamical criticality appears in the limit of an infinitely large ratio of the spin frequency with respect to the bosonic one. We determine its dynamical phase diagram and study the long-time averaged order parameters, whose semiclassical approximation yields a jump at the transition point. We find the critical times at which the rate function becomes nonanalytical, showing its associated critical exponent as well as the corrections introduced by a finite frequency ratio. Our results open the door for the study of DPTs without the need to scale up the number of components, thus allowing for their investigation in well controllable systems.
Dynamical quantum phase transitions (DQPTs) occur at times when a quantum state exhibits a nonanalytic change in its return probability. This can be viewed as the probability of collapsing the evolved state to the initial state by quantum measurement. However, the initial wave function usually has exponentially small amplitude in the late time evolved state. Here we perform statistical characterization for all the possible post-measurement states distributed according to the Born's rule, by sampling a one-dimensional quantum Ising chain after a quantum quench dynamics. The statistical ensemble can also be viewed as a mixed state when the time evolved state is subjected to maximally dephasing noise in a certain basis. We map the distribution to a statistical model and characterize its effective "energy" spectrum, and introduce the average dynamical free energy, establishing a framework for the statistical DQPTs. We show the recovering of DQPT under high-moment average and a delocalized level distribution following critical times. Through analytic continuation into the complex time plane, we demonstrate the vanishing of Yang-Lee-Fisher zeros and the emergent level crossing near critical times. Finally, we propose a measurement-based quantum computation protocol to simulate the unitary evolution via single-qubit measurements on a two-dimensional cluster state. Our results provide a way for experimentally investigating statistical DQPTs in quantum devices, shedding light on the structured circuit sampling with insights from DQPT and generalizing the understanding of mixed state due to decoherence beyond equilibrium.