Glassy dynamics with softened kinetic constraints on a noisy quantum computer

Introduction.— Quantum devices are attracting significant interest due to their potential to simulate large quantum many-body systems that would otherwise be inaccessible with classical computers 28; 62; 35; 45; 37. Mid-circuit measurements extend this paradigm beyond purely unitary dynamics, enabling the exploration of various facets of non-equilibrium quantum matter 3; 20; 56; 72; 22; 58. These operations not only influence the dynamics, but the associated outcomes—the quantum trajectories—reveal space-time resolved information about the many-body dynamics 6; 74; 73; 67. This perspective has already become a powerful tool for studying quantum-to-classical transitions 57; 38, entanglement phase transitions 50; 34; 2; 27, and quantum-enhanced sensing protocols 54; 63. The combination of mid-circuit measurements and feedback operations is furthermore a central aspect of ongoing efforts in quantum error correction 21; 65; 48.

A particularly interesting use case for mid-circuit measurements is the study of dissipative quantum systems exhibiting glassy dynamics, reminiscent of those found in kinetically constrained models 64; 10; 33; 5. In classical models, a hallmark of glassiness is dynamical heterogeneity, which is observable in individual stochastic realizations of the many-body dynamics. Here, different regions in space relax at vastly different rates due to a highly correlated approach towards stationarity. This occurs despite the fact that the stationary state itself may be entirely structureless 41; 31; 55; 40; 9; 43; 13; 46; 23; 68. Their quantum counterparts exhibit slow dynamics, non-thermal eigenstates and Hilbert space fragmentation 61; 66; 71; 7; 4, which can be further enriched by the interplay of coherent and dissipative processes 60; 11; 52; 53; 6. On classical computers, studying the ensuing quantum many-body dynamics is only possible with approximate methods, e.g. based on tensor networks  12; 15; 14; 16, that truncate the exponentially large state space. Quantum hardware, on the other hand, implements quantum dynamics and the associated state space natively. However, the current generation of quantum processors still suffers from ubiquitous noise, yielding random errors. Interestingly, these devices enable the investigation of highly nontrivial collective quantum many-body phenomena when one accepts this noise as part of the dynamics 24; 49; 36; 70.

In this work, we show this exemplarily for a Floquet-East model realized on the ibm_kingston quantum computer [cf. Fig. 1(a,b)]. Here, a set of ancilla qubits is coupled to a chain of system qubits that evolve under kinetically constrained dynamics. The hardware noise predominantly gives rise to random state changes of individual qubits. In the context of glassy dynamics studied here, such processes effectively determine a softening of the kinetic constraint 69; 25; 26. Monitoring the system dynamics takes place by periodically measuring the state of the ancillas [cf. Fig. 1(b)]. Gathering the binary measurement outcomes yields intricate space-time trajectories, an example of which is shown in Fig. 1(c). These trajectories display dynamical heterogeneity, reminiscent of glassy systems, which we characterize by analyzing the probability of finding an inactive cluster of a given spatial and temporal extent consisting exclusively of ‘0’ measurement outcomes [see Fig. 1(c) for an example]. From this probability, we define a dynamical free energy and show that it exhibits a crossover from an area- to perimeter-dominated scaling as the size of the inactive cluster increases. This behavior is the consequence of the competition between entropic and energetic contributions near a dynamical first-order phase transition. Our work thus highlights the utility of trajectory-level observables for investigating collective phenomena in quantum many-body systems on present-day quantum processors.

The model.— We implement the Floquet-East model on the heavy-hex topology of the ibm_kingston processor as shown in Fig. 1(a). A set of $L=48$ system (S) qubits is arranged in a 1D spin chain with periodic boundary conditions, where each qubit is described by the computational basis $\{\ket{0},\ket{1}\}$. Additionally, we include $L/2=24$ ancilla (A) qubits that are adjacent to every second system qubit and which are used to monitor their dynamics 16. The Floquet-East model that we consider here is then described by the quantum circuit shown in Fig. 1(b).

The elementary building block is a controlled single-qubit rotation that is implemented by the unitary

Here, $n=\outerproduct{1}{1}$ and $\sigma^{x}=\outerproduct{0}{1}+\mathrm{h.c.}$ denote the excitation number and the Pauli-X operators, respectively, and $\omega$ is the rotation angle. The gate in Eq. (1) is applied on all even and odd bonds of the spin chain in a brickwork structure. Notably, it only acts non-trivially on the system qubit at site $j$ if the control qubit at site $j-1$ is in state $\ket{1}$. This realizes the so-called facilitation condition, which is the kinetic constraint that governs the dynamics of the East model 41; 4.

The ancilla qubits are employed to monitor the excitation number. This is achieved by preparing the ancilla qubits in state $\ket{0_\mathrm{A}}$ and coupling them to the system qubits via the same gate as in Eq. (1). Measuring the ancilla qubits then realizes a generalized measurement of the corresponding system qubits described by the Kraus operators 59; 16

Here, $\gamma$ represents the measurement strength that interpolates between projective measurements ($\gamma=\pi/2$) and no measurements at all ($\gamma=0$). The measurement outcome $k_{i}\in\{0,1\}$ is obtained with probability $\pi_{k_{i}}=\|K_{k_{i}}\ket{\psi}\|^{2}$, and the system state is updated according to $\ket{\psi}\to K_{k_{i}}\ket{\psi}/\|K_{k_{i}}\ket{\psi}\|$. This generalized Born rule emphasizes the dual nature of the monitoring protocol, which simultaneously describes and steers a single stochastic realization. The time sequence of measurement outcomes yields the measurement record $\eta=[k_{i}(t)]_{i,t}$. This so-called quantum trajectory can exhibit a rich space-time structure even though the Floquet-East model has a structureless stationary state (see the Supplemental Material 142; 47; 39; 75; 32 for details). It further allows us to consider trajectory-level observables such as the activity of a given space-time region, which is defined as the total number of ‘1’ measurement outcomes within it. Trajectories of the Floquet-East model are characterized by the presence of inactive clusters, which have zero activity [cf. Fig. 1(c)].

When implementing the above circuit dynamics on the ibm_kingston quantum processor, the influence of the inherent hardware noise must be considered. Generally, noise arises from various error sources, such as imperfect gate operations, decoherence and even correlated errors, which render the characterization of corresponding noise models an ongoing challenge 47; 39; 75. However, the processes that are most relevant for the Floquet-East model are random bit flips of individual qubits. This is because they do not respect the kinetic constraint imposed by Eq. (1). The ensuing dynamics on the quantum computer is therefore that of a spin model subject to softened kinetic constraints 69; 25; 26. In the Supplemental Material 1, we show that our observations are consistent with an effective bit-flip probability of $p_{\mathrm{flip}}\approx 0.05$.

In the following, we investigate the space-time dynamics of this system, which features a total of 72 (system and ancilla) qubits. To this end, we analyze on the order of $10^{5}$ quantum trajectories that we obtained from the ibm_kingston quantum processor in less than 10 minutes of runtime. The possibility of gathering such an amount of data on quantum many-body dynamics is key to the potential of current quantum processors to investigate collective phenomena at the level of individual trajectories.

Dynamics at the classical deterministic point.— As a first step, we consider the dynamics at the special parameter set $\omega=\pi/2$ and $\gamma=\pi/2$, which corresponds to a classical deterministic Floquet-East model 46. Here, the system state is described by a classical product state at all times, which allows for efficient classical simulations. Also, the ancilla measurement outcomes directly reflect the state of the corresponding system qubits. This is because the unitary gate in Eq. (1) is equivalent, up to a phase, to a controlled-NOT gate that implements deterministic updates between these classical states. The Kraus operators in Eq. (2) instead describe projective measurements in the computational basis that extract the binary value of the corresponding system qubit without changing its state.

In Fig. 2(a), we show the locally resolved evolution of the excitation numbers $n_{j}(t)$ for a fixed, yet otherwise random, initial state in the computational basis. A characteristic feature of the East model is the formation of triangular void regions with $n_{j}(t)=0$. These regions are a manifestation of the dynamical heterogeneity: the kinetic constraint allows for these regions to only dissolve starting from their left boundary, which renders them long-lived. Regions with an extensive number of excitations relax much faster. As ancilla measurement outcomes $k_{i}(t)$ are obtained directly from the excitation number $n_{j}(t)$ at every second qubit, they display a very similar space-time structure [cf. Fig. 2(b)]. In particular, inactive clusters of exclusively ‘0’ measurement outcomes are clearly visible and mark space-time regions with different dynamical behavior.

Running the same circuit on the ibm_kingston, we obtain measurement trajectories such as the one shown in Fig. 2(c). We see that this measurement record exhibits individual defects that arise from the hardware noise, which makes the dynamics on the quantum processor no longer deterministic. This is also clearly seen in the average over 10 000 trajectories that we show in Fig. 2(d). Although deterministic and noise-free dynamics would produce an average identical to panel (b), here, the average ancilla excitation number $\expectationvalue{k_i(t)}$ rapidly approaches a uniform value that provides no information about potential collective phenomena. However, individual trajectories still display a highly correlated space-time structure that represents the dynamical heterogeneity. This is also the case away from this special point [cf. Fig. 1(c)], where, in addition to the stochasticity arising from hardware noise, we have quantum projection noise due to the fact that the system state is no longer represented by a classical product state.

Statistics of inactive clusters in space-time.— To analyze collective behavior, we specifically focus on the statistics of inactive clusters, which are defined as rectangular regions of size $\ell\times\tau$ in the measurement record $\eta$ that contain exclusively ‘0’ measurement outcomes [cf. Fig. 1(c)]. At the trajectory level, their probability $p_{\ell\times\tau}$ is a multi-point correlation function. In an uncorrelated trajectory, the probability of finding an inactive cluster of size $\ell\times\tau$ factorizes and hence decays exponentially with the area $\ell\tau$ of the cluster. We can draw a thermodynamic analogy, where we interpret the binary measurement record $\eta$ as a microscopic configuration of a 1+1D spin system 32. This allows us to define the dynamical free energy of an inactive cluster as $F_{\ell\times\tau}=-\log p_{\ell\times\tau}$. Here, the exponential decay of the probability with the area of the cluster can be understood as an entropic free-energy cost associated with the formation of this ordered structure, which is proportional to its area $\ell\tau$. Indeed, as we show in Fig. 3(a,c) for the classical deterministic point and the monitored quantum dynamics, the dynamical free energy exhibits this area-law scaling for clusters with small temporal extensions, where the slope of the dynamical free energy is proportional to the spatial extension $\ell$ of the cluster. However, as the temporal extension $\tau$ of the cluster increases, we observe that the dynamical free energy $F_{\ell\times\tau}$ grows more slowly, and its dependence on the spatial extension $\ell$ becomes progressively weaker. Hence, the probability is no longer suppressed exponentially with the area of the cluster, but rather with its perimeter. This crossover in the scaling behavior of inactive clusters is due to the fact that trajectories look random on a small scale, but are highly correlated on a mesoscopic scale.

The phenomenology is reminiscent of the so-called hydrophobic 19; 18, or generally orderphobic 44; 51, effect in the vicinity of a first-order phase transition. In this analogy, inactive clusters play the role of solutes or ordered regions. Small inactive clusters are formed by random fluctuations and are therefore described by an entropic free-energy cost. In contrast, large inactive clusters are stabilized by the reduction in free-energy cost associated with creating an interface between the ordered and disordered phases. For quantum trajectories that display unexpectedly large inactive clusters, this means that the system is dynamically close to a first-order phase transition between an active and an inactive phase in trajectory space 43. At the phase transition, macroscopically large active and inactive space-time regions coexist and give rise to the dynamical heterogeneity, which is a hallmark of glassy dynamics. Here, the appearance of large inactive clusters merely comes at the free-energy cost associated with the interface to the active phase, which determines the perimeter-law scaling of the dynamical free energy $F_{\ell\times\tau}$. For the classical and the monitored quantum Floquet-East model with perfect kinetic constraints, this dynamical first-order phase transition is known to exist at the level of trajectories 46; 23; 16.

In our setup, we have a softening of the East-model constraint due to the inherent noise on the ibm_kingston quantum processor, which is expected to move the system away from the exact phase coexistence point 25; 26. This leads to an intricate balance between entropic and energetic contributions to the dynamical free energy $F_{\ell\times\tau}$, which we quantify by extracting the temporal surface tension, i.e., the change of the dynamical free energy upon increasing the temporal extension by one unit. Concretely, we have

where $\alpha_{\tau}$ and $\beta_{\tau}$ are fit parameters encoding the area- and perimeter-related contributions to the dynamical free energy, respectively. Fig. 3(b,d) shows the extracted values of $\alpha_{\tau}$ and $\beta_{\tau}$ for the classical deterministic point and the monitored quantum dynamics. We observe that the crossover from area- to perimeter-dominated scaling occurs very rapidly at $\tau\approx 2$ for the classical deterministic point (see the Supplemental Material 1 for the analytic derivation of the noise-free case). While the crossover is more gradual for the monitored quantum dynamics, we still find that $\beta_{\tau}$ eventually exceeds $\alpha_{\tau}$, which indicates the emergence of a perimeter-dominated scaling reminiscent of strong spatiotemporal correlations at a mesoscopic scale. Together with the observation that the area contribution, as quantified by $\alpha_{\tau}$, is non-zero, this indicates that the system is not exactly at the phase coexistence point, but sufficiently close to witness the competition between the two scalings. These observations are in line with the tensor-network-based calculations of the dynamical free energy for a softened kinetic constraint with $p_{\mathrm{flip}}=0.05$ (see Fig. 3 and the Supplemental Material 1 for details). This supports the interpretation that the dominant effect of noise on these trajectory-level observables can be understood in terms of independent bit flips, thereby demonstrating that the quantum processor is effectively simulating softened kinetic constraints.

Discussion and Outlook.— In this paper, we demonstrated that quantum trajectories obtained from mid-circuit measurements on a noisy quantum processor exhibit strong spatiotemporal correlations characteristic of dynamical heterogeneity. The observed behavior reflects a nearby first-order phase transition and the associated coexistence of active and inactive phases in trajectory space. This is a hallmark of glassy systems and shows that such complex dynamics are accessible on current quantum processors, even in the presence of noise. Looking ahead, such trajectories provide a rich dataset for further data-driven analysis. Here, machine learning-based methods, such as those proposed in Refs. 8; 30, may complement hand-crafted observables and help to reveal unexpected features in the study of glassy quantum dynamics.

SUPPLEMENTAL MATERIAL

Glassy dynamics with softened kinetic constraints on a noisy quantum computer

Marcel Cech,¹ Igor Lesanovsky,^1,2 and Federico Carollo³

¹Institut für Theoretische Physik and Center for Integrated Quantum Science and Technology,
Universität Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany
²School of Physics and Astronomy and Centre for the Mathematics
and Theoretical Physics of Quantum Non-Equilibrium Systems,
The University of Nottingham, Nottingham, NG7 2RD, United Kingdom
³Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185 Rome, Italy