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Simulering av mätningsinducerade övergångar i sammanflätningsentropi

Evelina Stenseth

Published Jan 1, 2026• 0 citations
OpenAlex

Abstract

Measurement-induced entanglement transitions arise in quantum circuits of unitary gates and measurements, referred to as hybrid circuits. On average, unitary dynamics generate entanglement, and measurements suppress entanglement. In one-dimensional hybrid circuits, this competition leads to a transition between an area-law phase when the probability of measurements is higher, and a volume-law entangled phase when the probability is lower. In this thesis, the phase transition is studied using two different approaches. First, state vector simulations are performed for hybrid circuits composed of Haar-random unitary gates and random Clifford gates. This allows for direct access to the state vectors and entanglement entropy, but is only feasible for smaller size systems (up to 12 qubits). Second, a traffic flow model is implemented as an approximation to stabiliser dynamics of Clifford circuits, which is studied for larger systems (up to 2048 qubits). Scaling analyses are carried out for both approaches to investigate the dependence of the steady-state entropy on system size and measurement probability. Data collapses of simulation data for different circuit sizes are obtained using both a logarithmic form S−αlnN and by also including an additional correction S−αlnN+βN^(−ω) where ω = 1, leading to a more accurate collapse for smaller sizes. Both the state vector model and traffic flow model lead to a phase transition in entanglement entropy, but with different critical parameters. This indicates that the traffic flow model and the quantum circuit models do not belong to the same universality class. The results clarify the efficiency but also limitations of classical models for describing measurement-induced entanglement transitions.

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•Jan 1, 2025•0 citations
Measurement-induced entanglement transitions arise in quantum circuits of unitary gates and measurements, referred to as hybrid circuits. On average, unitary dynamics generate entanglement, and measurements suppress entanglement. In one-dimensional hybrid circuits, this competition leads to a transition between an area-law phase when the probability of measurements is higher, and a volume-law entangled phase when the probability is lower. In this thesis, the phase transition is studied using two different approaches. First, state vector simulations are performed for hybrid circuits composed of Haar-random unitary gates and random Clifford gates. This allows for direct access to the state vectors and entanglement entropy, but is only feasible for smaller size systems (up to 12 qubits). Second, a traffic flow model is implemented as an approximation to stabiliser dynamics of Clifford circuits, which is studied for larger systems (up to 2048 qubits). Scaling analyses are carried out for both approaches to investigate the dependence of the steady-state entropy on system size and measurement probability. Data collapses of simulation data for different circuit sizes are obtained using both a logarithmic form S−αlnN and by also including an additional correction S−αlnN+βN^(−ω) where ω = 1, leading to a more accurate collapse for smaller sizes. Both the state vector model and traffic flow model lead to a phase transition in entanglement entropy, but with different critical parameters. This indicates that the traffic flow model and the quantum circuit models do not belong to the same universality class. The results clarify the efficiency but also limitations of classical models for describing measurement-induced entanglement transitions.
We introduce and explore a one-dimensional ``hybrid'' quantum circuit model consisting of both unitary gates and projective measurements. While the unitary gates are drawn from a random distribution and act uniformly in the circuit, the measurements are made at random positions and times throughout the system. By varying the measurement rate we can tune between the volume law entangled phase for the random unitary circuit model (no measurements) and a ``quantum Zeno phase'' where strong measurements suppress the entanglement growth to saturate in an area law. Extensive numerical simulations of the quantum trajectories of the many-particle wave functions (exploiting Clifford circuitry to access systems up to 512 qubits) provide evidence for a stable ``weak measurement phase'' that exhibits volume-law entanglement entropy, with a coefficient decreasing with increasing measurement rate. We also present evidence for a continuous quantum dynamical phase transition between the ``weak measurement phase'' and the ``quantum Zeno phase,'' driven by a competition between the entangling tendencies of unitary evolution and the disentangling tendencies of projective measurements. Detailed steady-state and dynamic critical properties of this quantum entanglement transition are accessed.
Monitored random circuits, consisting of alternating layers of entangling two-qubit gates and projective single-qubit measurements applied to some fraction p of the qubits, have been a topic of recent interest. In particular, the resulting steady state exhibits a phase transition from highly correlated states with "volume-law" entanglement at p<p_{c} to localized states with "area-law" entanglement at p>p_{c}. It is hard to access this transition experimentally, as it cannot be seen at the ensemble level. Naively, to observe it one must repeat the experiment until the set of measurement results repeats itself, with likelihood that is exponentially small in the number of measurements. To overcome this issue, we present a hybrid quantum-classical algorithm which creates a matrix product state (MPS) based "unitary mirror" of the projected circuit. Polynomial-sized tensor networks can represent quantum states with area-law entanglement, and so the unitary mirror can well approximate the experimental state above p_{c} but fails exponentially below it. The breaking of this mirror can thus pinpoint the critical point. We outline the algorithm and how such results would be obtained. We present a bound on the maximum entanglement entropy of any given state that is well represented by an MPS, and from the bound suggest how the volume-law phase can be bounded. We consider whether the entanglement could similarly be bounded from below where the MPS fails. Finally, we present numerical results for small qubit numbers and for monitored circuits with random Clifford gates.
We investigate the dynamics of two-dimensional quantum spin systems under the combined effect of random unitary gates and local projective measurements. When considering steady states, a measurement-induced transition occurs between two distinct dynamical phases, one characterized by a volume-law scaling of entanglement entropy, the other by an area law. Employing stabilizer states and Clifford random unitary gates, we numerically investigate square lattices of linear dimension up to L=48L=48L=48 for two distinct measurement protocols. For both protocols, we observe a transition point where the dominant contribution in the entanglement entropy displays multiplicative logarithmic violations to the area law. We obtain estimates of the correlation length critical exponent at the percent level; these estimates suggest universal behavior and are incompatible with the universality class of 3D percolation.
Repeated projective measurements in unitary circuits can lead to an entanglement phase transition as the measurement rate is tuned. In this work, we consider a different setting in which the projective measurements are replaced by dynamically chosen unitary gates that minimize the entanglement. This can be seen as a one-dimensional unitary circuit game in which two players get to place unitary gates on randomly assigned bonds at different rates: the “entangler” applies a random local unitary gate with the aim of generating extensive (volume-law) entanglement. The “disentangler,” based on limited knowledge about the state, chooses a unitary gate to reduce the entanglement entropy on the assigned bond with the goal of limiting to only finite (area-law) entanglement. In order to elucidate the resulting entanglement dynamics, we consider three different scenarios: (i) a classical discrete height model, (ii) a Clifford circuit, and (iii) a general U(4) unitary circuit. We find that both the classical and Clifford circuit models exhibit phase transitions as a function of the rate that the disentangler places a gate, which have similar properties that can be understood through a connection to the stochastic Fredkin chain. In contrast, the entangler always wins when using Haar random unitary gates and we observe extensive, volume-law entanglement for all nonzero rates of entangling. Published by the American Physical Society 2024
We explore the dynamical phases of unitary Clifford circuits with variable-range interactions, coupled to a monitoring environment. We investigate two classes of models, distinguished by the action of the unitary gates, which either are organized in clusters of finite-range two-body gates, or are pair-wise interactions randomly distributed throughout the system with a power-law distribution. We find the range of the interactions plays a key role in characterizing both phases and their measurement-induced transitions. For the cluster unitary gates we find a transition between a phase with volume-law scaling of the entanglement entropy and a phase with area-law entanglement entropy. Our results indicate that the universality class of the phase transition is compatible to that of short range hybrid Clifford circuits. Oppositely, in the case of power-law distributed gates, we find the universality class of the phase transition changes continuously with the parameter controlling the range of interactions. In particular, for intermediate values of the control parameter, we find a non-conformal critical line which separates a phase with volume-law scaling of the entanglement entropy from one with sub-extensive scaling. Within this region, we find the entanglement entropy and the logarithmic negativity present a cross-over from a phase with algebraic growth of entanglement with system size, and an area-law phase.
Random quantum circuits provide a framework for probing universal dynamical phenomena in chaotic quantum many-body systems, by circumventing the microscopic details of Hamiltonian evolution. Bipartite entanglement measures can be used to characterize the dynamical phases that emerge in these systems. Typically, under random unitary evolution, bipartite entanglement entropies exhibit a linear growth, and attain steady-state values that scale extensively with the system volume. However, when local projective measurements are introduced, a suppression of steady-state bipartite entropies occurs, leading to a sub-extensive scaling with the system volume. This class of entanglement scaling transitions, from an extensive to a sub-extensive scaling regime, has sparked interest in a broader class of measurement-induced phase transitions, with a potential to further our understanding of fault-tolerance in quantum computing, and in realizing synthetic quantum phases on digital quantum simulators. In this work, we characterize entanglement scaling transitions in one-dimensional random quantum circuits composed of q-level systems, with the local Hilbert space dimension q ≥ 2. This predominantly numerical study employs generalizations of the stabilizer formalism and the Clifford group to higher-dimensional Hilbert spaces to simulate the measurement-induced transitions. We find that the nature of the entanglement scaling transition remains invariant under changes in the local Hilbert space dimension. However, the critical measurement rate inducing the transition increases with q, asymptotically approaching 1/2 in the limit q → ∞. To further explore this classical limit, we develop a minimal dynamical model that captures entanglement structures under unitary evolution. We analyze the entanglement structures in thermal stabilizer states using this model, and propose an approach for incorporating measurements within this framework.