Measurement-and Feedback-Driven Non-Equilibrium Phase Transitions on a Quantum Processor

TL;DR

The paper tackles how mid-circuit measurements and real-time feedback in adaptive quantum circuits can drive non-equilibrium phase transitions in quantum many-body dynamics. It introduces a flag-based adaptive circuit implemented on a superconducting processor, enabling both an absorbing-state transition in the quantum channel and a measurement-induced entanglement transition, governed by the measurement rate $p$. Critical behavior is characterized by DP universality with exponents $z \approx 1.58$, $\Theta \approx 0.31$, and $\alpha \approx -0.16$, with a separation between the absorbing-state critical point $p_c^{\rm abs} \approx 0.35$ and the MIPT point $p_c^{\rm MIPT} \approx 0.20$. The work demonstrates adaptive circuits as a powerful platform for exploring stochastic quantum dynamics and informs practical avenues for active quantum error correction and information flow control on programmable quantum devices.

Abstract

Mid-circuit measurements and feedback operations conditioned on the measurement outcomes are essential for implementing quantum error-correction on quantum hardware. When integrated in quantum many-body dynamics, they can give rise to novel non-equilibrium phase transitions both at the level of each individual quantum trajectory and the averaged quantum channel. Experimentally resolving both transitions on realistic devices has been challenging due to limitations on the fidelity and the significant latency for performing mid-circuit measurements and feedback operations in real time. Here, we develop a superconducting quantum processor that enables global mid-circuit measurement with an average quantum non-demolition (QND) fidelity of 98.7% and fast conditional feedback with a 200 ns real-time decision latency. Using this platform, we demonstrate the coexistence of an absorbing-state transition in the quantum channel and a measurement-induced entanglement transition at the level of individual quantum trajectories. For the absorbing-state transition, we experimentally extract a set of critical exponents at the transition point, which is in excellent agreement with the directed percolation universality class. Crucially, the two transitions occur at distinct values of the tuning parameter. Our results demonstrate that adaptive quantum circuits provide a powerful platform for exploring non-equilibrium quantum many-body dynamics.

Figures (12)

• Figure 1: Quantum processor and adaptive quantum circuit for measurement-feedback dynamics.a, The superconducting quantum processor used in this work. The device hosts 66 transmon qubits arranged in a $6 \times 11$ lattice, with 30 qubits (highlighted in blue) selected for the experiment to minimize crosstalk and enable high-fidelity mid-circuit operations. b, Schematics of the adaptive quantum circuit, consisting of alternating layers of two-qubit unitaries and mid-circuit measurements followed by real-time conditional feedback. The two-qubit unitaries are implemented using iSWAP-like entanglers dressed by random equatorial $\pi/2$ phase rotations (denoted by $U$). c, Phase diagram illustrating the measurement-and feedback-driven non-equilibrium phase transitions as the measurement rate $p$ is varied. d, Illustration for the experimentally executed sequence of the adaptive circuit for the initial state where all sites are occupied. During the evolution, the classical flag variables dynamically track active and inactive qubits under the adaptive feedback protocol. Each labeled step (i, ii, iii, …) consists of a unitary-evolution layer followed by a measurement–feedback layer. e, Cumulative error distributions for synchronized SQG, iSWAP, and mid-circuit RO operations, with median error rates of 0.07%, 0.7%, and 1.3%, respectively.
• Figure 2: Absorbing-state transition in the adaptive quantum circuit.a, Numerical simulation. Spatiotemporal profile of the local occupation $\langle n_i(t)\rangle$ for different measurement rates $p$ via tDMRG simulations. We choose the initial state with a single occupied site in the middle $|\psi_0\rangle=|0\cdots010\cdots 0\rangle$. Infrequent measurement ($p=0.10$) yields a ballistically spreading active cluster, while sufficiently frequent measurement ($p=0.45$) rapidly drives the system into the absorbing state with no particle. At the critical point ($p_c^{\rm abs}\simeq0.35$), the spreading becomes sub-ballistic with a lightcone satisfying $|r|\sim t^{1/z}$ with a dynamical exponent $z\approx 1.58$, consistent with DP universality class. b, Experiment. Measured spatiotemporal profile of $\langle n_i(t) \rangle$ on the 30-qubit processor shows excellent agreement with numerical simulations. Each figure was obtained by averaging over 100 random circuit instances and 10,000 trajectories for each circuit.
• Figure 3: Critical exponents in directed percolation dynamics.a, Time evolution of the average total particle number $N(t)$ for the initial state with a single occupied site in the middle $|\psi_0\rangle=|0\cdots010\cdots 0\rangle$. b, Time evolution of $N(t)$ for the fully occupied initial state $|\psi_0\rangle = |11\ldots 1\rangle$. In both panels, results are shown for three representative feedback rates: $p=0.10$ (active phase), $p\approx 0.35$ (critical point), and $p=0.45$ (absorbing phase). Colored symbols represent experimental data (averaged over 100 random circuit instances), and black crosses denote numerical simulations. Error bars indicate one standard error of the mean, obtained via bootstrap resampling, and are smaller than the symbol size when not visible. The fit for the critical exponents $\Theta$, $\alpha$ is most reliable for $t < 20$, with deviations at later times due to finite-size effects.
• Figure 4: Measurement-induced entanglement transition.a, Eight-qubit hybrid circuit used to probe the measurement-induced entanglement transition. It preserves the alternating unitary–measurement structure of the 30-qubit architecture, with idle qubits protected by dynamical decoupling during measurement windows. Mid-circuit measurement outcomes are recorded, and the reduced density matrix $\rho_A$ on a three-qubit subsystem $A$ conditioning on the measurement outcome is obtained via quantum state tomography. b, Average second Rényi entropy $\langle S^{(2)}_A \rangle$ versus measurement rate $p$. Experimental data (dots), corrected for readout errors and residual offsets, agree with numerical simulations (solid line) and show a monotonic suppression of entanglement with increasing $p$. c, Variance of $S^{(2)}_A$$\mathrm{Var}\!(S^{(2)}_A)$ across random circuit instances. The variance exhibits a clear peak near $p \approx 0.20$, which we identify as the measurement-induced entanglement transition. Error bars represent 90% bootstrap confidence intervals.
• Figure 5: Scaling of entanglement entropy of the steady states. Numerical tDMRG simulation of the averaged von-Neumann entanglement entropy $\langle S^{(1)}_A \rangle$ (a,c) and the second Rényi entropy $\langle S^{(2)}_A\rangle$ (b,d) as a function of subsystem size $|A|$, for the same 30-qubit quantum circuits used experimentally for probing absorbing-state transition. a,b, measurement rate $p=0.3$. c,d, measurement rate $p=0.35$. The entanglement entropy exhibits area-law scaling already at $p=0.30<p_c^{\rm abs}$. The results for different choices of bond dimension $\chi=64$ and $\chi=128$ are almost identical, confirming numerical convergence.