Nonequilibrium steady states induced by stochastic mid-circuit measurements and resets on a quantum computer

Technical Summary: Nonequilibrium Steady States Induced by Stochastic Mid-Circuit Measurements and Resets on a Quantum Computer

Problem Statement
Stochastic resetting is a versatile protocol where a process is interrupted at random times and restarted from an initial configuration, known to drive classical and quantum systems into non-equilibrium steady states (NESS) with specific correlations. While theoretical frameworks suggest that conditional resetting—where the reset state is determined by mid-circuit measurements—can generate collective NESS in interacting quantum many-body systems, a hardware validation of these states has been lacking. Realizing such states on quantum hardware is challenging due to mid-circuit readout errors, hardware noise, and decoherence, which can corrupt the reset operation and alter the resulting stationary state. This paper addresses the gap between theoretical predictions of stochastic-resetting-induced NESS and their experimental realization on noisy intermediate-scale quantum (NISQ) devices.

Methodology
The authors formulate a noisy discrete-time theory and implement it on a superconducting quantum processor (IBM's ibm_marrakesh) for systems of up to $N=7$ qubits.

1. Theoretical Framework:

   • Unconditional Resetting: The system evolves via unitary gates $U$ or is projected to a reset state $|0\rangle$ with probability $r$. The resulting NESS density matrix $\rho_{ness}$ is derived as a statistical mixture of survival probabilities.
   • Conditional Resetting: The reset state is chosen from a set of states (specifically $|+\rangle = |\uparrow\dots\uparrow\rangle$ or $|-\rangle = |\downarrow\dots\downarrow\rangle$) based on a global projective measurement. A "majority vote" protocol determines the subsequent reset state: if the majority of measured spins are up, the system resets to $|+\rangle$; otherwise, to $|-\rangle$. This defines a semi-Markov process.
   • Noise Modeling: To account for hardware imperfections, the authors introduce a noise model where reset operations are imperfect. Instead of pure reset states, the system is initialized in a mixed state $\rho_r$ containing bit-flip errors relative to the ideal ordered states. This model groups states by the number of bit-flips ($n_{bf}$) and utilizes $Z_2$ symmetry to reduce the number of fitting parameters.

2. Model System:

   • The unitary dynamics are governed by the interacting Floquet transverse-field Ising model (FTFIM). The unitary gate $U_\theta$ consists of a transverse field term ($U_x$) and an Ising interaction term ($U_{zz}$), parametrized by an angle $\theta$.
   • The order parameter for the phase transition is the longitudinal magnetization density $m$. For unconditional resetting, $\langle m \rangle_{ness}$ is studied; for conditional resetting, due to $Z_2$ symmetry, the average magnetization vanishes, so the squared magnetization density $\langle m^2 \rangle_{ness}$ is used as the observable.

3. Experimental Implementation:

   • Hardware: Experiments were conducted on the ibm_marrakesh processor (127-qubit heavy-hex lattice).
   • Unconditional Protocol: To avoid active mid-circuit resets, the authors exploited the renewal structure of the dynamics. They simulated trajectories by running pure unitary evolution for the time elapsed since a stochastically assigned reset event. This allowed for the extensive application of error mitigation techniques (dynamical decoupling, readout error mitigation, zero-noise extrapolation, and Pauli twirling).
   • Conditional Protocol: This required active mid-circuit interventions. The circuit included unitary evolution, mid-circuit $Z$-basis measurements, real-time classical processing (majority vote), and conditional single-qubit rotations ($X$ gates) to align the minority spins with the majority. Due to latency and coherence time constraints, advanced error mitigation was not integrated into the conditional feedback loops.
   • Data Collection: Results were averaged over $M=10^4$ readouts per reset realization and $10^3$ quantum trajectories.

Key Results

• Quantitative Agreement: The noisy conditional resetting theory shows excellent quantitative agreement with experimental data for $N=3, 5,$ and $7$ qubits. The theoretical model, incorporating the specific bit-flip noise model, successfully reproduces the experimental trends.
• Phase Transition Crossover: The experiments demonstrate a crossover behavior in the stationary state magnetization reminiscent of the equilibrium quantum phase transition of the FTFIM.
  • For unconditional resetting (with large $\theta=2.0$), the system exhibits features of digital quantum dynamics, including local maxima in magnetization due to Floquet quasi-energy degeneracies for small $N$.
  • For conditional resetting (with small $\theta=0.1$ approaching the continuum limit), the ferromagnetic order ($\langle m^2 \rangle_{ness}$) monotonically decreases as the transverse field $h$ increases, showing a smooth crossover around the critical point $h=1$.
• Noise Analysis:
  • Experimental data for unconditional resetting aligns well with noiseless theory, attributed to the effective use of error mitigation and the avoidance of dynamic circuits.
  • Conditional resetting data shows a significant downshift compared to noiseless theory. The proposed "reset-state error" model (accounting for bit-flips in the reset preparation and readout errors leading to incorrect conditional gates) provides a very good fit.
  • Comparisons with alternative noise models (depolarizing, phase-damping, amplitude-damping, and correlated $ZZ$ noise) in the Supplemental Material indicate that the reset-state error model is the most effective for $N=3$ and $N=5$, and remains competitive for $N=7$ despite having fewer fitting parameters.

Significance and Claims
The paper claims to provide the first demonstration of stochastic-resetting-induced non-equilibrium steady states on quantum hardware for interacting many-body systems. The results validate that collective stationary states arising from interacting dynamics and resets can be faithfully implemented on state-of-the-art noisy quantum processors.

The authors emphasize that their work connects stochastic resetting to the broader utility of classical feed-forward (via dynamic circuits and real-time logic) as a resource for state preparation on near-term quantum machines. They conclude that while quantum simulations for large system sizes remain challenging due to coherence limits and latency in feedback loops, their analysis paves the way for preparing collective stationary states on noisy devices and offers practical insights for developing quantum algorithms that utilize mid-circuit measurements. The paper modestly notes that future work will focus on integrating more sophisticated error mitigation protocols into conditional feedback loops to enable systematic scaling to larger system sizes.