Distinguishing synthetic unravelings on quantum computers

TL;DR

The GKSL master equation $\dot{\rho}_t=\mathcal{L}(\rho_t)$ fixes only the unconditional evolution, leaving the stochastic conditioned trajectories $\rho_t^{(r)}$ underdetermined. The authors design two synthetic, discrete-time unravelings—projective measurements and random-unitary kicks—that share the same average dynamics but yield different nonlinear trajectory statistics, enabling backaction to be observed beyond linear expectations. Implemented on IBM Quantum hardware for one- and two-qubit systems, they show that metrics like the trajectory-variance $\mathrm{Var}_{\mathrm{traj}}[\langle O\rangle^{(r)}]$ and the trajectory-averaged entropy $\mathbb{E}_r[S(\rho_t^{(r)})]$ distinguish unravelings, while linear averages remain identical. These results demonstrate that quantum trajectories encode measurement backaction in a way that is accessible experimentally and offer a scalable framework for studying measurement-driven phenomena and improving classical simulations of open quantum systems.

Abstract

Distinct monitoring or intervention schemes can produce different conditioned stochastic quantum trajectories while sharing the same unconditional (ensemble-averaged) dynamics. This is the essence of unravelings of a given Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation: any trajectory-ensemble average of a function that is linear in the conditional state is completely determined by the unconditional density matrix, whereas applying a nonlinear function before averaging can yield unraveling-dependent results beyond the average evolution. A paradigmatic example is resonance fluorescence, where direct photodetection (jump/Poisson) and homodyne or heterodyne detection (diffusive/Wiener) define inequivalent unravelings of the same GKSL dynamics. In earlier work, we showed that nonlinear trajectory averages can distinguish such unravelings, but observing the effect in that optical setting requires demanding experimental precision. Here we translate the same idea to a digital setting by introducing synthetic unravelings implemented as quantum circuits acting on one and two qubits. We design two unravelings - a projective measurement unraveling and a random-unitary "kick" unraveling - that share the same ensemble-averaged evolution while yielding different nonlinear conditional-state statistics. We implement the protocols on superconducting-qubit hardware provided by IBM Quantum to access trajectory-level information. We show that the variance across trajectories and the ensemble-averaged von Neumann entropy distinguish the unravelings in both theory and experiment, while the unconditional state and the ensemble-averaged expectation values that are linear in the state remain identical. Our results provide an accessible demonstration that quantum trajectories encode information about measurement backaction beyond what is fixed by the unconditional dynamics.

Figures (6)

• Figure 1: Conceptual overview of this work.(a)Theoretical framework: A GKSL master equation fixes the average evolution $\dot{\rho}_t=\mathcal{L}(\rho_t)$ but leaves the conditional dynamics undefined. Different monitoring/intervention schemes correspond to inequivalent unravelings, generating distinct trajectory ensembles $\{\rho_t^{(r)}\}$. (b)Quantum-circuit protocol: We engineer synthetic unravelings with discrete quantum circuits on one and two qubits. The unitary evolution is interrupted by interventions---either projective measurements or random-unitary "kicks" (applying $\mathbb{I}$ or $\sigma_z$ with 50% probability)---that generate distinct trajectory ensembles while preserving the same average state, $\rho_t$. The flowchart (right) summarizes the workflow: repeating shots to collect records $r$, reconstructing conditional states/values, and computing both linear and nonlinear statistics. (c)Distinguishing unravelings: Unraveling dependence appears only in nonlinear trajectory averages. While the linear average $\overline{\langle\sigma_z\rangle^{(r)}}(t)$ is identical for both schemes, the trajectory variance $\mathrm{Var}_\mathrm{traj}[\langle\sigma_z\rangle^{(r)}]$ depends on the unraveling, revealing the intervention- (or monitoring-) induced conditioning at the trajectory level (data shown for the 1Q protocols).
• Figure 2: Single-qubit trajectories on quantum hardware. (a) Projective unraveling: $\langle \sigma_z \rangle^{(r)}$ changes discontinuously at $t_1$ and $t_2$ due to the state update conditioned on the measurement outcome. The annotations next to the vertical intervention lines indicate the projector associated with that trajectory branch. Legend labels denote the sequence of outcomes (e.g., "01" corresponds to outcome $0$ at $t_1$ and $1$ at $t_2$). (b) Kick unraveling: the state evolves unitarily, with updates at the intervention times. The annotations indicate the applied gate ($\mathbb{I}$ or $\sigma_z$). Legend labels denote the sequence of kicks (e.g., "IZ" corresponds to $\mathbb{I}$ at $t_1$ and $\sigma_z$ at $t_2$). Lines show theory and markers show experimental data. Error bars are obtained via bootstrap resampling of the shot counts at each time point.
• Figure 3: Single-qubit protocol. (a) Trajectory-averaged expectation value $\overline{\langle\sigma_z\rangle^{(r)}}$. (b) Trajectory variance $\mathrm{Var}_{\mathrm{traj}}[\langle\sigma_z\rangle^{(r)}]$. The projective unraveling is shown as a solid blue curve with diamond markers, and the kick unraveling as a dashed orange curve with triangle markers. Curves show Qiskit local simulations and markers show quantum processor data. Vertical dotted lines indicate the interventions at $t_1$ and $t_2$. Error bars are obtained via bootstrap resampling. Parameters: $\Omega=4$, $\Delta=2$, $t_1=2$, $t_2=4$, $t_f=5$; 1000 shots per time point.
• Figure 4: Single-qubit protocol: entropy-based quantities as functions of time (von Neumann entropy computed with a base-2 logarithm). Purple: $S(\rho_t)$ for the ensemble-averaged state. Blue and orange: trajectory-averaged entropy $\mathbb{E}_r[S(\rho_t^{(r)})]$ for the projective and kick unravelings. Blue markers: estimate of $S(\rho_t)$ obtained by combining the theoretical state reconstruction with branch weights extracted from projective-hardware data. Error bars are obtained via bootstrap resampling of the shot counts at each time point. Vertical dotted lines indicate the interventions at $t_1$ and $t_2$. Parameters and intervention times are the same as in Fig. \ref{['fig:1Q_means_var']}.
• Figure 5: Two-qubit protocol. (a) Trajectory-averaged expectation value $\overline{\langle\sigma_z^{(1)}\rangle^{(r)}}$. (b) Trajectory variance $\mathrm{Var}_\mathrm{traj}[\langle\sigma_z^{(1)}\rangle^{(r)}]$. The projective unraveling is shown as a solid curve with diamond markers, and the kick unraveling as a dashed curve with triangle markers. Curves show Qiskit local simulations and markers show quantum processor data with readout-error mitigation. Vertical dotted lines indicate the interventions at $t_1$ and $t_2$. Error bars are obtained via bootstrap resampling. Parameters: $\Omega = 10$, $\Delta = 1$, $J = -0.5$, $t_1 = 0.6$, $t_2 = 1.4$, $t_f = 2.5$; 2000 shots per time point.