Quantum Trajectory Reconstruction

• Quantum Trajectory Reconstruction is a method for deducing the continuous evolution of quantum states using stochastic filtering and deterministic projection based on measurement records.
• It leverages stochastic differential equations and variational algorithms to model quantum jumps, diffusive trajectories, and mitigate errors in state estimation.
• The approach underpins real-time quantum feedback, sensor state design, and experimental investigations in superconducting circuits and related quantum systems.

Quantum trajectory reconstruction refers to the process of deducing or calculating the evolution of a quantum system’s state—either pure or mixed—conditioned on measurement records, environmental interactions, or imposed dynamical constraints. This topic encompasses a broad spectrum of approaches, from stochastic path integration with weak measurements to deterministic projection by imaginary time evolution, and links foundational quantum measurement theory, open systems, quantum control, and quantum information. Reconstruction techniques provide insight into quantum dynamics that are not accessible through ensemble averages or classical trajectory analogies, illuminate measurement-induced phenomena, and often serve as the basis for experimental quantum state tracking, feedback, and error mitigation.

1. Foundations of Quantum Trajectory Reconstruction

The quantum trajectory framework emerges from the need to model the conditional evolution of a system’s state under continuous measurement or environmental monitoring. In contrast to the conventional master equation or wavefunction collapse, quantum trajectory methods generate sample paths (“trajectories”) of the quantum state, corresponding to specific histories of measurement outcomes or environmental records (Kumar et al., 2018). The conditioned evolution is governed by stochastic differential equations (quantum filtering), unravelings of the Lindblad or master equation (quantum jumps or diffusive trajectories), or more general formalisms such as quantum state smoothing (Khademi et al., 19 Oct 2025).

For a quantum system described by the density matrix ρ(t) and subject to monitoring, the evolution may take the Stratonovich Itô form:

$$d\rho = -i[H, \rho]\,dt + \sum_k \gamma_k \mathcal{D}[L_k]\,\rho\,dt + \sum_j \mathcal{J}_j(\rho)\,dN_j$$

where $\mathcal{D}[L_k]$ are dissipators, and $\mathcal{J}_j$ are jump superoperators triggered by Poisson increments $dN_j$ (for discrete jumps) or by Wiener increments (for continuous measurement). Continuous weak measurements stretch out the wavefunction collapse, resulting in quantum trajectories that interpolate between pure states in a stochastic but physically interpretable manner. The approach is rigorously justified both theoretically and experimentally for a wide range of open quantum systems (Kumar et al., 2018).

2. Measurement, Control, and Conditional State Estimation

Quantum trajectory reconstruction has profound implications for real-time tracking and control of quantum systems. Weak and continuous measurements, as realized in superconducting circuit QED (Kumar et al., 2018, Weber et al., 2014), permit the real-time estimation of the system’s state—enabling feedback protocols, quantum state purification, and control of stochastic dynamics.

A key mathematical structure is the quantum filter, which updates the state estimate based on the measurement record up to (and, in smoothing, beyond) time t. The stochastic evolution can be captured by a nonlinear filter (for example, in the form of stochastic Bloch equations):

$$\begin{align*} \dot{x}(t) &= -\gamma x(t) + \Omega z(t) - \frac{x(t) z(t) r(t)}{\tau} \ \dot{z}(t) &= -\Omega x(t) + \frac{[1 - z^2(t)] r(t)}{\tau} \end{align*}$$

where $x(t)$ and $z(t)$ are Bloch vector components, $\Omega$ is the Rabi frequency, and $r(t)$ is the (dimensionless) measurement readout.

Importantly, the quantum trajectory approach generalizes to quantum state smoothing, where both past and future measurement records inform the reconstruction of the trajectory at time t (Khademi et al., 19 Oct 2025). The quantum state smoothing procedure combines forward-filtered and retrofiltered state estimates to yield a smoothed trajectory:

$$\langle x\rangle(t) = [V(t) - V^{ss}]\left( [V(t) - V^{ss}]^{-1} \langle x\rangle_F(t) + [V(t) + V^{ss}]^{-1}\langle x\rangle_R(t) \right)$$

This reduces statistical uncertainties but, unlike in classical smoothing, the quantum noise preserves the nondifferentiable (stochastic) nature of the trajectory.

3. Deterministic and Variational Approaches

Stochastic quantum trajectory reconstruction can suffer from computational intractabilities, especially in many-body or post-selected scenarios. The deterministic quantum trajectory via imaginary time evolution (Mittal et al., 31 Mar 2025) is designed to overcome the exponential post-selection barrier. By evolving an initial state $|\psi\rangle$ under a local Hamiltonian $H$ in imaginary time—the generator projecting onto the desired measurement outcome—the approach deterministically prepares the quantum trajectory corresponding to specific measurement results:

$$|\psi_\beta\rangle = \mathcal{N}_\beta e^{-\beta H}|\psi\rangle$$

with fidelity exponentially close to the target as $\beta$ increases. However, this method is confined to states exhibiting exponential clustering of correlations; in volume-law entangled states (long-range correlations), the method becomes ineffective.

Variational quantum algorithms extend trajectory reconstruction to high-dimensional systems such as structured quantum light (Koni et al., 26 Sep 2025). Here, the reconstruction is cast as the minimization of a quadratic cost function over measurement data, mapped onto an Ising Hamiltonian. Variational quantum eigensolvers combined with hardware-efficient circuits yield highly accurate state reconstructions even on noisy intermediate-scale devices by minimizing

$$E(\theta) = \langle\psi(\theta)| \mathcal{H} |\psi(\theta)\rangle$$

where $\mathcal{H}$ encodes the tomography cost function via affine Pauli mappings.

4. Trajectory-Based Theories and Quantum Hydrodynamics

Modern trajectory-based theories, particularly for relativistic systems, construct quantum mechanics without reference to complex amplitudes or wavefunctions (Poirier, 2012). In this picture, a quantum state is identified with an ensemble of real-valued trajectories—each a solution of a quantum Hamilton–Jacobi equation, typically incorporating a “quantum potential” constructed from spatial probability densities:

$$Q = - \frac{\hbar^2}{2m} \left(\frac{1}{\rho^{1/4}}\partial_i[\rho^{1/2}\,{}^{ij}\partial_j(\rho^{1/4})]\right)$$

This approach enforces causality and probability conservation at the trajectory level, supporting generalizations to curved spacetime and accelerated systems.

The hydrodynamic or Bohmian approach retains the wavefunction but interprets the phase and modulus as encoding velocity and probability, respectively. Recent work extends Bohmian mechanics to regimes such as quantum optical polarization (Luis et al., 2013), where time-evolving trajectories are derived from the local phase gradient of a stationary quantum distribution.

5. Quantum Trajectories in Experiment and Sensing

Quantum trajectory reconstruction is central to the experimental investigation of quantum dynamics and quantum sensing. Weak measurement protocols allow the experimental reassembly of “photon trajectories” in interferometers (Gosson et al., 2016, Zhou et al., 2017), verifying theoretical predictions of quantum probability flows and non-classical path phenomena (e.g., non-continuous or disconnected quantum evolutions as revealed in spectral marking of single-photon paths).

Trajectory sensing formalism explores the design of sensor states on qubit arrays to distinguish particle trajectories in one shot (Chin et al., 1 Oct 2024). The sensor state must ensure orthogonality of post-trajectory output states, typically constrained by a large, symmetry-exploited linear system. This formalism establishes a connection between trajectory sensing and quantum error correction: trajectory sensor states form novel quantum codes (including concatenations with known stabilizer codes), and noise-resilient trajectory sensing can be engineered via code concatenation and transversality.

In quantum information protocols, accurate estimation and reconstruction of quantum trajectories (for example, using post-processing, or error-mitigated estimators) enhances fidelity, reduces statistical uncertainty, and can improve the robustness of quantum error mitigation techniques (Donvil et al., 2023).

6. Analytical and Computational Challenges

The mathematical and computational landscape of quantum trajectory reconstruction involves (depending on context and complexity):

• Nonlinear and stochastic differential equations for trajectory propagation under measurement, often with drift and diffusion terms constrained by the Born rule and fluctuation-dissipation relations (Kumar et al., 2018).
• Variational and optimization-based approaches for trajectory reconstruction or tomography from incomplete or noisy data (Koni et al., 26 Sep 2025).
• Linear programming feasibility problems in the design of sensor states for trajectory discrimination (Chin et al., 1 Oct 2024).
• Simulation of quantum trajectories via engineered reservoirs or ancillary qubits, with classical post-processing via quasi-probability distributions (e.g., influence martingales) for error mitigation (Donvil et al., 2023).
• Limitations and complexity-theoretic boundaries: deterministic trajectory preparation is impossible for general states unless BQP = PP (Mittal et al., 31 Mar 2025), and practical implementations of sensor state design or trajectory-based codes often require additional constraints to manage resource scaling.

7. Impact and Outlook

Quantum trajectory reconstruction underpins both foundational and applied research across quantum physics:

• It provides a detailed description of stochastic state evolution under measurement and open system dynamics, enabling new approaches to quantum feedback, error correction, and quantum control.
• Methods integrating past and future measurement data (quantum state smoothing) allow higher-fidelity, lower-uncertainty state estimation, with observed departures from classical intuition due to persistent quantum noise and nondifferentiability (Khademi et al., 19 Oct 2025).
• Trajectory reconstruction is pivotal in resolving or interpreting phenomena such as superluminal neutrino propagation, nonlocal quantum dynamics, neutrino oscillations (Floyd, 2011), and the links between quantum interference and observable quantum “flows”.
• Strategies mapping reconstruction problems to quantum optimization (QUBO) for variational quantum eigensolvers—already demonstrated in high-energy track reconstruction and quantum optics—offer a pathway to exploit quantum devices for large-scale, real-time, or high-dimensional quantum trajectory inference (Crippa et al., 2022, Schwägerl et al., 2023, Koni et al., 26 Sep 2025).
• Connections between trajectory discrimination, quantum error correction, and code concatenation open avenues for quantum-enhanced sensing architectures with built-in resilience and tunable sensitivity (Chin et al., 1 Oct 2024).

Continued progress in experiment, theory, and quantum computation is expected to further unify deterministic and stochastic, continuous and discrete, and filtering and smoothing paradigms for quantum trajectory reconstruction, broadening the role of trajectories in the interpretation and exploitation of quantum dynamics across science and technology.