Simulation of Lindbladian dynamics via adaptive variational quantum trajectory compression

Technical Summary: Simulation of Lindbladian Dynamics via Adaptive Variational Quantum Trajectory Compression

Problem Statement
The quantum simulation of open quantum systems on Noisy Intermediate-Scale Quantum (NISQ) devices faces two primary hurdles: the non-unitary nature of dissipative dynamics and the severe limitations of available quantum resources, particularly circuit depth and qubit count. Existing methods for simulating Lindbladian dynamics often rely on embedding non-unitary evolution into enlarged Hilbert spaces (requiring auxiliary qubits) or decomposing dynamics into ensembles of trajectories that still necessitate deep circuits or complex post-processing. While Variational Quantum Algorithms (VQAs) have shown promise for closed-system dynamics, their application to open systems is hindered by the need for auxiliary qubits, growing circuit depths, and the accumulation of errors during classical post-processing.

Methodology
The authors propose a resource-efficient, ancilla-free algorithm that combines a compact adjoint channel representation with an adaptive variational quantum trajectory compression framework.

1. Compact Adjoint Channel Construction:
   For open systems with Pauli dissipations ($P_k \in \{I, X, Y, Z\}^{\otimes n}$), the authors derive a mixed-unitary adjoint channel that approximates the target dissipative dynamics with an error scaling of $O(\delta t^2)$. Unlike previous formulations that required classical post-processing to correct deviations, this new channel is constructed such that the target evolution is directly approximated by a probabilistic mixture of unitary operators:
   $$\mathcal{A}(\rho) = \sum_{k=0}^{D} p_k U_k \rho U_k^\dagger$$
   Here, $U_0$ represents a rescaled Hamiltonian evolution ($e^{-iH\Delta t}$), and $U_{k>0}$ are the Pauli dissipation operators. The probabilities $p_k$ are derived to ensure normalization. This structure allows the dynamics to be simulated via Monte Carlo sampling of quantum trajectories without auxiliary qubits.

2. Adaptive Variational Quantum Trajectory Compression:
   To address the circuit depth bottleneck caused by repeated applications of the no-jump operator $U_0$ in long-time simulations, the authors introduce a compression strategy.

   • Training: A series of depth-adaptive Parameterized Quantum Circuits (PQCs), denoted $U(\theta_s)$, are trained to approximate the $s$-step Hamiltonian evolution $(U_0)^s$.
   • Training Strategies: Two strategies are explored:
     • Direct Training: The PQC is trained directly against the exact $(U_0)^s$.
     • Iterative Training: The PQC for step $s$ is trained against the product of the previously trained circuit and a single step $U_0$ (i.e., $U(\theta_{s-1})U_0$), reducing the depth of the target circuit during training.
   • Loss Function: To avoid the need for auxiliary qubits required by Hilbert-Schmidt tests, the authors utilize a basis-sampled loss function. This evaluates the agreement between the PQC and the target unitary on a finite set of basis states, making it hardware-friendly.
   • Compression: In the sampled trajectories, consecutive sequences of $U_0$ blocks are identified and replaced by the corresponding trained PQC $U(\theta_s)$, significantly reducing the circuit depth while preserving the trajectory structure.

3. Simulation Protocol:
   The algorithm proceeds by sampling trajectories according to the probability distribution of the adjoint channel. Each trajectory is then compressed by replacing $U_0$ segments with the trained PQCs. Finally, the expectation values of observables are estimated by averaging the measurement outcomes over the ensemble of compressed trajectories.

Key Contributions

• Ancilla-Free Adjoint Channel: The derivation of a stable, compact mixed-unitary adjoint channel that eliminates the need for the additional classical post-processing step found in previous works, thereby reducing error amplification.
• Depth-Adaptive Compression: The introduction of a framework that uses variational quantum compiling to compress repeated Hamiltonian evolution blocks within sampled trajectories, directly addressing the circuit depth limitations of NISQ devices.
• Hardware-Efficient Training: The proposal of a basis-sampled loss function and iterative training strategies that allow the compression training to be performed without auxiliary qubits and with reduced target circuit depth.
• Integration of Structures: The work demonstrates how to jointly exploit the specific structure of Pauli dissipations (leading to frequent no-jump segments) and variational compression techniques to bridge the gap between open-system algorithms and near-term hardware implementation.

Results
The authors validated the method using numerical simulations of the dissipative quantum $XY$ model, specifically modeling the melting of a domain-wall state under local dephasing.

• Accuracy: The stochastic sampling of the derived adjoint channel successfully reproduced the exact Lindblad dynamics and the deterministic Kraus channel results. The sampling error scaled as $O(1/\sqrt{M})$ with the number of trajectories $M$.
• Compression Efficiency: The variational compression significantly reduced the circuit depth. For a simulation time $T=10$ with $M=128$ trajectories, the "direct" replacement strategy reduced the average single-qubit and two-qubit gate counts by approximately 43%.
• Trade-offs: The iterative training strategy showed that while it reduces the depth of the training target, it introduces accumulated approximation errors as the compression length ($s_{max}$) increases. However, even with these errors, the compressed trajectories maintained high fidelity to the physical dynamics, with the direct strategy offering the best balance between accuracy and resource reduction in the tested scenarios.

Significance and Claims
The paper claims to provide a practical route toward the ancilla-free and depth-reduced simulation of open quantum systems on near-term quantum hardware. The authors emphasize that their approach is particularly suited for the NISQ era because it:

1. Avoids the scalability issues of dilation-based methods by eliminating auxiliary qubits.
2. Mitigates the impact of hardware noise by reducing the circuit depth required for trajectory implementation.
3. Offers a flexible framework where the trade-off between algorithmic accuracy (discretization error) and hardware noise can be managed by choosing appropriate compression lengths and training strategies.

The authors modestly note that while higher-order channel constructions exist, they often lead to more complex trajectory structures that are less amenable to this specific compression technique. Therefore, for current devices, a lower-order but compact, ancilla-free, and depth-reduced algorithm is presented as a more practical solution than formally higher-order constructions with larger resource overheads. The work suggests that combining physically structured adjoint-channel representations with variational trajectory compression is a viable path for simulating realistic many-body open quantum systems.