Exact Stochastic Differential Equations for Quantum Reverse Diffusion

TL;DR

The paper addresses the apparent irreversibility of open quantum dynamics by formulating exact quantum reverse diffusion equations for forward processes driven by measurement-induced Pauli noise. It derives analytical forward and reverse SDEs and stochastic master equations, featuring a non-Markovian drift guided by a Brownian bridge that conditions on the measurement record to steer states back to their initial configuration, achieving almost-sure reversal. The results yield exact reversal for single Pauli channels and approximate reversal for depolarizing noise with fidelity approaching unity, and they propose real-time implementable schemes using measurement-based feedback and teleportation-assisted imaginar-time drift. This framework enables noise-resilient quantum gates, new tomographic protocols via forward–reverse cycles, and foundational ideas for diffusion-based quantum information processing and potential error-correction paradigms grounded in reverse diffusion.

Abstract

The ensemble-averaged dynamics of open quantum systems are typically irreversible. We show that this irreversibility need not hold at the level of individually monitored quantum trajectories. Our main results are analytical stochastic differential equations for quantum reverse diffusion, along with corresponding stochastic master equations. These equations describe the exact and approximate stochastic reverse processes for continuously monitored Pauli channels, including time-dependent depolarizing noise. We show that the reverse processes generalize the forward dynamics by combining the noise effects of the forward processes with an additional non-Markovian stochastic drift that dynamically steers a quantum state back to its initial configuration. Consequently, the exact SDEs admit closed-form solutions that can be implemented in real-time without the need for variational techniques. Our findings establish an analytical framework for quantum state recovery, noise-resilient quantum gates, quantum generative modelling, quantum tomography via forward-reverse cycles, and potential paradigms for quantum error correction based on reverse diffusion.

Figures (3)

• Figure 1: (a) Forward and reverse depolarizing noise processes on an ensemble of states. Both processes occur under the same noise conditions, with the reverse process incorporating a noise-aware stochastic drift, which steers the states back to their initial configuration. (b) Individual quantum-state trajectories for the forward (blue) and corresponding reverse (purple) processes. The initial and final states of the forward process are $\ket{\psi(0)}$ and $\ket{\psi(T)}$, respectively. The reverse process starts at time $T$ and evolves $\ket{\psi(T)}$ into $|\hat{\phi}(2T)\rangle \approx \ket{\psi(0)}$.
• Figure 2: The quantum state fidelity probability flow under forward and reverse dynamics with a representative single-trajectory realization shown in red. Here, $L = \sigma_x$ and $p=0.2$. The forward and reverse processes occur on the time intervals $[0,1]$ and $[1,2]$, respectively.
• Figure 3: The quantum circuit that near-deterministically implements the stochastic reverse drift $\mathcal{R}_1(t)$. The 2-qubit meter gates implement a Bell-basis measurement with binary outcomes $j$ and $\ell$ ($u$ and $v$). After the measurement, the state of interest acquires the drift $\mathcal{R}_1\sigma_3^{\ell}$, where $\sigma_3^{\ell}$ for $\ell=1$ is an undesirable byproduct. If $\ell=0$, the correct drift is implemented. If $\ell=1$, the process must be repeated.