Physically natural metric-measure Lindbladian ensembles and their learning hardness

TL;DR

This work analyzes how well one can infer the noise and dissipation structure in open quantum systems described by GKSL Lindbladians from finite-time measurement statistics. It introduces two physically motivated random Lindbladian ensembles via a linear parametrization around a reference generator and extends SQ/QPStat learning frameworks to open-system dynamics, proving exponential lower bounds on average-case learning in the parameter dimension $M$. The authors derive a Porter-Thomas-type open-system mean distance $m_0>0$ through linear-response theory and verify it in a local amplitude-damping model, linking high-dimensional concentration to intrinsic learning hardness. They further translate these results into cryptographic primitives, constructing two Lindbladian-PUF protocols (distribution-level and tomography-based) whose security follows from SQ/QPStat hardness. Collectively, the paper provides a rigorous open-system analogue of unitary design and pseudorandomness concepts, with concrete physical models and explicit relationships between perturbations, concentration, and learnability that have implications for noise characterization, quantum cryptography, and potential hardware validation.

Abstract

In open quantum systems, a basic question at the interface of quantum information, statistical physics, and many-body dynamics is how well can one infer the structure of noise and dissipation generators from finite-time measurement statistics alone. Motivated by this question, we study the learnability and cryptographic applications of random open-system dynamics generated by Lindblad-Gorini-Kossakowski-Sudarshan (GKSL) master equations. Working in the affine hull of the GKSL cone, we introduce physically motivated ensembles of random local Lindbladians via a linear parametrisation around a reference generator. On top of this geometric structure, we extend statistical query (SQ) and quantum-process statistical query (QPStat) frameworks to the open-system setting and prove exponential (in the parameter dimension $M$) lower bounds on the number of queries required to learn random Lindbladian dynamics. In particular, we establish average-case SQ-hardness for learning output distributions in total variation distance and average-case QPStat-hardness for learning Lindbladian channels in diamond norm. To support these results physically, we derive a linear-response expression for the ensemble-averaged total variation distance and verify the required nonvanishing scaling in a random local amplitude-damping chain. Finally, we design two Lindbladian physically unclonable function (Lindbladian-PUF) protocols based on random Lindbladian ensembles with distribution-level and tomography-based verification, thereby providing open-system examples where learning hardness can be translated into cryptographic security guarantees.

Figures (7)

• Figure 1: Schematic view of the metric probability spaces used to define random Lindbladian ensembles. Left: an abstract parameter space $(\Theta,d,\mu)$, where $\Theta$ is the parameter domain, $d$ is the chosen metric on $\Theta$, and $\mu$ is the ensemble (sampling) measure. Top right: the Haar-sphere ensemble, where parameters are sampled uniformly from the high-dimensional sphere $S^{M-1}:=\{\theta\in\mathbb R^M:\|\theta\|_2=1\}$ with respect to the rotation-invariant (Haar-induced) surface measure. The metric is the Euclidean distance $d_2(\theta,\theta')=\|\theta-\theta'\|_2$, and the Lindbladian is parameterized as $\mathcal{L}(\theta)=\mathcal{L}_{\mathrm{ref}}+\frac{\delta}{\sqrt{M}} \sum_j \theta_j G_j$. Bottom right: the product-measure ensemble, where each coupling $\theta_j$ is an independent random variable supported on $[-\delta/M,\delta/M]$, so that $\Theta_{\mathrm{cube}}:=[-\delta/M,\delta/M]^M$. Here the metric is the $\ell_1$ distance $d_1(\theta,\theta')=\|\theta-\theta'\|_1$, the measure is the product measure $\mu_{\mathrm{prod}}$, and the Lindbladian is parameterized as $\mathcal{L}(\theta)=\mathcal{L}_{\mathrm{ref}}+\sum_j \theta_j G_j$.
• Figure 2: Schematic setup of our random Lindbladian ensemble and the two information-access models considered in this work. Top: a Lindbladian family is parametrized by points $\theta$ on the high-dimensional sphere $S^{M-1}$ via the linear embedding $\mathcal{L}(\theta)=\mathcal{L}_{\mathrm{ref}}+ \frac{\delta}{\sqrt{M}}\sum_j \theta_j G_j$, and generates a quantum Markov semigroup $\Lambda_t^{(\theta)} = e^{t\mathcal{L}(\theta)}$. Bottom left: the distribution-level SQ interface, where a fixed input state $\rho_{\mathrm{in}}$ and POVM $\{M_x\}_{x\in X}$ define the output distribution $P_{\mathcal{L}(\theta)}(x)=\operatorname{Tr}\bigl(M_x\Lambda_t^{(\theta)}(\rho_{\mathrm{in}})\bigr)$, and the learner only accesses expectations $P_{\mathcal{L}(\theta)}[\varphi]$. Bottom right: the process-level QPStat interface for the channel $E_\theta:=\Lambda_t^{(\theta)}$, where a QPStat oracle, given a state $\rho$, an observable $O$ and a tolerance $\tau$, returns an approximation to $\operatorname{Tr}\bigl(O\,\Lambda_t^{(\theta)}(\rho)\bigr)$.
• Figure 3: Random local Lindbladian on a one-dimensional spin chain. Each circle represents a qubit on the chain. On site $i$, local dephasing is described by the jump operator $J_i^{(z)}$, indicated by the loop arrow. Excitation and relaxation processes are implemented by the amplitude-raising and amplitude-lowering jump operators $J_i^{(+)}$ and $J_i^{(-)}$, respectively. On each bond $(j,j+1)$ we include two-body dissipative channels $K_{j,j+1}^{(1)}$ and $K_{j,j+1}^{(2)}$, representing, for example, incoherent spin-exchange and correlated dephasing. In our construction, the corresponding dissipators are collected into a set of local directions $\{G_j\}$ that enter the spherical random-perturbation parametrization $\mathcal{L}(\theta)=\mathcal{L}_{\mathrm{ref}} + \frac{\delta}{\sqrt{M}}\sum_j \theta_j G_j$.
• Figure 4: Average TV distance $m(\delta;N) =\mathbb{E}_\theta d_{\mathrm{TV}}(P_\theta,Q)$ in the spherical random local Lindbladian model, for $N=2,\dots,6$ in the small-noise window $\delta\in[0.1,0.2]$. Solid dots are Monte Carlo data with error bars indicating the standard error. Solid lines are linear fits for each $N$, while the dashed line shows the theoretical linear response $m(\delta)=m_l\delta$ with $m_l=\sqrt{1/(2\pi)}\approx0.399$.
• Figure 5: Average TV distance $m(\delta;4)$ in the spherical random local Lindbladian model for a fixed system size $N=4$ over a wide noise range $\delta\in[0,10]$. Solid dots are Monte Carlo data with error bars, and the dashed line shows the linear-response prediction $m(\delta)=m_l\delta$. The two coincide very well in the small-noise regime, while for larger $\delta$ the curve $m(\delta)$ gradually bends away from the straight line and saturates, illustrating the nonlinear response of the Lindbladian dynamics. In particular, as long as $\delta>0$, $m(\delta;4)$ remains strictly positive and of order $\mathcal{O}(1)$.

Theorems & Definitions (170)

• Definition 1: Random local Lindbladian ensemble
• Theorem 1: Deterministic average-case SQ hardness for decision problems (informal)
• Theorem 2: Randomized average-case SQ hardness for decision problems (informal)
• Theorem 3: Average-case SQ hardness of learning open-system dynamics (informal)
• Theorem 4: Exponential QPStat hardness of learning random Lindbladian channels (informal)
• Theorem 5: Lindbladian-PUF security from learning hardness (informal)
• Theorem 6: Linear-response analytic scaling of the mean TV distance (informal)
• Theorem 7: Explicit computation of the mean scaling constant in a random local amplitude-damping model (informal)
• Definition 2: Euclidean norms on parameter space
• Definition 3: Schatten norms of operators