Technical Summary: Learning Arbitrary Lindbladians from Time Evolution

1. Problem Statement

The paper addresses the fundamental challenge of learning the dynamics of an unknown Markovian open quantum system. The system's evolution is generated by a Lindbladian $\mathcal{L}$, which describes both coherent (Hamiltonian) and dissipative interactions. In the Pauli basis, $\mathcal{L}$ is characterized by an exponentially large set of coefficients: Hamiltonian coefficients $\{h_a\}$ and dissipative coefficients $\{\gamma_{ab}\}$.

The learning objective is to estimate these coefficients to an entrywise accuracy $\varepsilon$ given black-box access to the time-evolution channel $e^{t\mathcal{L}}$ for arbitrary $t \ge 0$. The setting is ansatz-free: no prior knowledge of the system's locality, sparsity, or interaction structure is assumed. The only quantitative prior is a known upper bound $\Lambda$ on the dynamical strength ($\|\mathcal{L}^\dagger\|_{\infty \to \infty} \le \Lambda$), which fixes the natural time scale of the system.

The learning protocol must satisfy strict operational constraints:

• Control-free: No interleaving of control operations with the unknown evolution; the channel $e^{t\mathcal{L}}$ is applied once per experiment.
• Ancilla-free: No entanglement with auxiliary quantum systems (quantum memory) is permitted during state preparation, evolution, or measurement.
• Nonadaptive: The choice of inputs and measurements does not depend on previous outcomes.

2. Methodology

The proposed solution decomposes the learning task into two distinct, nonadaptive stages: Support Learning and Coefficient Learning. Both stages rely on estimating the endpoint derivatives ($t=0$) of specific signals derived from the Choi state of the evolution, utilizing Chebyshev interpolation to achieve high precision with minimal bias.

2.1 Core Primitive: Endpoint Differentiation

Direct finite-difference estimation of derivatives at $t=0$ suffers from high bias ($O(\Lambda^2 \tau)$) or requires prohibitively small time steps. The authors employ Chebyshev-Lobatto interpolation on a short interval $[0, T]$ where $T = 1/\Lambda$. By evaluating the signal at $q+1$ nodes and differentiating the interpolating polynomial at $t=0$, the bias is suppressed to $O(\Lambda/(q+1)!)$. This allows for a constant number of nodes $q = \tilde{\Theta}(1)$ to achieve bias $\varepsilon/2$, while the total absolute weight of the interpolation weights scales as $W = \tilde{O}(\Lambda)$.

2.2 Stage 1: Threshold Support Learning

The goal is to identify a candidate support set containing all coefficients with magnitude $\ge \eta$.

• Mechanism: The algorithm exploits the positivity of the Choi state $J_t$. Large coefficients in the Lindbladian manifest as "heavy" labels in a classical probability distribution $\mu(a)$ derived from the Pauli error rates of the evolution.
• Sampling: Instead of measuring Bell coherences directly (which would require entangled states), the algorithm implements a Pauli-twirled channel. By preparing random product Pauli eigenstates, applying random Pauli twirls, evolving, and measuring in complementary bases (computational and $X$-basis), the algorithm samples the $x$- and $z$-marginals of the distribution $\mu$.
• Recovery: The Cartesian product of the recovered marginal sets forms a candidate support $\hat{\mathcal{S}}$. This set is guaranteed to contain all $\eta$-heavy Hamiltonian and dissipative coordinates.
• Cost: Requires $\tilde{O}(\Lambda^2/\eta^2)$ experiments and $\tilde{O}(\Lambda/\eta^2)$ total evolution time.

2.3 Stage 2: Coefficient Learning via Clifford Probing

Given a candidate support of size $M$, the goal is to estimate all coefficients within this set to accuracy $\varepsilon$.

• Mechanism: The algorithm uses ancilla-free process shadows. It prepares random stabilizer states, evolves them, and measures in random Clifford bases. These measurements generate classical snapshots that unbiasedly estimate the Choi state $J_t$.
• Witnesses: Coefficients are recovered by differentiating "witness signals" $f_A(t) = \langle \Phi_A | J_t | \Phi_A \rangle$, where $|\Phi_A\rangle$ are Bell-basis vectors constructed from Pauli operators. Polarization identities allow off-diagonal matrix elements (and thus dissipative coefficients) to be reconstructed from diagonal expectations.
• Variance Control: A key technical result is that the second moment of the shadow estimator for these witnesses is bounded by a constant ($\le 160$), independent of the system size $n$. This enables the use of the median-of-means estimator to control statistical errors efficiently.
• Cost: Requires $\tilde{O}(\Lambda^2 \log M / \varepsilon^2)$ experiments and $\tilde{O}(\Lambda \log M / \varepsilon^2)$ total evolution time. The classical post-processing scales polynomially in $M$ and $n$.

3. Key Results

By composing the two stages (setting the support threshold $\eta = \varepsilon$), the authors present an algorithm for learning arbitrary Lindbladians with the following guarantees:

• Sample Complexity: The total number of experiments is $\tilde{O}(\Lambda^2/\varepsilon^2)$.
• Total Evolution Time: The total time the system evolves is $\tilde{O}(\Lambda/\varepsilon^2)$.
• Classical Complexity: The classical running time is polynomial in $n$, $\Lambda$, and $1/\varepsilon$.
• Optimality: These scalings match the information-theoretic lower bounds established in [ACG+26] up to polylogarithmic factors.
• Generality: The algorithm works for completely arbitrary Lindbladians without assumptions on locality, sparsity, or coefficient gaps. It also efficiently learns generic $k$-local Lindbladians.

4. Significance and Claims

The paper claims several significant theoretical advances:

1. Closing the Gap Between Closed and Open Systems: In the in situ (control-free, ancilla-free) setting, learning open-system dynamics (Lindbladians) is no harder than learning closed-system dynamics (Hamiltonians). Both achieve the same optimal scaling in total evolution time, differing only by polylogarithmic factors. This contrasts with settings allowing coherent control, where Hamiltonian learning can achieve Heisenberg scaling ($1/\varepsilon$) while Lindbladian learning is limited to the standard quantum limit ($1/\varepsilon^2$).
2. No Advantage from Quantum Memory: The authors demonstrate that for in situ Lindbladian learning, the use of ancillary systems (quantum memory) does not provide a significant advantage. While quantum memory offers exponential advantages in tasks like state tomography or Pauli-channel learning, the bottleneck for Lindbladian learning is the signal accumulation rate of a single short-time experiment, which cannot be amplified by entanglement. The proposed ancilla-free algorithm matches the lower bounds that hold even for protocols with arbitrary ancillas.
3. Feasibility of Arbitrary Learning: The work resolves the central question of whether arbitrary Lindbladians can be learned efficiently. It shows that despite the exponential size of the parameter space, the dynamical strength bound $\Lambda$ controls the number of "heavy" coefficients, making efficient recovery possible without structural assumptions.

The paper concludes that dissipation is not an obstruction to efficient learning in the in situ regime, provided the dynamical strength is bounded, and that the proposed two-stage algorithm is nearly optimal in its resource usage.