Efficiently learning non-Markovian noise in many-body quantum simulators

TL;DR

The paper tackles learning non-Markovian noise in many-body quantum simulators under a stationary Gaussian environment by focusing memory kernels that encode environment correlations. It develops a measurement protocol using short-time evolution, product-state preparation, a single layer of local gates, and Pauli measurements to extract derivatives of memory kernels, with provable sample complexity that scales favorably with system size for fixed derivative order M. The approach handles both non-Markovian noise and an ensemble-Hamiltonian model with Gaussian coefficients, giving logarithmic (or near-logarithmic) sample complexity in N for the former and efficient means to recover means and covariances in the latter. Numerically, the method demonstrates accurate recovery of kernel parameters and Hamiltonian means for models with finite memory and dense correlations, and shows error scaling behaving as 1/√S with shot noise while remaining robust to increasing system size under reasonable locality assumptions.

Abstract

As quantum simulators are scaled up to larger system sizes and lower noise rates, non-Markovian noise channels are expected to become dominant. While provably efficient protocols for Markovian models of quantum simulators, either closed system models (described by a Hamiltonian) or open system models (described by a Lindbladian), have been developed, it remains less well understood whether similar protocols for non-Markovian models exist. In this paper, we consider geometrically local lattice models with both quantum and classical non-Markovian noise and show that, under a Gaussian assumption on the noise, we can learn the noise with sample complexity scaling logarithmically with the system size. Our protocol requires preparing the simulator qubits initially in a product state, introducing a layer of single-qubit Clifford gates and measuring product observables.

Figures (12)

• Figure 1: Open system models considered in this paper. (a) The system $S$ (blue circle lattice) is assumed to interact with an environment $E$ (pink), with a coupling $P_a A_a(t)$ between a Pauli string $P_a$ acting on the system and a time-dependent Hermitian operator $A_a(t)$ acting on the environment. (b) A non-dissipative environment, where the environment operators are commuting $[A_a(t), A_{b}(s)] = 0$, can be equivalently described by a noisy term in the system Hamiltonian $\alpha_a(t) P_a$, where $\alpha_a(t)$ is a classical noise.
• Figure 2: a) Single-qubit gate $W$ applied on the support of $P_a$ when trying to estimate kernel $K_{a,b}(t)$. b) Quantum channel used to prepare the state: an initial state $\rho(0)=\rho_S\otimes \gamma_E$ is prepared, where $\rho_S$ is a product state chosen depending on the kernels we are trying to learn while $\gamma_E$ is a Gaussian state. The state is time evolved under $U(t,0)$ and a unitary $W$, consisting of single-qubit Clifford gates on certain sites, is applied. The state is further evolved under $U(2t,t)$ and the environment is traced out.
• Figure 3: a) In the ensemble Hamiltonian model, while the Pauli strings are geometrically local, the correlations between their coefficients can be all-to-all: $\Lambda_a,\Lambda_b$ can have covariance $\Sigma_{ab} = \Omega(1)$ even if the supports of $P_a,P_b$ are far away. b) Quantum channel used to prepare the sate: a state $\rho(0)=\rho_S\otimes \gamma_E$ is prepared, where $\rho_S$ is a product state chosen depending on the kernel we are trying to learn while $\gamma_E$ is a Gaussian state. The state is time evolved under $\mathcal{U}_\Lambda (t)(\rho_S)=U_\Lambda(2t,0)\rho_S U_\Lambda(2t,0)^\dagger$ with $U_\Lambda(t,0) = e^{-itH_\Lambda}$, without introducing any intermediate gate $W$. We take the expectation value with respect to the jointly Gaussian random variables $\Lambda$.
• Figure 4: Schematic for the procedure to learn the kernel derivatives $K_{a,b}^{(0)}(0),K_{a,b}^{(1)}(0),...$. Given the Pauli strings $P_a,P_b$ associated to the kernel $K_{a,b}(t)$, we perform tomography on a region $\mathcal{I}_{a,b}$ slightly larger than the supports of $P_a,P_b$. We estimate $\frac{1}{2^N}\mathrm{Tr}(P_O \mathcal{E}_W(t)(P_I))$ by sampling eigenstates of $P_I$, which is supported only on $\mathcal{I}_{a,b}$, evolving for time $t$, introducing a layer of single-qubit gates $W$, evolving further until time $2t$ and finally measuring $P_O$, which is only supported on $\mathcal{I}_{a,b}$. The intermediate $W$ depends on $P_a,P_b$ and can be chosen to be a single-qubit gate according to Table \ref{['tab:W_gate']}. Repeating this procedure for several $t$ yields a time trace $B_{W,(O,I)}(t)$. We similarly obtain the time trace $B_{\mathds{1}^{\otimes N},(O,I)}(t)$, this time without introducing any gate $W$. We estimate the $m$-th derivative at $t=0$ of these time traces using polynomial interpolation: these yield systems of equations $T_{\mathds{1}^{\otimes N}}^{(m)}$ ($m$ even), $T_{W}^{(m)}$ ($m$ odd) that are linear in the $(m-2)$-th kernel derivative, using Eq. \ref{['eq:time_traces_equations']}. In order to obtain $T_{\mathds{1}^{\otimes N}}^{(2)}$ from $\partial_t^2B_{\mathds{1}^{\otimes},(O,I)}(t)|_{t=0}$ we need to use estimates $\lambda_a$ for the system coefficients. Tomography allows us to obtain an estimate for $K_{a,b}^{(0)}(0)$ from the linear map $T_{\mathds{1}^{\otimes N}}^{(2)}$ using Eq.\ref{['eq:xi_case1']}. Similarly, we obtain $T_{W}^{(3)}$ from $\partial_t^3B_{W,(O,I)}(t)|_{t=0}$ and recover $K_{a,b}^{(1)}(0)$ using Eqs.\ref{['eq:xi_case2']},\ref{['eq:xi_case3']}. The fourth derivative $\partial_t^4B_{\mathds{1}^{\otimes N},(O,I)}(t)|_{t=0}$ will depend nonlinearly on $K_{a,b}^{(0)}(0)$, as well as the system parameters $\lambda_a$. Thus, we use our estimates obtained in previous steps to obtain the linear map $T_{\mathds{1}^{\otimes N}}^{(4)}$, which allows us to obtain an estimate for $K_{a,b}^{(2)}(0)$. Similarly, higher derivatives of the time trace depend linearly on the kernel derivatives not yet estimated, and nonlinearly on those kernel derivatives and system parameters already estimated.
• Figure 5: We show how to find a tomography support $\mathcal{I}_{a,b}$ that allows us learn the derivatives $K_{a,b}^{(m)}(0)$. The target Pauli strings $P_a,P_b$, shown in the top left panel, act on the joint support $\mathcal{S}_{a,b}$ (dark blue). The conflicting pairs of Pauli strings $(P_{a_1},P_{b_1}),(P_{a_2},P_{b_2})$, depicted in the bottom panels, satisfy Eqs.\ref{['eq:enlarged_support_cond_1']},\ref{['eq:enlarged_support_cond_2']}, so their kernel coefficients would mix with those of $K_{a,b}^{(m)}(0)$ if we only performed tomography on $\mathcal{S}_{a,b}$. We add one extra qubit from the region $\mathcal{S}_{a,b}^c$ that lies in the support of each pair (red), obtaining $\mathcal{Q}_{a,b}$. Our tomography region is $\mathcal{I}_{a,b}=\mathcal{S}_{a,b}\cup\mathcal{Q}_{a,b}$, the dark blue and red sites in the top right panel.

Theorems & Definitions (38)

• Proposition 1
• Proposition 2
• Lemma 1: Adapted Theorem D.2, stilck2024efficient
• Lemma 2: Adapted Theorem E.1, Proposition E.1, Corollary E.1 stilck2024efficient, Theorem 1.2 arora2024outlier
• Lemma 3: Enlarged support
• proof
• Lemma 4
• proof
• Lemma 5
• Lemma 6