Demonstrating and Benchmarking Classical Shadows for Lindblad Tomography

Abstract

Spurious couplings and decoherence degrade the performance of solid-state quantum processors, demanding careful design, calibration, and mitigation protocols. These strategies often rely on characterization of the idling processor, but tomographic recovery of (time-independent) Lindblad dynamics scales exponentially with qubit count. Here, we experimentally benchmark and demonstrate that randomized ("shadow") measurements accelerate Lindblad tomography on a superconducting transmon processor. We first implement extensible Lindblad tomography, which estimates Lindblad parameters using a complete tomographic dataset, and use it as a baseline to benchmark a shadow tomography approach, shadow Lindblad tomography. The shadow approach recycles randomized configurations to estimate the same Lindblad parameters using far fewer resources under physically motivated locality assumptions. We experimentally verify these assumptions in our processor by implementing the protocols on one- and three-qubit subsystems; here, shadow Lindblad tomography reproduces extensible Lindblad tomography within uncertainties while using exponentially fewer configurations. Leveraging this efficiency, we apply shadow Lindblad tomography to the full five-qubit processor and recover all single qubit dissipation and two-qubit coupling parameters in 9 hours of acquisition time compared to an estimated 58 hours for extensible Lindblad tomography. Additionally, our shadow implementation is compatible with conventional Gaussian error propagation, avoiding the use of median-of-means estimators. Together, these results demonstrate how randomized shadow tomography protocols can be practically implemented to learn quantum processor dynamics at an increasing qubit count.

Figures (9)

• Figure 1: ELT and SLT blueprint. (a) Optical image of a lithographically identical 5-qubit processor to the one used in this work with schematics of coupling and loss mechanisms (left panel). A general qubit processor can be described by a Markovian LME with a Lindblad operator, here written out in the Pauli basis (right panel). (b) The LME can be learned via ELT or SLT. Both procedures are, in the worst case, exponentially expensive in system size but become feasible under locality assumptions of the underlying LME. (c) ELT extracts Lindblad parameters using a data set that loops over a tomographically complete set of $j$ input and output states and letting the system evolve for varying lengths of time $t$. (d) SLT comprises of initializing a random input state and measurement basis and letting the system evolve for varying lengths of time $t$ and then performing shadow tomographical postprocessing (see Sec. \ref{['sec:SLT_theory']} for details) to extract Lindblad parameters.
• Figure 2: Extracting Lindblad parameters using ELT and SLT for two qubits. (a) Pauli expectation values as a function of time. Each of the $j$ expectation values is found from a single choice of initialization and finalization gates. (b) Transfer matrix entries as a function of time. Each expectation value is found from all randomization experiments (details in Sec. \ref{['sec:SLT_theory']}). (c) The tomographic data is linearly transformed and fitted to a low-order polynomial (here a linear fit, see further details in Sec. \ref{['sec:LindbladTomography']}). The Lindblad parameters are then extracted as the derivative of the fitted curve at $t=0$.
• Figure 3: ELT and SLT on a single qubit. (a) Magnitude of the estimated parameters from both the ELT and SLT procedure. (b) A visual representation of the fitted Lindblad parameters from ELT. The ground state of the Hamiltonian is shown on the Bloch sphere and the three extracted Lindblad jumping operators are represented by how they deform the Bloch sphere, along with their respective rates. (c) A subset of the data used for ELT/SLT can be interpreted as a $T_1$-experiment. The data and a traditional $T_1$ experiment is shown together with the extracted $T_1$ from ELT and SLT. (d) Difference in all parameters extracted from the two methods (in red points) plotted against the number of random instances used to do SLT.
• Figure 4: Extensible Lindblad tomography and shadow Lindblad tomography for three qubits. (a) The magnitude of the fitted Lindblad parameters both for ELT and SLT ordered by their size estimated in ELT. We have here omitted 4016 parameters for clarity. Error bars are 16 and 84 percent quantiles of the magnitude. (b) Difference in extracted parameters of the two methods (in red points) plotted against the number of shadows used to do SLT. The plot is capped to omit the largest deviating residuals for clarity. (c) The standard deviation of the residual plotted against the relative size of the number of shadows used for SLT compared to the full number of shots done in ELT for both a single qubit and 3 qubits. The dashed horizontal lines show the average uncertainty from ELT on three qubits and a single qubit respectively. Both residual standard deviations go down with the number of randomizations before saturating around the horizontal lines in the plot. We see that the saturation point is relatively lower for 3 qubits than for a single qubit, a property which is consistent with being in the $k$-local regime.
• Figure 5: SLT on a 5-qubit processor. (a) Characterizing the Lindblad dynamics for the entire five qubit device assuming only single qubit loss and two-qubit interactions. (b) The magnitude of extracted on-site Lindblad parameters. The x-axis is the same as in Fig. \ref{['fig: Figure3']}(a). (c), (d) The magnitude of all extracted nearest neighbor and next-nearest neighbor interactions. The x-ticks are $a_{ij}$ with $i,j \in \{x,y,z\}$ in lexicographic order.