Large-scale implementation of quantum subspace expansion with classical shadows

Abstract

Quantum subspace expansion (QSE) offers promising avenues to perform spectral calculations on quantum processors but comes with a large measurement overhead. Informationally complete (IC) measurements, such as classical shadows, were recently proposed to overcome this bottleneck. Here, we report the first large-scale implementation of QSE with IC measurements. In particular, we probe the quantum phase transition of a spin model with three-body interactions, for which we observe accurate ground state energy recovery and mitigation of local order parameters across system sizes of up to 80 qubits. We achieve this by reformulating QSE as a constrained optimization problem, obtaining rigorous statistical error estimates and avoiding numerical ill-conditioning. With over $3 \times 10^4$ measurement basis randomizations per circuit and the evaluation of $O(10^{14})$ Pauli traces, this represents one of the most significant experimental realizations of classical shadows to date.

Figures (5)

• Figure 1: Subspace expansion as a constrained optimization problem. The z axis shows the estimated energy $\hat{H}(\vec{c})$ when varying two dimensions of the subspace coefficients $\vec{c}$, while color indicates the statistical error $\hat{\epsilon}(\vec{c})$ of the energy estimation. Naive minimization of the energy (red arrow) may yield an energy with high statistical error that can severely violate the variational principle (black circle indicates the true ground state energy). Constrained optimization with a maximum allowed error $\epsilon_\text{max}$ avoids statistically unstable regions (blue path). Data corresponds to experiments presented in Fig. \ref{['fig:PSE_MPS_results']} for $N=48$ and $g=-0.5$.
• Figure 2: Error mitigation with subspace expansion based on IC measurements of ground state preparation circuits for a spin model. The qubit number increases from left to right from $N=16$ to $N=80$ while the three rows show the energy density and the order parameters of the traversed phase transition. Error bars represent the estimated statistical error $\hat{\epsilon} (\vec{c})$ and the inlays show the chosen physical qubits of the 156-qubit device ibm_fez. For $N=16$, grey shaded curves show the theoretical values for the lowest 25 excited states obtained with exact diagonalization, with a color gradient from black for the ground state to lighter shades for progressively higher-excited states.
• Figure 3: Quantum circuit that prepares an approximate ground state of the Hamiltonian from Eq. \ref{['eq:Hamiltonian_MPS_toy_model']} for $N=16$ with $g = -0.7$ through a renormalization-group-based MPS ansatz (without readout basis rotations used for classical shadow measurements). The left panel shows the physical qubit layout used in the experimental demonstration. Red boxes represent single-qubit $\sqrt{X}$ gates while blue boxes represent parameterized $R_z$ gates.
• Figure 4: Bias-variance-tradeoff in the constrained optimization problem. a) Obtained subspace energy density when solving the constrained optimization for increasing error budgets $\epsilon_\text{max}$. The horizontal lines are the unmitigated energy estimate $\hat{H}_\text{unmit}$ (red) and the true ground state energy (black), while the vertical dashed line signals the statistical error of $\hat{H}_\text{unmit}$. b) Decrease of the signal-to-noise-ratio (SNR) of the mitigated energy density with $\epsilon_\text{max}$. Data corresponds to experiments with $N=48$ and $g=-0.5$.
• Figure 5: Suppressing non-physical energies through regularization of the overlap matrix on experimental data for $N=48$. Data points with solid lines show the smallest pseudoeigenvalue when discarding the indicated number of the smallest singular values (SVs). Data points with error bars correspond to the "Krylov+" subspace results from Fig. \ref{['fig:PSE_MPS_results']}. Note the non-linear y-scale for better visibility.