Physics-Informed Bayesian Optimization of Variational Quantum Circuits

TL;DR

A novel and powerful method to harness Bayesian optimization for Variational Quantum Eigensolvers (VQEs) -- a hybrid quantum-classical protocol used to approximate the ground state of a quantum Hamiltonian by derive a VQE-kernel which incorporates important prior information about quantum circuits.

Abstract

In this paper, we propose a novel and powerful method to harness Bayesian optimization for Variational Quantum Eigensolvers (VQEs) -- a hybrid quantum-classical protocol used to approximate the ground state of a quantum Hamiltonian. Specifically, we derive a VQE-kernel which incorporates important prior information about quantum circuits: the kernel feature map of the VQE-kernel exactly matches the known functional form of the VQE's objective function and thereby significantly reduces the posterior uncertainty. Moreover, we propose a novel acquisition function for Bayesian optimization called Expected Maximum Improvement over Confident Regions (EMICoRe) which can actively exploit the inductive bias of the VQE-kernel by treating regions with low predictive uncertainty as indirectly ``observed''. As a result, observations at as few as three points in the search domain are sufficient to determine the complete objective function along an entire one-dimensional subspace of the optimization landscape. Our numerical experiments demonstrate that our approach improves over state-of-the-art baselines.

Figures (17)

• Figure 1: Illustration of EMICoRe (ours) and NFT nakanishi20 (baseline) procedures. In each step of NFT (bottom row), a) given the current best point $\hat{\boldsymbol{x}}^t$ and the direction $d^t$ to be explored, b) next observation points are chosen in a deterministic fashion. Then, c) the minimum along the line is found by the sinusoidal function fitting to the three points, and d) the next optimization step for the new direction $d^{t+1}$ starts from the found minimum $\hat{\boldsymbol{x}}^{t+1} = \hat{\boldsymbol{x}}^t_{\mathrm{min}}$. The EMICoRe procedure (top row) uses GP regression and BO in the steps highlighted by the light blue box: b) the next observation points are chosen by BO with the EMICoRe acquisition function based on the GP trained on the previous observations, and c) minimizing the predictive mean function of the updated GP with the new observations gives the best point $\hat{\boldsymbol{x}}^t_{\mathrm{min}}$.
• Figure 2: Comparison of our VQE-kernel (red) to the RBF and the periodic kernel benchmarks (blue and orange) in the VQE optimization using the standard BO procedure, for the Ising Hamiltonian with the $(L=3)$-layered $(Q=3)$-qubits quantum circuit. The search domain dimension is $D=24$, and $N_\mathrm{{shots}}=1024$ readout shots are taken for each observation. The energy (left) and the fidelity (right) are plotted, and in each plot, optimization progress is shown with the median (solid) and the 25- and 75-th percentiles (shadows) over 50 trials. The portrait square shows the distribution of the final solution after 150 observations have been performed.
• Figure 3: Comparison (in the same format as \ref{['fig:vqekernel']}) between our EMICoRe (red) and the NFT baselines (green and purple) in the VQE for the Ising (top row) and Heisenberg (bottom row) Hamiltonians with the $(L=3)$-layered $(Q=5)$-qubits quantum circuit (thus, $D =40$) and $N_\mathrm{{shots}}=1024$. We confirmed for the Ising Hamiltonian that longer optimization by EMICoRe up to 6000 observed points reaches the ground state with $98\%$ fidelity (see \ref{['sec:A.ablationLongRuns']}).
• Figure 4: Evolution of the GP in NFT-with-EMICoRe. The posterior mean (blue solid) with uncertainty (blue shadow) along the direction $d$ to be optimized in the steps $t=0, 1, 293, 294$ (columns) are shown before (top row) and after (bottom row) the chosen two new points $\boldsymbol{X}' = (\boldsymbol{x}_1', \boldsymbol{x}_2')$ are observed. The red solid curve shows the true energy $f^*(\boldsymbol{x})$. On the top, the number of observed samples $|\boldsymbol{X}|$ until step $t-1$ and the direction $d$ are shown.
• Figure 5: Illustration of the VQE workflow. In the first step, highly complicated optimization problems can be translated into the Hamiltonian formulation (see, e.g., Mohammadbagherpoor:2021zqaChai:2023ixt). The Hamiltonian $H$ and an initial state $\vert\psi_0\rangle$ are plugged into the VQE block (light purple), where the variational quantum circuit is instantiated with random angular parameters $\mathbf{x}^0$. In the VQE block, the top vertex of the triangle represents the quantum computer, the red arrows and blocks refer to operations running on a quantum computer, while the green parts refer to classical steps. In the bottom-left green box, the current parameters $\mathbf{x}$ are updated with the new best parameters $\hat{\mathbf{x}}$ found during classical optimization routines. Then, the quantum circuit $G(\mathbf{x})$ is updated using the new optimum point, $\mathbf{x}\to \hat{\mathbf{x}}$, and the energy $E({\mathbf{x}})$ for the updated variational wave function $\vert\psi_{{\mathbf{x}}}\rangle$ is measured. The VQE block is executed for $T$ iterations and finally outputs the solution to the task as a variational approximation $\vert{\psi_{\hat{\mathbf{x}}^T}}\rangle$ of the ground state and the corresponding optimal parameters $\hat{\mathbf{x}}^T$ for the quantum circuit.

Theorems & Definitions (10)

• Proposition 1
• Proposition 2
• Theorem 1
• Theorem 2
• Definition 1
• proof
• Proposition 3
• Theorem 3
• proof
• proof