Optimal Clifford Initial States for Ising Hamiltonians

TL;DR

This work extends CAFQA-style classical bootstrapping to Transverse Field Ising Hamiltonians by reframing the Clifford-state ground-state search as a polynomial-time submodular minimization over vertex subsets: minimize $f(V)=-|E(V)|-g|V_G\setminus V|$ to obtain a Clifford initialization $|\varphi_0\rangle$ with energy $\langle\varphi_0|H|\varphi_0\rangle = -|E(V)|-g|V_G\setminus V|$. The construction leverages stabilizer theory to relate Clifford states to graph-theoretic structures, enabling an explicit Clifford state via $|\varphi\rangle = \prod_{q_j\notin V} R_Y\left(\frac{\pi}{2}\right)_j|0\rangle$. Numerical experiments across linear, fully connected, and random graphs show relative errors typically between 0% and 25%, with distinct regimes as $g\to0$ or $g\to\infty$ and transition behaviors captured by the edge function $\mathcal{E}(n)$. The framework generalizes to weighted Ising and certain Heisenberg models, highlighting a path toward efficient, principled VQE initializations on NISQ devices. Overall, the paper provides a rigorous, scalable method to obtain high-quality Clifford initial states that can bootstrap quantum simulations for Ising-type problems.

Abstract

Evaluating quantum circuits is currently very noisy. Therefore, developing classical bootstraps that help minimize the number of times quantum circuits have to be executed on noisy quantum devices is a powerful technique for improving the practicality of Variational Quantum Algorithms. CAFQA is a previously proposed classical bootstrap for VQAs that uses an initial ansatz that reduces to Clifford operators. CAFQA has been shown to produce fairly accurate initialization for VQA applied to molecular chemistry Hamiltonians. Motivated by this result, in this paper we seek to analyze the Clifford states that optimize the cost function for a new type of Hamiltonian, namely Transverse Field Ising Hamiltonians. Our primary result connects the problem of finding the optimal CAFQA initialization to a submodular minimization problem which in turn can be solved in polynomial time.

Figures (8)

• Figure 1: Comparison of the optimal Clifford state to the true ground state expectation for the selected $9$ qubit graphs. Notice that the maximal errors line up with the transition values and that all of these graphs are two segmented.
• Figure 2: Plot of the relative error for $L_n,P_n,$ and $K_n$ for a range of values for $n$. Notice that all of the relative errors decrease with $n$ and eventually begin plateau.
• Figure 3: Plot of the mean relative error for different amounts of vertices. Notice that the value of $g$ for which the maximum mean relative error is obtained increases with $N$ as expected.
• Figure 4: Comparison of the optimal Clifford state to the true ground state expectation for the selected graphs. Notice that none of these graphs are two segmented.
• Figure 5: First $5$ open linear chains

Theorems & Definitions (26)

• Definition 3.1
• Definition 3.2
• Definition 3.3
• Definition 3.4
• Corollary 3.5
• Corollary 3.6
• Definition 3.7
• Lemma 3.8
• Definition 3.9
• Definition 3.10