Learning Hamiltonians for $O(1)$ Oracle-Query Quantum State Preparation

TL;DR

This work introduces a Hamiltonian-based framework for quantum state preparation that shifts the bulk of computation to classical preprocessing by learning a diagonal Hamiltonian whose fixed-depth quantum evolution encodes the target amplitudes. By either oracle-accessing the learned diagonal or expanding it in a Walsh basis with a polynomial number of terms, the method achieves O(1) quantum query complexity, with a classical cost of O(N log N) and hardware-efficient, one-/two-local circuit implementations. The results show that two-layer Hamiltonian evolutions are necessary for accurate amplitude synthesis, with favorable fidelity and scalable runtimes on structured datasets when using Walsh truncation. The approach demonstrates a practical pathway to quantum advantage on near-term devices through complexity transfer and data-structure exploitation.

Abstract

We propose a Hamiltonian-based quantum state preparation method implemented via a shallow parametrized quantum circuit. The approach learns the parameters of a diagonal Hamiltonian through a classical training phase, while the quantum circuit itself performs only fixed-depth Hamiltonian evolution and mixing operations. With oracle access to the learned Hamiltonian parameters, $N$ classical data values can be encoded into $n=\log_2{N}$ qubits using $O(1)$ quantum queries, shifting the overall computational cost to an $O(N\log{N})$ classical preprocessing stage. For structured datasets generated by an underlying function, oracle access can be avoided by expressing the Hamiltonian in the Walsh basis and retaining only a polynomial number of significant terms. In this regime, quantum state preparation is achieved in $\text{poly}(n)$ time using $\text{poly}(n)$ parameters, reaching infidelities on the order of $10^{-5}$. By restricting the Hamiltonian to one-local and two-local terms, the method naturally yields hardware-efficient circuits suitable for near-term quantum devices.

Figures (6)

• Figure 1: Quantum circuit of the proposed Hamiltonian‑simulation‑based state preparation algorithm.
• Figure 2: Quantum circuit for simulating diagonal Hamiltonians using oracle access. The oracle $U_{k}$ encodes the coefficient $h_{k,j}$ into an ancillary register, single‑qubit phase gates $P(\theta)$ apply the corresponding evolution $e^{-ih_{k,j}}$ to the computational basis state, and the inverse oracle $U_{k}^{\dagger}$ uncomputes the ancilla to $\ket{0}$
• Figure 3: Quantum circuit implementing the unitary $e^{-ic_{11}W_{11}}$ corresponding to the Walsh operator ${W_{11}=Z^{(3)}\otimes Z^{(1)}\otimes Z^{(0)}}$. Because the binary representation of $r=11$ is $1011$, the operation acts nontrivially on qubits 0, 1, and 3. A more depth‑efficient decomposition achieving the same gate count can be found in sriluckshmy2023optimal.
• Figure 4: Summary of training performance, sample‑size dependence, and classical runtime scaling.
• Figure 5: Simulation results for structured datasets. Panels (a–d) show linear and sine state preparations using Two‑Local and Hardware‑Efficient Hamiltonians, illustrating amplitude reconstruction and fidelity comparison.