Quantum Amplitude-Amplification Eigensolver: A State-Learning-Assisted Approach beyond Energy-Gradient-Based Heuristics

TL;DR

The paper tackles robust ground-state estimation for near-term quantum simulators by introducing the quantum amplitude-amplification eigensolver (QAAE), a non-variational, hardware-aware approach that coherently amplifies the ground-state component and then learns a re-encoded target without energy-gradient optimization. Each round executes an amplitude-amplification step with $\hat{T}(\boldsymbol\theta)=\hat{R}(\boldsymbol\theta)\hat{U}\hat{R}(\boldsymbol\theta)\hat{U}^\dagger$, where $\hat{U}=\sum_{k=0}^1 i^k |k\rangle\langle k| \otimes e^{(-1)^k i \omega \hat{H}}$ and $\omega=\tfrac{\pi}{4}$, followed by a state-learning update that re-encodes the amplified state into the next-round trial. The authors prove monotone convergence of the ground-state overlap per round under standard assumptions, derive a polynomial-depth bound per round, and provide a stability result that tolerates learning imperfection. They validate QAAE on IBMQ hardware for simple and Ising-type systems and benchmark on H$_2$, LiH, and a 10-qubit LTFIM, showing improved accuracy and robustness relative to gradient-based VQE while remaining hardware-conscious via modular ansatz design and short-time Hamiltonian simulation. The results suggest QAAE as a principled, non-variational route to near-term quantum simulation, with potential extensions in adaptive learning, symmetry-aware encoding, and advanced short-time simulation techniques.

Abstract

Ground-state estimation lies at the heart of a broad range of quantum simulations. Most near-term approaches are cast as variational energy minimization and thus inherit the challenges of problem-specific energy landscapes. We develop the quantum amplitude-amplification eigensolver (QAAE), which departs from the variational paradigm and instead coherently drives a trial state toward the ground state via quantum amplitude amplification. Each amplitude-amplification round interleaves a reflection about the learned trial state with a controlled short-time evolution under a normalized Hamiltonian; an ancilla readout yields an amplitude-amplified pure target state that a state-learning step then re-encodes into an ansatz circuit for the next round -- without evaluating the energy gradients. Under standard assumptions (normalized $\hat{H}$, a nondegenerate ground-state, and a learning update), the ground-state overlap increases monotonically per round and the procedure converges; here, a per-round depth bound in terms of the ansatz depth and Hamiltonian-simulation cost establishes hardware compatibility. Cloud experiments on IBMQ processor verify our amplification mechanism on a two-level Hamiltonian and a two-qubit Ising model, and numerical benchmarks on $\mathrm{H}_2$, $\mathrm{LiH}$, and a $10$-qubit longitudinal-and-transverse-field Ising model show that QAAE integrates with chemistry-inspired and hardware-efficient circuits and can surpass gradient-based VQE in accuracy and stability. These results position QAAE as a variational-free and hardware-compatible route to ground-state estimation for near-term quantum simulation.

Figures (11)

• Figure 1: Schematic of our QAAE setup. (a) The circuits for generating the trial state $\left|\Psi(\boldsymbol\theta)\right>$ and implementing the ground-state amplitude amplification $\hat{T}(\boldsymbol\theta)$. (b) The reflecting operation $\hat{R}(\boldsymbol\theta)$ is implemented by using $\hat{A}(\boldsymbol\theta)$ and some fundamental gates (Hadamard, Pauli-X($\hat{\sigma}_x$), Pauli-Z($\hat{\sigma}_z$), and $q$-Toffoli, etc.). (c) The state-learning [as in Eq. (\ref{['eq:opt']})] is performed for $\left|\varphi_{\text{out},k}\right>$. Then, a new vector $\boldsymbol\theta'$ is identified and the circuit elements $\hat{A}(\boldsymbol\theta)$ and $\hat{R}(\boldsymbol\theta)$ are (re)dialed: $\boldsymbol\theta \to \boldsymbol\theta'$.
• Figure 2: The predicted behavior of converging to the ground state in our QAAE. Let us start by a system output state $\left|\varphi_{\text{out},k}(r)\right>$ at any $r$-th round, which is closer to the ground state than those from the previous rounds. At this round, the state learning is completed and the control parameter vector is updated as $\boldsymbol\theta_{r} \to \boldsymbol\theta_{r+1}$, and the PQC states $\left|\alpha(\boldsymbol\theta_{r+1})\right>$ is prepared (see the blue solid line). By iterating these processes, the PQC states evolves into the ground state, which is depicted as a trajectory along the red dashed line.
• Figure 3: Ground‑state fidelity vs. number of QAAE rounds for the two‑level Hamiltonian. Both Qiskit emulation and ibm_yonsei experiments show monotone improvement consistent with Proposition \ref{['prop:gsAA']}.
• Figure 4: Expectation values $\left<\hat{\sigma_x}\right>$ and $\left<\hat{\sigma_z}\right>$ from (a) Qiskit emulation and (b) ibm_yonsei experiments (orange), and corresponding normalized Bloch vectors used for re‑encoding (blue). The red star marks the ground state. The normalization projects noisy Bloch vectors back to purity, mitigating isotropic‑type noise and improving update robustness.
• Figure 5: Single‑ancilla QAAE circuit for the two‑level Hamiltonian. One round consists of the reflection--evolution--reflection--inverse‑evolution pattern of Sec. \ref{['Sec:2']}, followed by ancilla measurement and re‑encoding of the amplified pure state.

Theorems & Definitions (6)

• Proposition 1: Ground-state amplitude amplification
• proof
• Lemma 1: Stability under learning error
• proof
• Proposition 2: Per‑round circuit depth
• proof : Proof sketch