Quantum Amplitude-Amplification Eigensolver

• Quantum Amplitude-Amplification Eigensolver is a class of quantum algorithms that combine state preparation, phase estimation, and Grover-type iterations to achieve quadratic speedup in eigenstate and eigenvalue extraction.
• It employs block-encoding and probabilistic imaginary-time evolution to mitigate numerical instabilities and reduce overhead in large, ill-conditioned systems.
• Applications span quantum chemistry, materials science, and vibrational analysis, enabling efficient simulations by dramatically cutting measurement and runtime costs.

The Quantum Amplitude-Amplification Eigensolver (QAAE) is a class of quantum algorithms that exploit amplitude amplification—a generalization of Grover’s search—to accelerate the preparation or identification of desired eigenstates or eigenvalues of large matrices, most commonly quantum Hamiltonians and generalized eigenproblems. QAAE frameworks combine state preparation, quantum phase estimation, and amplitude amplifier circuits to quadratically reduce runtime and measurement overhead relative to both direct probabilistic quantum algorithms and classical eigensolvers. Applications span quantum chemistry, materials modeling, vibrational analysis, and situations demanding robust access to excited and ground states, especially in settings marked by ill-conditioned matrices or exponentially large Hilbert spaces (Rajchel-Mieldzioć et al., 16 Jun 2025, Danz, 18 Sep 2025, Nishi et al., 2022, Baek et al., 15 Nov 2025, Nishi et al., 2023).

1. Fundamental Principles and Algorithmic Structure

QAAE algorithms are constructed by embedding traditional quantum eigenpair-solving subroutines within an amplitude-amplification loop. The general workflow comprises:

• State Preparation: A multipartite register is initialized as an equal-weighted or targeted superposition over system and auxiliary (parameter, ancilla) degrees of freedom.
• Operator Application: Block-encoded operators represent parameter-dependent matrix families $M(\alpha)$ or Hamiltonians $H$. For generalized eigenproblems, $M(\alpha)=H-\alpha S$ or $M(\alpha)=\sum_j \alpha^j M_j$ appears explicitly (Rajchel-Mieldzioć et al., 16 Jun 2025). Imaginary-time evolution may be simulated via probabilistic (nonunitary) maps (Nishi et al., 2022, Nishi et al., 2023).
• Quantum Phase Estimation (QPE): Used in CES-type solvers to extract spectral information (phases) corresponding to eigenvalues or singular values, often with error reduction by median-of-$c$ parallel QPEs (Rajchel-Mieldzioć et al., 16 Jun 2025, Danz, 18 Sep 2025).
• Region Oracle/Reflection: Ancilla qubits (oracles) mark “good” subspaces (e.g., eigenvalues in a desired window, ancillas in the all-zero state) for reflection.
• Grover-Type Amplitude Amplification: Alternating reflections (about the initial state and the good subspace) generate iterative rotations in a two-dimensional Hilbert subspace, quadratically increasing the amplitude (and thus the measurement probability) of the solution subspace.
• Measurement and Post-Processing: The measurement of marked qubits provides, with amplified probability, labels or projections of the desired eigenstates or eigenvalues.

These ingredients admit several concrete algorithmic instantiations, such as for generalized eigenproblems via collocation (Rajchel-Mieldzioć et al., 16 Jun 2025), “complete eigenpair solver” (CES) filtering (Danz, 18 Sep 2025), and probabilistic imaginary-time evolution (PITE) with QAA (Nishi et al., 2022, Nishi et al., 2023).

2. Quantum Amplitude-Amplification for Generalized Eigenproblems

QAAE for generalized eigenvalue problems operates by discretizing the parameter $\alpha$ and block-encoding the family $M(\alpha)$ on a combined register. The procedure is as follows (Rajchel-Mieldzioć et al., 16 Jun 2025):

1. Block-Diagonal Embedding: Construct $\tilde{M} = \sum_{j=0}^J D^j(\alpha) \otimes M_j$, with $D(\alpha)$ diagonal on the “parameter” register and $M_j$ on the “system” register, yielding a joint Hilbert space of dimension $NK$ ($N$ system size, $K$ parameter discretization).
2. PREP Circuit: Prepare $$(1/\sqrt{NK}) \sum_{i=0}^{NK-1} |i\rangle_k \otimes |i\rangle_n$$.
3. Quantum Phase Estimation: Apply to $\tilde{M}$, with multiple QPE repetitions ($c+1$ clock registers) and median calculation to robustly estimate singular or eigenvalues.
4. Region Oracle: Flip an ancilla if $|\tilde{\lambda}_i - \lambda_0| < \epsilon$.
5. Amplitude Amplification: Define good/bad subspaces, use Grover iterate $G=DR$ (where $R=I-2|\chi_+\rangle\langle\chi_+|$ and $D=2|\Psi\rangle\langle\Psi| - I$), and iterate $k=O(\sqrt{NK})$ times.
6. Measurement: Collapse to the “good” $\alpha_i$ with success probability $\sin^2((2k+1)\theta)$, $\sin^2\theta = m/(NK)$ ($m$ targets).

QAAE can avoid the numerical instabilities of classical collocation by minimizing singular values directly, eliminating matrix inversion and its associated condition-number $\kappa$ amplification (Rajchel-Mieldzioć et al., 16 Jun 2025).

3. Integration with Imaginary-Time Evolution and Probabilistic Algorithms

Imaginary-time evolution algorithms based on nonunitary updates are inherently probabilistic; their post-selection success probabilities decay exponentially with system size or number of steps—an impediment for deep state preparation (Nishi et al., 2022, Nishi et al., 2023). QAAE achieves quadratic speedup by:

• Block-Encoding the PITE Operator: Implement $U_{\rm PITE}$ as a controlled unitary acting via Kraus construction, yielding a success amplitude $a=\sqrt{p(\Delta\tau)}$ on the “good” state (ancilla $\ket{0}$).
• Generalized Grover Iterate: Compose the preamplified operator $\widetilde{Q}$, designed to minimize circuit depth per iteration.
• Amplification: Repeat $O(1/a)$ times, raising the overall success probability from $\sim a^2$ to near-unity, thereby transforming quadratic measurement overhead into linear.
• Deterministic Imaginary-Time Evolution: By tuning parameters (notably $\gamma$), the algorithm can guarantee unity success with a fixed number of amplification steps. This removes the requirement for repeated post-selection.

This acceleration transforms the cost of ground-state preparation from $O(1/|c_1|^2)$ to $O(1/|c_1|)$ in overlap $|c_1|^2$ with the target state, outperforming both probabilistic and standard QPE approaches in this regime (Nishi et al., 2023).

4. Amplitude-Amplified Eigensolvers with State Learning

A recent algorithmic branch introduces a hybrid quantum-classical scheme, where each amplitude amplification round is followed by a classical state-learning step to re-encode the system register into an efficient ansatz. The procedure iteratively amplifies the ground-state component using “Grover-like” operators and then learns the output state for re-use in the next round (Baek et al., 15 Nov 2025):

1. Trial State Preparation: $$\ket{\Psi(\boldsymbol{\theta})} = \ket{+}_{\rm anc}\otimes\ket{\alpha(\boldsymbol{\theta})}$$, where the ansatz $\hat{A}(\boldsymbol{\theta})$ is parameterized.
2. Controlled Short-Time Evolution: Apply a controlled-$e^{\pm i\omega\hat{H}}$ operation.
3. Reflection about the Trial State: Implemented as a Householder reflection.
4. Grover-Like Iteration: $$\hat{T}(\boldsymbol{\theta}) = \hat{R}(\boldsymbol{\theta}) \hat{U} \hat{R}(\boldsymbol{\theta}) \hat{U}^\dagger$$.
5. Measurement and State Learning: Upon measurement, the amplified state is learned (compiled) into the ansatz to form the input for the next round.
6. Monotonic Overlap Gain: Under standard conditions, each round strictly increases ground-state overlap. Convergence is guaranteed if the learning error is below a computable threshold.

Empirical results on both IBMQ hardware and simulated molecular systems (e.g., H$_2$, LiH, 10-qubit Ising models) confirm disciplinary advantages over traditional VQE, notably monotonic convergence, immunity to gradient-based barren plateaus, and adaptability to hardware-efficient or chemistry-inspired ansätze (Baek et al., 15 Nov 2025).

5. Complexity, Resource Estimates, and Scaling

QAAE achieves substantial resource gains over both classical and standard quantum approaches:

• Circuit Resources:
  • Block-encoding overhead: $O(\log N+\log K)$ qubits per term in $M_j$.
  • QPE cost per precision: $O(\|\tilde{M}\| t + \log(1/\epsilon))$ calls (Rajchel-Mieldzioć et al., 16 Jun 2025).
  • Amplitude amplification: Grover iteration count $O(\sqrt{NK})$ in the generalized eigenproblem context; $O(1/\sqrt{P_K})$ for PITE with QAA.
• Overall Scaling:
  • For quantum collocation: $T_{\rm total} = O(\zeta\,\sqrt{NK}/\epsilon)$.
  • When $K\sim1/\epsilon$, total $T$-count is $O(N/\epsilon\,\mathrm{polylog}(N,1/\epsilon))$, representing a quartic improvement in $N$ (from $N^2$ to $\sqrt{N}$) and quadratic in $K$ (from $K$ to $\sqrt{K}$) versus classical approaches (Rajchel-Mieldzioć et al., 16 Jun 2025).
• Memory:
  • Memory requirement is $O(\log N+\log(1/\epsilon))$ qubits for Hamiltonians and $O(\log NK+\log(1/\epsilon))$ for block-diagonal parameterizations.
• Sampling Overhead:
  • Post-processing sampling overhead for eigenstate extraction is reduced from $O(N^2)$ to $O(k^2)$ (for $k$ target eigenstates) by amplitude amplification (Danz, 18 Sep 2025).
• Robustness:
  • Avoids error amplification from high condition numbers ($\kappa$) intrinsic to matrix inversion-based classical approaches.

6. Applications and Implementation Considerations

QAAE is applicable in:

• Quantum Chemistry and Materials Science: Efficient computation of multiple eigenvalues (beyond ground states), and the paper of dense spectra in high-dimensional or ill-conditioned systems (Rajchel-Mieldzioć et al., 16 Jun 2025, Baek et al., 15 Nov 2025).
• Vibrational and Structural Analysis: Localization of band-limited modes in large mechanical and electrical networks (Danz, 18 Sep 2025).
• Quantum State Preparation: Deterministic preparation of ground states and low-lying excited states with tunable fidelity, particularly suitable for scenarios where initial-state overlap is difficult to guarantee.
• Performance Gains in Measurement/Extraction: Dramatic reduction in the number of quantum measurements for rare-event eigenvalue extraction, making output-restricted quantum computations competitive in real-world applications (Danz, 18 Sep 2025).

Circuit depth per amplification round is typically a sum of ansatz depth, Hamiltonian simulation depth (e.g., Trotter, QSP), and reflection/measurement subroutine depths. Implementation is compatible with block-encoding, hardware-efficient, or UCCSD ansätze, and reflectivity oracles can be efficiently constructed for practical eigenwindow or ancilla-flag marking (Rajchel-Mieldzioć et al., 16 Jun 2025, Baek et al., 15 Nov 2025).

7. Limitations and Future Directions

QAAE’s main limitations stem from:

• Ansatz Expressivity: In state-learning-integrated QAAE, failure to represent the amplified target within the chosen ansatz manifold stalls convergence (Baek et al., 15 Nov 2025).
• Classical State Learning Overhead: Quantum-classical hybrid loops introduce non-negligible classical optimization costs.
• Depth Constraints: For large systems (e.g., chemistry Hamiltonians), full amplitude amplification iterations or deep QPE may challenge near-term hardware.
• Knowledge of Overlap Parameters: For optimal scheduling and deterministic versions, prior estimation of ground-state overlap may be needed (Nishi et al., 2022, Nishi et al., 2023).
• Error Accumulation: While amplitude amplification is robust to moderate circuit noise, accumulation of small errors necessitates careful fault-tolerant design in deep applications.

Anticipated avenues include: integration with fixed-point QAA or quantum signal processing to bypass overlap estimation, hybrid amplitude amplification/variational strategies, improved error mitigation, warm-start techniques for higher initial overlap, and adaptation to larger systems via advanced simulation primitives and symmetry exploitation.

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Table: QAAE Variants and Principal Features

Variant / Reference	System Class	Key Feature
(Rajchel-Mieldzioć et al., 16 Jun 2025)	Generalized eigenproblem, collocation	Direct singular value minimization, robust to $\kappa$
(Danz, 18 Sep 2025)	Hermitian, sparse	AA post-filtering after QPE
(Nishi et al., 2022, Nishi et al., 2023)	Imaginary-time/PITE	Deterministic ITE via QAA, quadratic speedup
(Baek et al., 15 Nov 2025)	Hamiltonian ground states	Hybrid amplitude amplification & state-learning

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QAAE, through the synergistic deployment of amplitude amplification and quantum spectral routines, serves as a scalable, robust, and extensible framework for eigenvalue and eigenstate computation at the heart of quantum simulation, numerical analysis, and quantum-enhanced scientific computing.