Quantum Algorithm for Low Energy Effective Hamiltonian and Quasi-Degenerate Eigenvalue Problem

1. Problem Statement

The paper addresses the challenge of solving quasi-degenerate eigenvalue problems in quantum chemistry and condensed-matter physics.

• Context: In many physical systems (e.g., near quantum phase transitions, in open-shell molecules, or topological phases), the relevant physics is governed not by a single isolated ground state, but by a low-energy manifold of nearly degenerate states.
• Limitations of Existing Methods:
  • Classical: Effective Hamiltonian (downfolding) methods exist but suffer from exponential memory costs when calculating exact resolvents or require perturbative expansions that break down in strongly correlated regimes.
  • Quantum: Existing fault-tolerant algorithms (e.g., Quantum Phase Estimation, QSVT-based filtering) are primarily designed for single-target eigenstate preparation. They lack a systematic workflow to:
    1. Determine the dimension of a quasi-degenerate subspace.
    2. Produce a controlled, orthonormal basis spanning that subspace.
    3. Handle cases where the initial guess has poor overlap with specific orthogonal states within the degenerate cluster.

2. Methodology

The authors propose a fault-tolerant quantum algorithm that operates directly at the subspace level using the Feshbach/Schur-complement formalism. The core idea is to map the full high-dimensional eigenproblem onto a low-dimensional reference subspace ($P$-space) and solve a nonlinear eigenproblem there.

Key Technical Components:

1. Feshbach Projection & Effective Hamiltonian:

   • The Hilbert space is split into a reference subspace $PH$ (dimension $d$) and its complement $QH$.
   • The full Hamiltonian $H$ is projected to an energy-dependent effective Hamiltonian acting on $PH$:
     $$H_{\text{eff}}(\lambda) = H_{PP} - H_{PQ}(H_{QQ} - \lambda I)^{-1}H_{QP}$$
   • The eigenvalues of $H$ correspond to the fixed points where the eigenvalues of $H_{\text{eff}}(\lambda)$, denoted $\xi_i(\lambda)$, satisfy $\xi_i(\lambda) = \lambda$.

2. Quantum Implementation via Block-Encoding and QSVT:

   • Resolvent Approximation: The term $(H_{QQ} - \lambda I)^{-1}$ (the resolvent) is approximated using Quantum Singular Value Transformation (QSVT). A polynomial $f(x) \approx 1/x$ is applied to the block-encoded operator $H_{QQ} - \lambda I$.
   • Self-Energy Construction: The "self-energy" term $\Sigma(\lambda) = -H_{PQ}(H_{QQ} - \lambda I)^{-1}H_{QP}$ is constructed as a block-encoded operator.
   • P-Controlled NOT Gate: A crucial primitive is a $P$-controlled NOT gate, which allows the algorithm to coherently access the projected blocks $H_{PQ}$ and $H_{QP}$ without explicitly constructing the full matrices. This is efficiently implemented if the reference subspace is spanned by classically tractable states (e.g., Slater determinants).

3. Algorithm Workflow:

   • Eigenvalue Estimation:
     1. Use generalized Hadamard tests to estimate the $d \times d$ matrix elements of $H_{\text{eff}}(\lambda)$ on a classical computer.
     2. Diagonalize the estimated matrix to find eigenbranches $\hat{\xi}_i(\lambda)$.
     3. Perform a noisy bisection (root-finding) on the residual $\hat{\mu}_i(\lambda) = \hat{\xi}_i(\lambda) - \lambda$ to find the fixed points $\hat{\lambda}_k \approx \lambda_k$.
   • Eigenstate Preparation:
     1. Once eigenvalues are found, the corresponding eigenvectors in the $P$-space are "lifted" to the full Hilbert space using a block-encoded wave operator $\Omega(\lambda)$.
     2. Löwdin Orthonormalization: Since the lifted states may not be orthogonal, a Gram matrix is estimated and inverted (symmetrically) to produce an orthonormal basis for the quasi-degenerate manifold.

3. Key Contributions & Theoretical Guarantees

The paper provides rigorous error bounds and complexity guarantees, which are often missing in classical subspace methods.

• Eigenvalue Accuracy:

  • The error scales as $|\hat{\lambda}_k - \lambda_k| = O(\gamma_k^2 \epsilon)$, where $\gamma_k = \sqrt{\langle \Psi_k | P | \Psi_k \rangle}$ is the overlap between the true state and the reference subspace, and $\epsilon$ is the estimation precision.
  • Query Complexity: Scales as $\tilde{O}(1/\epsilon)$, which is nearly optimal.

• State Preparation Fidelity (Second-Order Robustness):

  • Non-degenerate case: The infidelity scales quadratically with the error parameter: $1 - F = O(\gamma_k^2 \epsilon^2 / \Delta^2)$, where $\Delta$ is the spectral gap.
  • Quasi-degenerate case: The algorithm guarantees the preparation of the entire eigenspace with a minimum fidelity bound scaling as $O(\gamma_{\max}^2 \epsilon^2 / \Delta^2)$.
  • Significance: This "second-order robustness" means that small errors in the effective Hamiltonian entries result in even smaller errors in the state fidelity.

• Handling Intruder States:

  • The complexity depends on a spectral distance parameter $g = \text{dist}(\lambda_k, \text{spec}(H_{QQ}))$. If the target energy is close to a "pole" (an intruder state in the $Q$-space), the polynomial degree required for QSVT increases ($O(1/g)$). The paper analyzes the trade-off between the overlap $\gamma_k$ and the distance $g$.

4. Numerical Results

The algorithm was benchmarked on three diverse systems, demonstrating its ability to resolve low-energy manifolds and track eigenstates through crossings:

1. 3×3 Fermi-Hubbard Model:

   • Simulated a system with level crossings and degeneracies.
   • Result: Accurately tracked energy spectra and resolved degeneracies (2-fold and 4-fold) with relative errors $< 10^{-3}$ and infidelities $< 10^{-5}$.

2. Lithium Hydride (LiH) Molecule:

   • Tested along a bond-stretching coordinate where quasi-degeneracy becomes severe near dissociation.
   • Result: Achieved chemical accuracy ($< 1$ kcal/mol) for ground and excited states across the bond length, successfully resolving a 6-fold degenerate excited state without spurious splitting.

3. Transition Metal Complex [Ru(bpy)$_3$]$^{2+}$:

   • A realistic complex with a dense manifold of low-lying excited states (Metal-to-Ligand Charge Transfer).
   • Result: Reproduced Full Configuration Interaction (FCI) quality energies within the active space with deviations $\sim 10^{-4}$ Hartree. Correctly identified the 3-fold degenerate triplet $T_1$ state.

5. Significance and Impact

• Systematic Subspace Approach: Unlike previous quantum algorithms that treat eigenstates individually, this method treats the manifold as the fundamental object, providing a systematic way to detect degeneracy and output a basis.
• Bypassing Classical Bottlenecks: By using QSVT to implicitly evaluate the exact resolvent, the algorithm avoids the exponential memory cost of classical downfolding and the breakdown of perturbative expansions in strongly correlated regimes.
• Robustness: The quadratic scaling of state infidelity with respect to eigenvalue error makes the method highly robust against noise and approximation errors in the effective Hamiltonian.
• Broad Applicability: The framework is applicable to topological phases, non-adiabatic dynamics, and multireference quantum chemistry problems where low-energy physics is dominated by clustered states.

In summary, this work presents a rigorous, fault-tolerant framework for solving the most challenging class of eigenvalue problems in quantum simulation—those involving quasi-degenerate manifolds—bridging the gap between effective Hamiltonian theory and modern quantum algorithms.