Variational subspace methods and application to improving variational Monte Carlo dynamics

1. Problem Statement

Variational methods, particularly those using Neural Quantum States (NQS) or tensor networks, are powerful tools for approximating quantum many-body ground states and dynamics. However, simulating quantum dynamics via the Time-Dependent Variational Principle (TDVP) or Projected TDVP (p-tVMC) suffers from two primary sources of error:

1. Optimization Error: The inability of the variational ansatz to perfectly represent the true state at a given time step due to limited expressivity or finite sampling.
2. Discretization Error: Errors introduced by the time-stepping scheme (e.g., Taylor expansion of the time-evolution operator) used to advance the state.

Standard subspace methods (like Lanczos or Krylov subspace techniques) can improve accuracy by considering linear combinations of basis states. However, existing implementations in Variational Monte Carlo (VMC) typically optimize individual states sequentially to build a basis. This sequential approach leads to an accumulation of errors in later basis states and lacks a formal language to operate directly on the subspace as a single entity, rather than a collection of individual vectors.

2. Methodology

The authors propose a unified framework based on the Determinant State Mapping and a post-processing algorithm called Bridge.

A. The Determinant State Formalism

The core theoretical contribution is a formalism that treats a subspace $V$ of dimension $m$ as a single quantum state in a larger Hilbert space $\mathcal{H}^{\otimes m}$.

• Encoding: A subspace spanned by a basis $\{|\phi_k\rangle\}_{k=1}^m$ is encoded into a determinant state $|\phi_A\rangle$ via antisymmetrization:
  $$|\phi_A\rangle = \mathcal{S}_- |\phi_1\rangle \otimes \dots \otimes |\phi_m\rangle$$
  where $\mathcal{S}_-$ is the antisymmetrization operator.
• Basis Independence: Crucially, if the basis changes via an invertible matrix $B$, the determinant state changes only by a scalar factor $\det(B)$. This makes the mapping well-defined on the Grassmannian (the space of subspaces), effectively treating the subspace as a single quantum state up to a global phase.
• Generalized Operators:
  • Unitary Operators: Generalized as tensor products $\hat{U}^{\otimes m}$.
  • Hermitian Operators (Hamiltonians): Generalized as sums $\hat{H} = \sum_k \hat{H}_k$, where $\hat{H}_k$ acts on the $k$-th copy of the Hilbert space.
• Energy and Distance:
  • The energy of a subspace is defined as the expectation value of the generalized Hamiltonian on the determinant state. This recovers the trace of the Rayleigh matrix ($G^{-1}G(H)$), where $G$ is the Gram matrix and $G(H)$ is the projected Hamiltonian.
  • The distance between subspaces is defined via the Fubini-Study metric between their corresponding determinant states.

B. The "Bridge" Algorithm

Building on this formalism, the authors introduce Bridge, a computationally inexpensive post-processing method to improve variational dynamics.

• Input: A set of $m$ variational states $\{|\phi_k\rangle\}$ obtained at discrete time steps $t = k\delta$ from a standard method (e.g., p-tVMC).
• Process:
  1. Construct the subspace spanned by these states.
  2. Estimate the Rayleigh matrix $R = G^{-1}G(H)$ using a Monte Carlo estimator.
  3. Solve an effective Schrödinger equation within this subspace:
     $$\dot{\alpha}(t) = -i R \alpha(t)$$
     where $\alpha(t)$ are the time-dependent coefficients of the linear combination $|\psi_\alpha(t)\rangle = \sum_k \alpha_k(t) |\phi_k\rangle$.
  4. The solution is $|\psi_\alpha(t)\rangle = e^{-iRt} |\psi_\alpha(0)\rangle$.
• Capabilities: Bridge allows for:
  • Refinement: Improving the accuracy of states at the original time steps $t=k\delta$.
  • Interpolation: Generating accurate states at intermediate times between $k\delta$.
  • Extrapolation: Extending dynamics slightly beyond the last computed time step.

C. Estimators for the Rayleigh Matrix

The paper compares two estimators for the Rayleigh matrix:

1. Sum of States (SOS) Estimator: Estimates $G$ and $G(H)$ separately and combines them. It is cheaper per sample but suffers from numerical instability when basis states are nearly linearly dependent (common in dynamics), requiring pseudo-inversion with singular value cutoffs.
2. Determinant State Estimator: Directly estimates $G^{-1}G(H)$ using the determinant state formalism (Eq. 28 in the paper). It is more robust to linear dependence and yields lower variance, though it is computationally more expensive per sample.

3. Key Contributions

1. Formalism: Established a rigorous mathematical framework (Determinant State Mapping) to treat subspaces as fundamental objects, generalizing single-state variational principles to subspaces. This recontextualizes existing excited-state methods (like those by Pfau et al.) as natural generalizations.
2. Bridge Algorithm: Introduced a practical, low-cost post-processing tool that significantly mitigates discretization errors in variational dynamics without re-optimizing the neural network parameters.
3. Robust Estimation: Demonstrated that the Determinant State Estimator is superior to the Sum of States estimator for dynamics applications, particularly when dealing with near-linearly dependent basis states, ensuring consistent performance as the subspace size increases.

4. Results

The authors tested Bridge on the Transverse-Field Ising Model on square lattices ($4\times4$ and $8\times8$).

• Error Reduction:
  • In regimes dominated by discretization error (e.g., $h = 2h_c$), Bridge improved the infidelity of the dynamics by 3 to 4 orders of magnitude compared to the original p-tVMC states.
  • In regimes dominated by optimization error, Bridge provided limited improvement, confirming that it specifically targets discretization artifacts (which have a linear structure correctable by linear combinations) rather than ansatz limitations.
• Interpolation/Extrapolation: Bridge successfully interpolated between time steps and provided reasonable extrapolation, creating a continuous approximation of the dynamics.
• System Size: On an $8\times8$ lattice (beyond exact diagonalization capabilities), Bridge faithfully reproduced magnetization dynamics that the original p-tVMC states failed to capture accurately.
• Computational Cost: The cost of running Bridge is negligible compared to generating the original states. For example, on an $8\times8$ lattice, generating p-tVMC states took ~12–17 hours on 4–12 H100 GPUs, while running Bridge took only 15–60 minutes on a single A100 GPU.

5. Significance

• Systematic Post-Processing: Bridge offers a systematic way to "clean up" variational dynamics results. Since it is cheap, it can be applied routinely to any variational dynamics simulation to recover accuracy lost to time-step discretization.
• Theoretical Unification: The determinant state formalism provides a unified language for subspace methods, bridging the gap between ground-state optimization, excited-state estimation, and time-dependent dynamics.
• Scalability: The method is highly scalable and compatible with modern machine learning quantum states (CNNs, Transformers), making it a viable tool for large-scale quantum simulations where exact methods are impossible.
• Future Directions: The paper highlights the potential for "restarting" dynamics by compressing the improved linear combination state back into a single variational state, a challenge that could further enhance long-time simulations.

In summary, the paper presents a powerful theoretical framework and a practical algorithm that significantly enhances the reliability of variational quantum dynamics, specifically addressing the critical issue of time-discretization errors.