Full Configuration Interaction Quantum Monte Carlo

• FCIQMC is a stochastic projector method that accurately samples the full configuration interaction wavefunction using a population of signed walkers.
• The initiator adaptation reduces the fermionic sign problem by restricting walker spawning to well-sampled determinants, ensuring efficient convergence.
• Benchmark studies, such as on the homogeneous electron gas, demonstrate FCIQMC’s capability to deliver variational energies that rival or surpass fixed-node DMC results.

Full Configuration Interaction Quantum Monte Carlo (FCIQMC) is a stochastic projector method developed to sample the exact ground-state wavefunction—expressed as a linear combination of all possible Slater determinants—for correlated quantum many-body systems. Unlike diffusion Monte Carlo (DMC), which is constrained by the fixed-node approximation, FCIQMC operates directly in determinant space and eschews any explicit nodal constraint, thus yielding variational energies that do not depend on trial nodal surfaces. FCIQMC achieves high accuracy with modest computational resources by harnessing a discrete population of signed walkers to stochastically evolve the imaginary-time Schrödinger equation. Through innovations such as the initiator adaptation, semi-stochastic propagation, and explicit correlation treatments, FCIQMC has established itself as a benchmark tool for ground and excited states in molecular and extended systems, even those with determinant spaces exceeding 10^100.

1. Theoretical Foundations and Algorithmic Structure

At its core, FCIQMC samples the FCI wavefunction

$|\Psi\rangle = \sum_i C_i |D_i\rangle$

where $|D_i\rangle$ spans all determinants formed from a finite one-particle basis. The evolution in imaginary time,

$$-\frac{dC_i}{d\tau} = (H_{ii} - S)C_i + \sum_{j \neq i} H_{ij} C_j,$$

is simulated using a population of “walkers” that reside on determinants, carrying discrete amplitudes (sign and magnitude) and representing the expansion coefficients $C_i$.

Each discrete iteration involves:

• Spawning: Walkers on determinant $i$ attempt to create offspring on connected determinants $j$, with probability proportional to the off-diagonal Hamiltonian element $H_{ji}$.
• Death/Cloning: Each walker is subject to an adjustment determined by $H_{ii} - S$, controlling overall population change.
• Annihilation: Walkers of opposite sign occupying the same determinant are cancelled, addressing the fermionic sign problem.
• Population control: The energy shift $S$ is dynamically tuned to stabilize the walker population.

This protocol enables the simulation to project stochastically onto the ground state of a Hamiltonian acting in determinant space, even in Hilbert spaces with intractable explicit storage requirements.

2. Initiator Adaptation and Control of the Sign Problem

For large systems, the exponential growth of the determinant space makes unbiased FCIQMC simulations computationally prohibitive due to the sign problem: noise and population noise can overwhelm signal before the sign structure is established.

The initiator adaptation (iFCIQMC) addresses this by introducing a mechanism to restrict walker spawning from under-sampled regions:

• Initiators: Determinants populated above a threshold ($n_{\mathrm{add}}$) can spawn onto any unoccupied determinant.
• Non-initiators: Can only spawn onto already-occupied determinants.

The initiator criterion curtails “spurious” population growth into the exponentially vast determinant space, effectively focusing sampling on a manageable subspace. In the infinite walker limit, iFCIQMC recovers the exact FCI solution for the finite basis. This adaptation has been shown to drastically ameliorate the sign problem and enables accurate calculations in large periodic systems (e.g., the 54-electron homogeneous electron gas with $10^{108}$ determinants) (Shepherd et al., 2011), with errors decaying exponentially with walker population.

3. Application to the Homogeneous Electron Gas and Periodic Systems

A benchmark application is the 54-electron homogeneous electron gas (HEG), where the Hamiltonian,

$$H = \sum_\alpha (-\nabla_\alpha^2) + \sum_{\alpha \neq \beta} \hat{v}_{\alpha\beta} + N v_M,$$

is represented in a finite basis of plane-wave Slater determinants: $$\psi_j(r, \sigma) = (1/\sqrt{\Omega}) e^{i k_j \cdot r} \delta_{\sigma, \sigma_j}$$ with a kinetic energy cutoff. This system is challenging due to the extensive Hilbert space and need for high accuracy to compare with DMC results.

For the HEG at $r_s = 1.0$ a.u., finite-basis FCIQMC returns energies slightly below fixed-node DMC but, after extrapolation, matches DMC backflow results, indicating residual DMC fixed-node errors are small at these densities. At higher densities ($r_s = 0.5$ a.u.), FCIQMC energies are variationally lower than DMC with backflow, suggesting non-negligible fixed-node bias even for optimized DMC trial nodes. Notably, sampling a tiny fraction of the $10^{108}$-dimensional space with hundreds of millions of walkers is computationally tractable (300,000 core hours).

4. Basis Set Extrapolation and Incompleteness Correction

Plane-wave methods for periodic systems introduce basis set incompleteness errors due to the finite kinetic energy cutoff. FCIQMC outputs for a given cutoff provide rigorous upper bounds to the true ground state, but systematic removal of basis errors is required.

The error in the correlation energy scales as $1/M$, with $M$ the number of spatial orbitals; thus, extrapolating energies as a function of $1/M$ yields the complete basis set (CBS) limit. FCIQMC results, once extrapolated, consistently align with or outperform the best DMC data—for both total and correlation energies—demonstrating the capacity of stochastic determinant-space methods to deliver ground-state benchmarks for periodic electron gases.

5. Efficiency, Population Control, and Bias

FCIQMC achieves high efficiency provided attention is paid to walker population control. The Markov chain analysis of FCIQMC (Vigor et al., 2014) characterizes the bias in energy estimation arising from population control: it decays as $1/\langle N \rangle$, with smaller damping in energy shift updates (parameter $\xi$) reducing the bias. Post-processing reweighting schemes developed for DMC can also be used in FCIQMC to remove this bias without affecting the statistical error, especially important in simulations targeting high accuracy for subtle energy differences.

Resource requirements are determined by the walker population necessary to equilibrate across the sign problem plateau, which is a function of the system's correlation strength and the efficiency of the annihilation process. Notably, the computational cost does not scale with the total number of Slater determinants, but rather with the minimum population of walkers required to faithfully resolve the sign structure.

6. Comparison with Diffusion Monte Carlo and Broader Implications

A critical comparison with fixed-node DMC positions FCIQMC as a variational, nodal-surface-free method. Where DMC is limited by the quality of its trial nodal surfaces, FCIQMC provides an alternative route, yielding upper bounds within the chosen basis. The extrapolated FCIQMC energies in HEG sometimes lie below DMC, highlighting persistent fixed-node error in DMC for some densities even with sophisticated trial wavefunctions. This positions FCIQMC as an invaluable tool for benchmarking not just DMC but also for assessing methods constrained by reference biases or approximations.

The ability to produce definitive results for large, periodic, and metallic systems opens new opportunities for studying collective electronic properties—momentum distribution, Fermi liquid parameters—and benchmarking quantum Monte Carlo, coupled cluster, and density functional methods. The extension of FCIQMC to metals or excited-state response properties via further algorithmic developments is anticipated.

7. Prospects and Ongoing Developments

The findings in the homogeneous electron gas drive several future research avenues:

• Algorithmic advances: Further taming of the sign problem, more effective semi-stochastic expansions, and parallelization to facilitate ever-larger systems.
• Property calculations: Beyond energies, FCIQMC’s unbiased wavefunction sampling enables computation of momentum distributions, density response, and other correlation functions for periodic systems.
• Improved extrapolation: Refined basis set extrapolation techniques, potentially leveraging analytic results for scaling with cutoff or spatial orbitals, could further enhance result accuracy.
• Application scope: FCIQMC’s flexibility positions it for benchmarking and method development in real metals, correlated insulators, and model Hamiltonians, particularly where conventional methods fail or where nodal constraints are difficult to control.

The method's demonstrated ability to deliver variationally accurate, unbiased energies in determinant spaces that far exceed the reach of explicit diagonalization marks FCIQMC as a reference platform for correlated electron benchmarks in condensed matter and quantum chemistry (Shepherd et al., 2011).