method-qmc

name

method-qmc

description

Use when a quantum Monte Carlo (QMC) reproduction needs method-level route and tool selection — stochastic series expansion (SSE) for sign-free spin/boson finite-temperature targets, constrained-path / phaseless auxiliary-field QMC (CPMC/AFQMC) for fermionic Hubbard ground states, or fixed-node diffusion Monte Carlo (DMC) for plane-wave solids using Quantum ESPRESSO orbitals. Triggers include "QMC", "SSE", "stochastic series expansion", "finite-temperature susceptibility", "sign problem", "AFQMC", "CPMC", "constrained path", "Hubbard ground state by Monte Carlo", "DMC", "diffusion Monte Carlo", "QMCPACK", "Slater-Jastrow", "twist averaging", "fixed-node".

Method QMC

QMC is the stochastic-sampling method class. This card owns method selection (step 1), software routing (step 2), and method-level setup (step 3, method side). Method internals are in ## Details; software parameter values live in the tool skills; paper- and model-specific facts live in /reproduce-paper and .knowledge/models/.

Three routes:

• SSE — stochastic series expansion for sign-problem-free spin/bosonic lattices, finite-temperature curves (and ground states via large β).
• CPMC/AFQMC — constrained-path / phaseless auxiliary-field QMC for interacting fermions. The survey covers both the zero-temperature (ground-state projection) and finite-temperature (grand-canonical) forms; the harness route — the official CPMC-Lab package — implements the ground-state form for repulsive Hubbard-type models.
• DMC — real-space fixed-node diffusion Monte Carlo for plane-wave solids, using Quantum ESPRESSO orbitals for a Slater-Jastrow trial wavefunction with twist averaging. Handled as a self-contained pipeline in the DMC route section below (tool skills /using-quantum-espresso + /using-qmcpack).

Sources

• Methodology reference (reproduction-grade algorithm, parameters, validation, gap analysis): references/qmc-methodology.md
• Track README: tracks/qmc/README.md
• Tool skills: /using-sse (SSE route), /using-cpmc-lab (CPMC/AFQMC route)
• Tool skills (DMC route): /using-quantum-espresso (QE orbital generation) and /using-qmcpack (Slater-Jastrow VMC/DMC)
• Survey (primary, AFQMC): Zhang, Auxiliary-Field Quantum Monte Carlo at Zero- and Finite-Temperature (2019) .knowledge/literature/quantum-monte-carlo/zhang_2019_auxiliary-field-quantum-monte-carlo.md — the methodology reference for the CPMC/AFQMC route.
• Reproduction target + CP algorithm details: Nguyen, Shi, Xu, Zhang, CPMC-Lab (2014) .knowledge/literature/quantum-monte-carlo/1407.7967_cpmc-lab-a-matlab-package-for-constrained-path-monte-carlo-c.md.
• SSE methodology: Sandvik, Computational Studies of Quantum Spin Systems (2010); QMC textbook: Becca & Sorella (2017).
• Software landscape (step 2): ipie (github.com/JoonhoLee-Group/ipie), QMCPACK AFQMC module, ALF, SmoQyDQMC.jl.

Select method — step 1

Suited for

• CPMC/AFQMC: ground states (and, in the grand-canonical form, finite temperature) of interacting many-fermion systems with two-body interactions — in any one-particle basis (lattice sites, plane waves, Gaussians), spanning correlated-electron models, cold Fermi gases, solids, and quantum chemistry. The constraint restores low-polynomial (~cube-of-size) scaling, exact only if the trial wavefunction is exact; production AFQMC routinely treats O(1000) electrons at accuracy comparable to CCSD(T) near equilibrium. In this harness the only AFQMC tool is CPMC-Lab — single-band repulsive Hubbard ground state; broader targets need a production code (route out).
• SSE: sign-problem-free spin and bosonic lattice models in a fixed basis — finite-temperature observables (susceptibility, magnetization, structure factor, stiffness, energy, Binder ratios), and ground states by taking β large.
• Sizes reached: CPMC-Lab demonstrates 1D chains up to 128 sites (65↑/63↓) and 2D up to 4×4 (5↑5↓, 7↑7↓), any filling; thermodynamic limit via a size series + twist averaging. SSE scales to ~10³ sign-free spins.

Route elsewhere when

• Frustrated or fermionic model with an uncontrolled sign problem outside a supported package route → official code / web search / another track.
• Real-time dynamics, generic unconstrained determinant QMC, or continuous-time impurity QMC → out of scope for this card.
• A 1D / quasi-1D ground state → /method-mps (DMRG) is usually cheaper and near-exact.

Options & trade-offs

Route / method	Good at	Weak at	Typical size
SSE (this card)	sign-free spin/boson, finite-T, large sizes	fermions, frustration (sign problem)	10²–10³ spins
CPMC/AFQMC (this card)	2D fermion ground states, no area-law bias	constraint bias; non-commuting observables need back-propagation	10s–100+ sites
DMRG (/method-mps)	1D/quasi-1D ground states, near-exact	2D area-law cost	wide cylinders
ED (/method-ed)	exact, any observable, the cross-check	exponential in size	~tens of sites
free-projection QMC	unbiased	variance grows exponentially (sign problem)	small / short

Select software — step 2

Tools — the route, the maintained SOTA, and a cross-check

• SSE route: StochasticSeriesExpansion.jl on Carlo.jl (Julia) — the harness default for sign-free spin/boson SSE.
• CPMC/AFQMC — harness tool: the official CPMC-Lab MATLAB package — pedagogical, frozen since 2014, single-band-Hubbard ground-state constrained-path. Easy to read and run, not production-scale. Tool specifics live in /using-cpmc-lab.
• CPMC/AFQMC — maintained production SOTA (not in the harness; route out / install on demand):
  • ipie (Apache-2.0; Python + C/C++; active through 2026; github.com/JoonhoLee-Group/ipie) — the leading phaseless AFQMC code: ground-state and finite-T, lattice and ab-initio (PySCF interface), single/multi-determinant and selected-CI trials, back-propagation, GPU + MPI. Best general-purpose choice once CPMC-Lab is outgrown.
  • QMCPACK AFQMC module (open / NCSA-style; C++; v4.2.0, 2026) — mature ph-AFQMC, ab-initio focused, NOMSD/PHMSD trials, back-propagated RDM observables, PySCF front end, leadership-HPC scale; GPU for single-determinant trials.
  • PAUXY is deprecated → use ipie.
• Independent cross-check (different method, not constrained-path): unconstrained finite-T determinant QMC — ALF (Fortran, lattice) or SmoQyDQMC.jl (Julia, very active) — numerically exact but sign-problem-limited; cross-validate, don't substitute.

Features to confirm

• SSE: thermalization, sweeps, chains, bins, β/temperature grid, autocorrelation/binning diagnostics, MPI — owned by /using-sse.
• CPMC-Lab: signature, parameters, .mat outputs, install/run mechanics — all owned by /using-cpmc-lab.

Options & trade-offs

Tool	Maintained	Lang	CP / ph	T=0 / T>0	Lattice / ab-initio	GPU	Use when
CPMC-Lab (harness)	frozen 2014	MATLAB	constrained-path	T=0	Hubbard	no	learning / small Hubbard ground states
ipie	yes (2026)	Python+C/C++	phaseless (+ free-proj)	both	both	yes	production AFQMC, ab-initio, GPU
QMCPACK AFQMC	yes (2026)	C++	phaseless	T=0	mainly ab-initio	partial	ab-initio at HPC scale
ALF / SmoQyDQMC.jl	yes (2026)	Fortran / Julia	unconstrained DQMC	T>0	lattice	no	finite-T lattice cross-check (sign-limited)

The SSE route uses StochasticSeriesExpansion.jl / Carlo.jl (Julia; native, MPI-parallel).

Efficiency & parallelism

Cost is ~O(N³) per propagation step (N = sites/basis) × walkers × steps, memory ~O(N²) per walker; AFQMC is embarrassingly parallel over walkers and over independent points (twists, sizes, Δτ). What each tool does with that:

Tool	Parallelism	Reach
CPMC-Lab (harness)	single-core MATLAB — no MPI/GPU/threading; parallel only by farming independent runs as array jobs (/using-slurm)	~100 sites locally (≈32 min @ 4×4 5↑5↓, ≈460 min @ 128×1)
ipie	MPI + GPU (batched propagation / force bias)	O(1000) electrons on GPU clusters
QMCPACK AFQMC	MPI + GPU (single-determinant trials)	leadership-HPC scale
ALF / SmoQyDQMC.jl	MPI, lattice DQMC	finite-T lattice; sign-limited

Handoff

• Invoke /using-sse once an SSE route is chosen — it owns thermalization, sweeps, chains, bins, the β grid, autocorrelation/binning diagnostics, MPI, and time estimate. The sign criterion stays in this card's Verification.
• Invoke /using-cpmc-lab once a CPMC/AFQMC route is chosen — it owns MATLAB setup, package install/discovery, the CPMC_Lab signature, .mat output extraction, and package-level timing.
• This card owns the method-level why (route choice, sign-problem control, Trotter/constraint bias); the tool skills own software parameter values; the model/paper skills own scientific values.

Method setup — step 3 (method side)

Conceptual knobs and the tricks behind them — for each, the intuition for choosing it and how it moves the result; these have no one-size default, so the trick is the guidance. Concrete software values live in /using-sse and /using-cpmc-lab.

CPMC/AFQMC — method-level knobs (concrete CPMC-Lab parameter names, starting points, and convergence values live in /using-cpmc-lab):

Knob (method-level)	Controls	Trick / how it affects results
Imaginary-time step	Trotter discretization	error ∝ step²; extrapolate step→0. The constraint also carries a small finite-step error
Population control	walker-weight stability	stops weight blow-up but biases the result when total weight is modified — a bias/variance trade; extrapolate or carry a weight-history correction
Importance sampling / force bias	sampling efficiency	shift the field distribution by the optimal force bias (∝ √step) to minimize weight fluctuation; for complex fields this is the phaseless force bias
Trial wavefunction + constraint	the CP / phaseless bias	set by trial quality; lower it via mean-field background subtraction, symmetry restoration, or multi-determinant / symmetry-projected trials
Boundary / twist averaging	finite-size error	PBC has large shell effects; twist-average (TABC) to reach the thermodynamic limit faster; keep size × #twists ≈ const
Estimator (mixed vs back-propagated)	observable accuracy	mixed is exact for the energy, biased for non-commuting observables → back-propagation for pure estimates
Constraint release / self-consistency	systematic bias removal	free-projection to gauge/remove CP bias, or feed the AFQMC density matrix back as a self-consistent constraint

SSE: β = 1/T grid (a ground-state claim needs a β sweep, not one low-T point); thermalization sweeps (drop early bins); bin size (raise near criticality where autocorrelation grows); sweeps/chains (error bars); MPI chain count.

DMC route (VMC/DMC for plane-wave solids)

Software-stack skill for using QMCPACK DMC with Quantum ESPRESSO orbitals to study periodic solids. It owns the package-level run mechanics: generate plane-wave orbitals with Quantum ESPRESSO, convert them with pw2qmcpack.x, build QMCPACK XML inputs, optimize the Jastrow, run fixed-node DMC, and summarize twist-averaged energies with defensible uncertainty and convergence caveats.

Operating principle

Pseudopotential-consistent, twist-first, stage-gated.

• Pseudopotential-consistent - use a norm-conserving QE UPF and the matching QMCPACK XML pseudopotential from the same source. Do not mix a convenient QE ultrasoft/PAW potential with a different QMC pseudopotential.
• Twist-first - choose the QMC supercell and twist grid before generating QE NSCF orbitals. A dense QE k grid is not a substitute for a QMC twist plan.
• Stage-gated - prove each stage with a tiny smoke before expanding: QE SCF/NSCF, pw2qmcpack, one QMCPACK twist, all twists, then production DMC.

The workflow spine

Use this sequence unless the active project convention gives a validated reason to deviate:

choose structure, supercell, twist grid, and pseudopotential pair
pw.x scf                    # charge density
pw.x nscf                   # exact full twist grid, nosym/noinv
pw2qmcpack.x                # writes ESHDF under QE outdir
build QMCPACK XML           # Slater determinant + Jastrow + Hamiltonian
tiny VMC smoke              # one twist, short blocks
Jastrow optimization        # linear method loops
final VMC / SJ statistics   # fixed optimized Jastrow
DMC                         # fixed-node, timestep/warmup/replica plan
twist average and compare   # size-matched structures only

-> per-stage input files, exact invocation, and run-directory layout in using-qmcpack and using-quantum-espresso

Pipeline orchestration: run the QE-stage script using-quantum-espresso/references/qmcpack-orbitals/run_qe_orbitals.template.sh to generate the trial orbitals, then the QMCPACK-stage script using-qmcpack/references/run_qmcpack_dmc.template.sh.

Method setup (judgment on the knobs)

Generate QE orbitals

-> QE SCF / NSCF / pw2qmcpack conversion mechanics passages in using-quantum-espresso (twist-first judgment is in Operating principle above)

Prepare QMCPACK XML (method judgment)

Required consistency checks:

• twistnum matches the per-file twist index.
• Electron group sizes match the valence electron count implied by the QMC pseudopotentials.
• Ionic positions and species match the QE structure.
• Jastrow coefficients marked optimize="yes" are only optimized during the intended optimization stage.

Run one short VMC smoke before any optimization or DMC. Treat parser failures, missing HDF5, wrong electron counts, or pseudopotential mismatches as hard workflow errors; do not paper over them by changing DMC parameters.

-> literal QMCPACK XML keywords, templates, and values in using-qmcpack

Optimize and run DMC (method judgment)

Use a short staged XML during development:

tiny VMC preface
linear-method Jastrow optimization loops
final VMC with optimized Slater-Jastrow
short DMC smoke

For production:

• Reuse optimized Slater-Jastrow files only when the structure, ESHDF, twist grid, pseudopotential pair, and relevant XML conventions match.
• Keep timestep-bias checks explicit, usually at multiple timestep values.
• Run independent DMC replicas with different seeds when statistics matter.
• Define warmup/discard policy before summarizing scalar data.
• Record blocks, steps, timestep, targetwalkers, warmup, seed, and twistnum in a machine-readable timing or metadata file.

Project-specific defaults belong to the project workflow, not this generic skill. For example, the post-HCP reference workflow uses fixed optimized DMC replicas and a 50-block production discard after warmup for long 3000-block runs; do not silently transfer that exact rule to unrelated systems without a fresh scalar-data check.

-> literal DMC/VMC/optimization XML values and run mechanics in using-qmcpack

Details

Stochastic sampling of a quantum partition function or ground-state projection.

This card is generic methodology. Paper-specific Hamiltonian choices, figure protocols, and target claims belong in /reproduce-paper; model facts belong in .knowledge/models/.

Notation

• β = 1/T: inverse temperature.
• Sweep / Monte Carlo step: one update pass.
• Thermalization (equilibration): updates discarded before measurement.
• Bin: sweeps averaged into one saved statistical sample.
• Autocorrelation time: correlation scale that sets honest error bars.
• Sign (SSE): average Monte Carlo sign; should stay near 1.
• Mixed estimator (CPMC): ⟨Ψ_T|O|Φ⟩ / ⟨Ψ_T|Φ⟩ — exact for the energy, biased for observables that do not commute with H (need back-propagation).
• Constrained path (CPMC): random-walk paths kept on one side of the trial-WF node to tame the sign/phase problem; the resulting energy is non-variational and biased.
• Phaseless (AFQMC): the complex-auxiliary-field generalization of the constraint (force bias + cosine projection), for systems with a phase problem (Coulomb / ab-initio).

Routes

• SSE: expand Z = Tr e^{−βH} as a power series and sample operator strings; measure finite-T observables in a fixed basis. Sign-freeness needs a bipartite / unfrustrated structure (e.g. a sublattice rotation).
• CPMC/AFQMC: Hubbard-Stratonovich the two-body interaction into auxiliary fields, then sample paths in over-complete Slater-determinant space — imaginary-time projection of a trial WF (ground state) or a grand-canonical path integral det[I + B_L⋯B_1] with fluctuating particle number (finite T). The sign/phase problem comes from the |Ψ₀⟩ ↔ −|Ψ₀⟩ symmetry: determinant space splits at the unknown node ⟨Ψ₀|φ⟩ = 0, and paths reaching it contribute only noise. An exact boundary condition (discard paths crossing the node) keeps the estimate exact; in practice the node is approximated by the trial WF — the constraint — removing the sign/phase decay at the cost of a bias. Real fields (short-range Hubbard) → sign problem → CPMC; complex fields (Coulomb / ab-initio) → phase problem → phaseless AFQMC (force bias). Ground-state energy via the mixed estimator.

Verification — implementation stage

Intermediate (mid-run)

• CPMC: the energy vs imaginary-time projection should flatten once equilibrated (read τ_eq off it); per-block energies should stabilize; growing free-projection fluctuations signal the sign problem.
• SSE: the average sign must stay near 1 — a decaying sign means the route is not controlled.

Final verification + expert criticism

• Exact small-system cross-check: CPMC vs ED on the same Hamiltonian/boundary (the CPMC-Lab paper tabulates exact small-lattice energies; reproduction values live in /reproduce-paper).
• Imaginary-time-step → 0 extrapolation; population-control-bias check (vary walker count); block decorrelation.
• Twist averaging + energy/site vs 1/L² to the thermodynamic limit; benchmark vs Bethe-ansatz / ED where available.
• Constraint-bias control: free-projection / constraint release, a better or symmetry-projected trial, or a self-consistent constraint — the CP energy sitting below the exact value is expected, not a correctness guarantee.
• SSE: β convergence for any ground-state claim; finite-size scaling across several L; compare square-lattice Heisenberg to published reference values.
• Criticize: quoting the constrained-path energy as if it were variational (it is not); no Δτ→0 or N_wlk extrapolation; non-commuting observables read off the mixed estimator without back-propagation; a single-determinant trial with no symmetry restoration and no release/free-projection check on the constraint bias; PBC-only finite-size claims with no twist averaging.

Citations

• .knowledge/literature/quantum-monte-carlo/zhang_2019_auxiliary-field-quantum-monte-carlo.md — Zhang, AFQMC at Zero- and Finite-Temperature (2019). Survey: §2 formalism (HS transformation, ground-state projection, §2.3 finite-T grand-canonical), §3 sign/phase problem, exact boundary condition, constrained-path & phaseless approximations.
• .knowledge/literature/quantum-monte-carlo/1407.7967_cpmc-lab-a-matlab-package-for-constrained-path-monte-carlo-c.md — Nguyen, Shi, Xu, Zhang, CPMC-Lab (2014). CPMC method, parameters, Table I exact energies, §V timing.
• .knowledge/literature/quantum-monte-carlo/1101.3281_computational-studies-of-quantum-spin-systems.md — Sandvik (2010). SSE methodology.
• .knowledge/literature/quantum-monte-carlo/10-1017-9781316417041.md — Becca & Sorella (2017).