[challenge]: An open reference implementation of connected-determinant diagrammatic Monte Carlo (CDet), plus two unpublished benchmarks

Released by

Kun Chen, Institute of Theoretical Physics, Chinese Academy of Sciences

Contact email

chenkun0228@gmail.com

Method

Other

Challenge issue

Background and the exact object to implement

At perturbation order (n), conventional diagrammatic Monte Carlo faces factorially many fermionic diagram topologies. CDet groups all diagrams at fixed space-time vertices into determinants and removes disconnected contributions by subset recursion, reducing the algebraic cost to (O(3^n+n^3 2^n)).

The benchmark model is the grand-canonical square-lattice Hubbard Hamiltonian

[
H=-t\sum_{\langle ij\rangle,\sigma}(c^\dagger_{i\sigma}c_{j\sigma}+{\rm h.c.})
+U\sum_i n_{i\uparrow}n_{i\downarrow}
-\mu\sum_{i,\sigma}n_{i\sigma},\qquad t=1 .
]

The benchmark below follows Rossi's first-order semi-bold convention, (\mu(U)=\mu_0+Un_0/2). A bare expansion is allowed for the structural tests, but bare and semi-bold coefficients must never be compared order by order without conversion.

The recurrence must be rooted. For a vacuum/free-energy contribution choose a distinguished vertex (v_0\in V):

[
c(V;v_0)=a(V)-\sum_{\substack{S\subsetneq V\v_0\in S}}c(S;v_0),a(V\setminus S).
]

Without the (v_0\in S) restriction, disconnected partitions are subtracted more than once. For a correlation function, the external set supplies the root and the equivalent recurrence is (c_E(V)=a_E(V)-\sum_{S\subsetneq V}c_E(S)a_\varnothing(V\setminus S)). For the balanced Hubbard model, up to the explicitly pinned interaction/sign convention,

[
a(V)\propto \det G^0_\uparrow(V)\det G^0_\downarrow(V)
]

(the two determinants are equal in the spin-balanced case), with (a(\varnothing)=1). Every implementation must state where factors of ((-U)^n), (1/n!), spin, and the fixed root are applied.

What already exists, and what may still be new

1. Production CDet already exists. Rossi and Šimkovic's MIT-licensed C++20 repository, FastFeynmanDiagrammatics, was used for the Science 2024 pseudogap work. A clean-room rewrite is therefore a software/pedagogy contribution, not a first CDet implementation.
2. TCI-vs-CDet appears open as of 25 July 2026. We found TCI applications to impurity, Keldysh, parquet, and general high-dimensional integration, but no published head-to-head comparison for the same thermodynamic-limit lattice-Hubbard connected integrals. This is a time-stamped literature assessment, not a proof of absence.
3. The tree-determinant algorithm is a 2025 theoretical proposal. arXiv:2512.12010 gives a randomized weak-coupling algorithm using Prüfer-tree sampling and belief propagation. We found no public numerical implementation as of the same cutoff. A toy implementation is potentially first-in-class, but a full lattice implementation is a stretch goal, not the five-day floor.

Day-0 prerequisites

Organizers must provide:

• a tested (G^0(\mathbf r,\tau)) table/FFT scaffold and sign/anti-periodicity fixtures;
• deterministic low-order reference values for several fixed vertex configurations;
• the precise Hubbard Hamiltonian, chemical-potential convention, tadpole/semi-bold convention, observable estimators, and units;
• a small reference dataset or executable from FastFeynmanDiagrammatics.

A (64\times64) momentum grid is a quadrature choice, not automatically the thermodynamic limit. Teams must compare at least (64^2) and (128^2) grids (or an adaptive Brillouin-zone integral) and report the resulting systematic.

Five-day milestone ladder

1. Structural validation - required

• Configuration-level oracle: for orders (n=1)-5, compare the rooted (3^n) recursion with explicit topology/partition enumeration configuration by configuration, to near machine precision.
• Hubbard atom: compare deterministically integrated low-order coefficients with the closed-form atom result. "Machine precision" applies to the algebraic recurrence at fixed vertices; integrated coefficients need a stated quadrature tolerance and must not be validated with the same MC estimator.
• Second-order lattice term: direct deterministic summation/integration provides an independent, zero-MC-noise check of (G^0), determinants, normalization, and observables.
• Unit tests must fail when the root restriction, spin determinant, factorial, or anti-periodicity convention is deliberately removed.

2. Published (U/t=2,\ \beta t=8) benchmark - required

Rossi's original calculation used a first-order semi-bold/tadpole-resummed convention and tuned the physical chemical potential at fixed density (n=0.875). The published references are

[
\mu/t=0.55978(7),\quad E/{\rm site}=-1.25992(6),\quad S/{\rm site}=0.1958(4).
]

The primary comparison is order resolved:

• double occupancy versus truncation order (Fig. 2, orders through 8);
• pressure versus order (Fig. 3, through order 11);
• coefficient decay and convergence radius (R=5.1(1)) (Fig. 4).

For a five-day implementation, the acceptance floor is:

1. reproduce deterministic orders 1-2 and stochastic orders 3-4 within combined uncertainties;
2. show stable signs, normalization, and error bars under independent seeds;
3. with the organizer scaffold, reach order 5-6 and compare the endpoint to the published thermodynamic values at a realistic (10^{-3})-level target.

The five-digit paper error bars are reference values, not a mandatory school-scale target: the published order-11 pressure calculation used roughly 7000 CPU hours. Fixing (\mu/t=0.55978) and predicting (n), or retuning (\mu) to (n=0.875), are both allowed, but the chosen ensemble and propagated (\mu) uncertainty must be stated.

3. Pick one extension only after milestones 1-2 pass

Extension A - TCI comparison

Evaluate exactly the same rooted connected integrand with MC and xfac/TensorCrossInterpolation.jl.

• Required extension floor: order 4, identical grids/conventions, agreement within combined numerical uncertainty.
• Stretch: orders 5-6.
• Report error versus integrand evaluations, wall-clock, peak rank, pivot tolerance, and discretization. MC supplies the (N^{-1/2}) baseline; TCI has no preregistered universal (N^{-2}) law. Rank explosion is a valid negative result only if the tolerance/grid sweep is released.

Extension B - tree-determinant algorithm

Implement a minimal version of the Prüfer-tree sampler and belief-propagation estimator from arXiv:2512.12010.

• Required extension floor: reproduce the low-order cumulant/tree identity and test BP on a small high-temperature, weak-(U) Hubbard instance.
• Full-spectrum finite-(T) ED is the exact oracle only for clusters of roughly (\le8) sites.
• A (4\times4) Hubbard cluster has Hilbert-space dimension (4^{16}\approx4.3\times10^9); use FTLM/TPQ there and carry its statistical/systematic error. Do not label it exact ED.
• The comparison window must lie in both the numerically accessible regime and the paper's provably weak-coupling regime. A full end-to-end (4\times4) implementation is stretch.

Extension C - (U/t=4)

Compare coefficient-level and resummed observables with LeBlanc et al. (arXiv:1505.02290) and Šimkovic-Rossi-Ferrero (arXiv:2110.05863). Use SmoQyDQMC.jl only as a finite-lattice cross-check and extrapolate finite size before comparison with thermodynamic-limit data. Because (U/R\approx0.78), report bare-coefficient agreement separately from Padé/conformal-Borel values, with the resummation spread as a systematic.

Suggested schedule

• Day 1: conventions frozen; (G^0), determinant, rooted recursion, and deliberate-failure unit tests.
• Day 2: atom and deterministic second order; (k)-grid/(\tau)-grid convergence.
• Day 3: stochastic orders 3-4; preregister fit/resummation/error rules.
• Day 4: order 5-6 if stable; otherwise finish the pedagogical benchmark. Start one extension only after the core gate passes.
• Day 5: reproducible repository, raw samples, order-resolved plots, wall-clock/memory audit, and limitations statement.

Acceptance and overclaiming rules

• Guaranteed deliverable: a readable rooted CDet implementation, unit tests, low-order exact tables, and at least one order-resolved Hubbard benchmark.
• Not guaranteed: five-digit thermodynamic endpoints, a working full tree-determinant solver, or a favorable TCI speedup.
• A clean-room rewrite is not claimed as algorithmic novelty.
• A TCI or tree-determinant result is called new only after a submission-day prior-art check and release of code, raw data, and the negative-result criteria.

References

• R. Rossi, PRL 119, 045701 (2017), arXiv:1612.05184
• R. Rossi et al., EPL 118, 10004 (2017), arXiv:1703.10141
• F. Šimkovic et al., Science 385, eade9194 (2024), arXiv:2209.09237
• Y. Núñez-Fernández et al., PRX 12, 041018 (2022), arXiv:2207.06135; SciPost Phys. 18, 104 (2025), arXiv:2407.02454
• H. Chen et al., arXiv:2512.12010
• J. P. F. LeBlanc et al., PRX 5, 041041 (2015), arXiv:1505.02290
• F. Šimkovic, R. Rossi, M. Ferrero, PRR 4, 043201 (2022), arXiv:2110.05863

No repository files, tests, or implementation entry points are named in the issue, so first obtain the required G0 scaffold, reference dataset, conventions, and executable from the organizers. Start with the rooted recursion and structural-validation requirements, then use the atom and second-order checks as the first definition of done before attempting the Hubbard benchmark or an extension.