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arxiv: 2511.09943 · v2 · submitted 2025-11-13 · 💻 cs.MS · cs.SC· physics.chem-ph· physics.comp-ph· quant-ph

SeQuant Framework for Symbolic and Numerical Tensor Algebra. I. Core Capabilities

Bimal Gaudel , Robert G. Adam , Ajay Melekamburath , Conner Masteran , Nakul Teke , Azam Besharatnik , Andreas K\"ohn , Edward F. Valeev This is my paper

Pith reviewed 2026-05-17 22:55 UTC · model grok-4.3

classification 💻 cs.MS cs.SCphysics.chem-phphysics.comp-phquant-ph
keywords tensor algebrasymbolic computationtensor networksWick's theoremcanonicalizationgraph theoryquantum chemistrynumerical evaluation
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The pith

SeQuant uses a graph-theoretic canonicalizer to handle symmetric tensor networks faster than group-theoretic methods.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper describes SeQuant, an open-source library for performing symbolic algebra on tensors that may represent scalars or operators. At its heart is a graph-theoretic canonicalizer for tensor networks that accounts for symmetries more quickly than group theory based methods. This enables faster routine simplification of tensor expressions, optimized application of Wick's theorem to operator products, and better preparation of expressions for numerical evaluation. The library also accommodates noncovariant networks and nested tensor structures where some modes depend on indices from others. If these capabilities hold, they would allow more efficient symbolic processing in fields like quantum chemistry that rely on complex tensor manipulations.

Core claim

The central discovery is that a graph-theoretic approach to canonicalizing tensor networks with symmetries can outperform standard group-theoretic methods in speed while supporting the core operations needed for symbolic tensor algebra. This includes simplifying conventional expressions, optimizing Wick contractions for non-commutative operator tensors, and handling intermediates for numerical work. SeQuant additionally provides features for noncovariant networks and for tensors with parametrically dependent modes, which can be seen as tensors nested inside other tensors.

What carries the argument

The graph-theoretic tensor network canonicalizer, which models tensor networks as graphs to achieve faster canonicalization of symmetric cases compared to group theory.

If this is right

  • Routine simplification of conventional tensor expressions becomes faster for networks with symmetries.
  • Application of Wick's theorem for canonicalizing products of tensors over operator fields is optimized.
  • Manipulation of intermediate representations for numerical evaluation is handled more efficiently.
  • Noncovariant tensor networks arising from decompositions are supported.
  • Tensors with modes that depend parametrically on other indices, viewed as nested structures, can be manipulated.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The speed gain could make symbolic preprocessing practical for larger-scale quantum simulations.
  • Hybrid symbolic-numeric workflows might become easier to build when connecting to external numerical tensor libraries.
  • Similar graph methods could be tested on tensor networks in statistical mechanics or machine learning.

Load-bearing premise

The graph-theoretic canonicalizer must preserve all algebraic equivalences of the tensor networks while providing speedups over group-theoretic methods for the symmetries arising in quantum chemistry and physics expressions.

What would settle it

A direct comparison on benchmark symmetric tensor expressions from quantum chemistry where the graph-theoretic canonicalizer either produces different results from a group-theoretic method or shows no speedup would disprove the central claim.

Figures

Figures reproduced from arXiv: 2511.09943 by Ajay Melekamburath, Andreas K\"ohn, Azam Besharatnik, Bimal Gaudel, Conner Masteran, Edward F. Valeev, Nakul Teke, Robert G. Adam.

Figure 1
Figure 1. Figure 1: illustrates how these expression types are used to represent the square of a Fock-space 2-body Hamilto￾nian. Here is a brief walkthrough of this example. • Just like in the formal equations in this paper, Ein￾stein’s summation convention is used for SeQuant expressions involving AbstractTensor objects, i.e., summation is implied for all indices that appear on two different tensors within a Product of two o… view at source ↗
Figure 2
Figure 2. Figure 2: Graphical representation of TN in Eq. (20). External edges are shown in maroon, internal edges are shown in orange. C. SQ/core: Tensor Network Canonicalizer 1. Problem Statement A key building block for many algorithms in sym￾bolic tensor algebra involves rewriting tensor networks in their canonical forms by applying symbolic transfor￾mations that leave the tensor network invariant. Sym￾bolic transformatio… view at source ↗
Figure 3
Figure 3. Figure 3: An illustrative example of the colored graph representation of a tensor network containing tensors with [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of TN canonicalization for two simple TNs, [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of representative tensor canonicalizers (Wolfram: default Butler-Portugal canonicalizer in [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Wall time required to derive the t cluster amplitudes of the standard and cluster-specific (PNO-like) truncated coupled-cluster models. All results were obtained on an Apple M2 processor using 4 threads. arising from series expansion of time propagators and partition functions in field theoretic context as well as in many-body methods of more “traditional” quantum mechanics. Diagrams also are useful in the… view at source ↗
Figure 7
Figure 7. Figure 7: Use of IndexSpaceRegistry to define a set of two base index spaces. III. SQ/EVAL: EVALUATING TENSOR ALGEBRA For most users, symbolic manipulation of tensor ex￾pressions is only the first step; the ultimate goal is to evaluate the expressions numerically. Efficient evaluation of tensor expressions usually requires significant trans￾formations, such as factorization, common subexpression elimination, etc., w… view at source ↗
read the original abstract

SeQuant is an open-source library for symbolic algebra of tensors over commutative (scalar) and non-commutative (operator) rings. The key innovation supporting most of its functionality is a graph-theoretic tensor network (TN) canonicalizer that can handle tensor networks with symmetries faster than their standard group-theoretic counterparts. The TN canonicalizer is used for routine simplification of conventional tensor expressions, for optimizing application of Wick's theorem (used to canonicalize products of tensors over operator fields), and for manipulation of the intermediate representation leading to the numerical evaluation. Notable features of SeQuant include support for noncovariant tensor networks (which often arise from tensor decompositions) and for tensors with modes that depend parametrically on indices of other tensor modes (such dependencies between degrees of freedom are naturally viewed as nesting of tensors, "tensors of tensors" arising in block-wise data compressions in data science and modern quantum simulation). SeQuant blurs the line between pure symbolic manipulation/code generation and numerical evaluation by including compiler-like components to optimize and directly interpret tensor expressions using external numerical tensor algebra frameworks. The SeQuant source code is available at https://github.com/ValeevGroup/SeQuant.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript presents SeQuant, an open-source library for symbolic algebra of tensors over commutative (scalar) and non-commutative (operator) rings. The central innovation is a graph-theoretic tensor network (TN) canonicalizer claimed to handle symmetric tensor networks faster than standard group-theoretic methods; this component underpins simplification of tensor expressions, optimization of Wick contractions, and manipulation of the intermediate representation for numerical evaluation. The library additionally supports noncovariant tensor networks and parametrically nested tensors (tensors of tensors), and integrates compiler-like features for direct interpretation with external numerical tensor frameworks. Source code is provided at https://github.com/ValeevGroup/SeQuant.

Significance. If the graph-theoretic canonicalizer correctly preserves algebraic equivalences (including index permutations, antisymmetry signs, and noncovariant cases) while delivering the claimed speedups, the work would provide a practical advance for symbolic tensor manipulations in quantum chemistry and physics. The explicit support for noncovariant and nested structures addresses real needs arising from tensor decompositions and block compressions. Open-source release and the blurring of symbolic/numerical boundaries are strengths that aid reproducibility and adoption.

major comments (1)
  1. [TN canonicalizer description] The section describing the TN canonicalizer: the central claim that the graph-theoretic approach correctly maps algebraically equivalent networks (accounting for symmetries, signs, and noncovariant/nested structures) to identical canonical forms is load-bearing for all downstream uses (simplification, Wick optimization, IR manipulation), yet the manuscript provides neither a formal equivalence proof nor exhaustive cross-validation against group-theoretic results on representative quantum-chemistry expressions; without this, the performance advantage cannot be reliably assessed.
minor comments (1)
  1. [Abstract] The abstract states that the canonicalizer 'can handle tensor networks with symmetries faster' but does not reference specific benchmarks, test suites, or comparison baselines; adding these details would strengthen the presentation.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for their careful and constructive review of the manuscript. The major comment concerning validation of the tensor network canonicalizer is addressed point by point below. We agree that this aspect is central to the work and will strengthen the revised version accordingly.

read point-by-point responses

  1. Referee: [TN canonicalizer description] The section describing the TN canonicalizer: the central claim that the graph-theoretic approach correctly maps algebraically equivalent networks (accounting for symmetries, signs, and noncovariant/nested structures) to identical canonical forms is load-bearing for all downstream uses (simplification, Wick optimization, IR manipulation), yet the manuscript provides neither a formal equivalence proof nor exhaustive cross-validation against group-theoretic results on representative quantum-chemistry expressions; without this, the performance advantage cannot be reliably assessed.

    Authors: We agree that establishing the correctness of the TN canonicalizer is essential, as it underpins the library's core operations. The manuscript presents the graph-theoretic method as a practical implementation that extends standard tensor network isomorphism techniques to accommodate symmetries, signs from antisymmetries, noncovariant indices, and parametric nesting. To directly address the concern, the revised manuscript will add a dedicated subsection in the TN canonicalizer description that reports cross-validation results. These will compare canonical forms generated by SeQuant against independent group-theoretic canonicalizers on representative quantum-chemistry tensor expressions, including cases with permutational symmetries, sign tracking for antisymmetric tensors, noncovariant structures, and nested tensors. The validation will confirm that algebraically equivalent networks produce identical canonical representations. Performance benchmarks will be restricted to these validated cases. A complete formal proof of equivalence across all conceivable symmetry groups and non-commutative operator rings is a substantial theoretical undertaking that lies outside the scope of this software-focused paper; we will explicitly note this limitation while emphasizing the empirical support from the added validation suite. revision: yes

standing simulated objections not resolved
  • A formal mathematical proof of equivalence between the graph-theoretic canonicalizer and group-theoretic methods for arbitrary tensor networks, including all symmetry groups, sign conventions, noncovariant cases, and nested structures.

Circularity Check

0 steps flagged

No circularity: SeQuant presents new software implementation without self-referential derivation

full rationale

The manuscript describes an open-source library implementing symbolic tensor algebra, with the graph-theoretic TN canonicalizer introduced as a novel capability for handling symmetries, noncovariant networks, and nested tensors. No load-bearing step reduces by construction to a fitted parameter, self-defined quantity, or self-citation chain; the core claims concern software features, performance comparisons, and use cases in quantum chemistry, which are independent of the paper's own inputs. The derivation of functionality is via explicit implementation and external numerical frameworks rather than internal equivalence. This is the expected non-finding for a library paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The contribution is a new software implementation rather than new physical parameters or entities. The graph-theoretic representation of tensor networks is treated as a methodological innovation grounded in standard computer science techniques for handling symmetries.

axioms (1)
  • domain assumption Tensor networks with symmetries can be canonically represented and simplified using graph-theoretic methods that outperform group-theoretic approaches.
    This underpins the central claim about the canonicalizer's advantages and is invoked to support simplification, Wick's theorem optimization, and numerical evaluation.

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Reference graph

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