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A Quantum Information Theoretic Approach to Tractable Probabilistic Models

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Positive unital circuits build normalized non-monotone distributions from quantum measurements, and their decomposable-only variant D-PUnCs claims to be the first non-monotone tractable class needing only decomposability.

desk verdict Good core idea, but the main theorem is false as written: adding smoothness to Definition 5.7 fixes it, and the POVM framing still deserves a referee's time. read the letter →

arxiv 2506.01824 v2 pith:QRY36NQI submitted 2025-06-02 quant-ph cs.AI

Pedro Zuidberg Dos Martires This is my paper
classification quant-phcs.AI
keywords probabilisticcircuitspositiveunitalPOVMtractablemodelsnon-monotonedecomposabilitystructuredquantuminformationtheory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper aims to show that a standard object from quantum mechanics, the positive operator-valued measure (POVM), can serve as the evaluation algebra for tractable probabilistic models. It constructs positive unital circuits (PUnCs), whose leaves are positive semidefinite matrices from a POVM and whose internal nodes are unital quantum operations, and proves that for any density matrix $\rho$ the map $p_X(x)=\mathrm{Tr}[O(x)\rho]$ is a proper, normalized probability distribution. Because matrix-valued outcomes can encode negative correlations, PUnCs are non-monotone, yet they avoid the NP-hard distribution-validity check that generally plagues non-monotone circuits. The paper's sharpest claim is that decomposable PUnCs (D-PUnCs) give the first non-monotone tractable circuit class that requires only decomposability rather than structured decomposability. If correct, this extends tractable probabilistic modeling to the larger DNNF-style circuit language.

What carries the argument

The load-bearing object is the positive operator-valued measure (POVM): a family of positive semidefinite matrices that sum to the identity matrix. In a PUnC, the Kronecker product at product nodes combines subsystems, and each sum or internal node applies a unital quantum operation $\Phi_k$ with $\Phi_k(1)=1$, so that summing the circuit output over all instantiations pushes down to the leaves and yields the identity. The trace with a density matrix $\rho$ then converts the matrix-valued output into a probability. For the decomposable D-PUnCs of Section 5, the same pushing argument is used to prove that normalization still holds, which is the mechanism behind the claimed new circuit class.

What would settle it

Build a two-variable D-PUnC whose root sum unit mixes a normalized distribution over $X_1$ with a normalized distribution over $X_2$, each carried by a unital operation, and compute $\sum_{x_1,x_2} \mathrm{Tr}[O(x_1,x_2)\rho]$. For binary variables a direct calculation gives a total of $2$, not $1$, showing that non-smooth D-PUnCs do not in general encode proper distributions.

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Extended reading notes

Core claim

On the author's own terms, the central discovery is that PUnCs strictly generalize both probabilistic circuits and PSD circuits, and the generalization is supported by the algebraic structure of quantum measurements. Each leaf of a PUnC carries a positive semidefinite effect matrix $E_{x_k}$, with the effects for a variable summing to the identity; each internal node is a completely positive unital map $\Phi_k$, so the identity is preserved. Consequently the family $\{O(x)\}$ over all instantiations is itself a POVM, and with any density matrix $\rho$ the trace $p(x)=\mathrm{Tr}[O(x)\rho]$ gives a normalized distribution with polynomial-time marginalization. Restricting to pure states recovers PSD circuits, restricting to diagonal matrices recovers ordinary probabilistic circuits, and restricting to block-diagonal noisy forms recovers the product-of-squares models called $\mu$SOCS. The headline extension is Theorem 5.8, which claims the same POVM argument works for D-PUnCs, making decomposable non-monotone circuits the first such class that does not need structured decomposability.

Load-bearing premise

The load-bearing premise is that each sum unit combines inputs over exactly the same set of variables, so that marginalizing a variable pushes cleanly to the leaves; the paper's formal definition of D-PUnCs leaves this smoothness condition out of the theorem's stated hypotheses.

Editorial extensions

If this is right

  • PUnCs strictly generalize probabilistic circuits and PSD circuits, so every distribution representable by those existing families is also representable as a PUnC, with a strictly larger model class.
  • D-PUnCs allow polynomial-time marginalization while remaining non-monotone, so negative correlations are expressible without the structured-decomposability restriction that all earlier non-monotone circuit classes required.
  • Because normalization is guaranteed by construction, learning the parameters of a PUnC does not require a separate validity certificate, sidestepping the NP-hard verification bottleneck identified for earlier non-monotone circuits.
  • Specializing PUnCs to diagonal operators recovers probabilistic circuits and specializing to pure states recovers PSD circuits, unifying these previously separate circuit families in one quantum-information framework.
  • The paper conjectures an exponential separation in expressive efficiency between D-PUnCs and SD-PUnCs, which would give a non-monotone analogue of the known DNNF-versus-SDNNF gap.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The proof of Theorem 5.8 pushes summations through sum units as though their inputs share a common variable scope; making that smoothness condition explicit would yield the slightly narrower but still new statement that smooth decomposable PUnCs are tractable non-monotone circuits.
  • The POVM formulation suggests a parameterization-by-construction route: any parameterization that keeps leaf matrices positive semidefinite and sum-unit operations unital automatically stays inside the space of valid distributions, which could simplify constrained training.
  • If the paper's quantum-computation conjecture is right, some PUnC marginals could be evaluated faster on a quantum computer than classically; the Fourier-transform mechanism the paper mentions is a concrete candidate to test for such a speedup.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper introduces positive unital circuits (PUnCs), a circuit model whose leaves are POVM elements and whose internal units are unital completely positive maps, so that evaluating the circuit against a density matrix yields nonnegative numbers. The main claim is that these numbers sum to one, i.e., the circuit defines a proper probability distribution. The paper then shows that several known tractable circuit classes—probabilistic circuits, PSD circuits, and µSOCS circuits—arise as special cases of PUnCs by restricting the matrices to be pure, diagonal, or block-diagonal. In Section 5, the author drops structured decomposability and defines decomposable PUnCs (D-PUnCs), claiming that they define proper distributions without requiring structured decomposability, thereby giving the first non-monotone tractable circuit class based on plain decomposability.

Significance. If established, the POVM-based construction would provide a clean information-theoretic explanation for the normalization of non-monotone circuits and a new perspective on tractable probabilistic models. The explicit reductions of PSD circuits, probabilistic circuits, and µSOCS to special cases of PUnCs are a useful contribution and are carried out by explicit isomorphisms rather than by parameter fitting. The D-PUnC extension, if corrected, would be a genuinely new result: a non-monotone circuit family that avoids structured decomposability. However, the central theorem for D-PUnCs is currently false as stated because the definitions omit the smoothness condition, and a supporting proposition about unital quantum operations is also false. These issues are local and fixable, but they are load-bearing, so the present version does not yet substantiate the paper's main claims.

major comments (3)
  1. [Definition 5.7, Theorem 5.8, Appendix E.1] Theorem 5.8 is false as stated because Definition 5.7 does not require sum-unit inputs to share a common scope. Consider binary variables X1, X2 and a D-PUnC whose root is a sum unit with two leaf inputs of scopes {X1} and {X2}, weights 1/2, identity unital operations, and binary POVM leaves {|0><0|, |1><1|}. Then sum_{x1,x2} o(x1,x2) = (1/2)*2I + (1/2)*2I = 2I, so Tr[o(x)rho] sums to 2 rather than 1. The proof in Appendix E.1 pushes the summation down to the leaves without accounting for the fact that summing an operator that is constant outside its input scope introduces extra |Omega| factors. The fix is to add the smoothness condition (all inputs of each sum unit have the same scope) to Definition 5.7; the paper's own Definition 5.5 and the remark that smoothness is usually assumed suggest that this is the intended hypothesis, and the example in Figure 2 satisfies it. With smoothness added, the push-down argument goes through. As it stands, the paper's central claim that D-PUnCs define proper probability distributions is not established.
  2. [Proposition 3.5 and Appendix B.1] The claim that every unital quantum operation satisfies sum_j K_j^* K_j <= 1 is false. Concrete counterexample: let H = C^2 and G = C^3, with Kraus operators K1 = [[1,0],[0,0],[0,0]] and K2 = [[0,0],[1,0],[0,1]]. Then K1 K1^* + K2 K2^* = I_3, so Phi is unital, but K1^* K1 + K2^* K2 = diag(2,1) has an eigenvalue 2 and is not <= I_2. The proof in Appendix B.1 derives only that Tr[Phi(sigma)rho] <= 1 for sigma equal to a sum of a subset of a POVM, which does not imply the operator inequality. This proposition is not actually needed for Theorem 3.8, which uses only complete positivity and unitality; the author should replace the term 'quantum operation' with 'completely positive unital map' and remove or correct Proposition 3.5.
  3. [Definition 5.1 and Proposition 5.2] The same smoothness omission affects the paper's definition of probabilistic circuits. Definition 5.1 allows a sum unit to combine inputs with different scopes, yet Proposition 5.2 claims every such circuit defines a proper probability distribution. The binary example above, with leaves of scopes {X1} and {X2} and weights 1/2, sums to 2 rather than 1. Thus Proposition 5.2 is false as stated. This also weakens Proposition 5.10: if smoothness is added to Definition 5.7, then D-PUnCs do not contain all circuits of Definition 5.1, only the smooth ones. The paper should add smoothness to Definition 5.1 (and Proposition 5.2) or specify that all circuits are assumed smooth, which is standard in the tractable-circuits literature.
minor comments (5)
  1. [Appendix A.1, after Eq. (24)] The line 'p(i) >= 1' should read 'p(i) >= 0'.
  2. [Definition 5.5] The quantifier 'forall j1, j2 in in(k)' is missing braces around the set; it should read 'forall j1, j2 in in(k)' with appropriate set notation.
  3. [Figure 2 caption] The phrase 'from lest to right' contains a typo and should be 'from left to right'.
  4. [Appendix C.4, proof of Proposition 4.9] The display 'Ok = X j JjDkj J * j kj' contains a stray 'kj' and should be rewritten cleanly as 'Ok = sum_j Jj Dkj J_j^*' (or similar); the following lines should be checked for consistency.
  5. [Definitions 3.2, 5.1, and 5.7] The notation xk is used both for a set of variables and for a concrete assignment, which is ambiguous. Using a bold symbol for the scope and a plain symbol for the assignment would improve readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: PUnC normalization follows from POVM/unitality axioms and the special-case reductions are explicit isomorphisms.

full rationale

I walked the derivation chain from the POVM axioms (Definition 2.2), event probabilities (Definition 2.4), unital quantum operations (Definition 3.4), and the PUnC construction (Definitions 3.2 and 3.6). Theorem 3.8 is an inductive consequence of those stated axioms: leaf operators form a POVM and unital operations preserve the identity when summing over all instantiations. The conclusion is not secretly used as an input; it is derived from the defining properties. The special-case reductions in Section 4 are also non-circular: Proposition 4.5 gives an explicit vectorization argument identifying pure PUnC evaluations with PSD circuit evaluations, and Proposition 4.10 builds an explicit bijection between diagonal PUnC computation units and probabilistic circuit computation units (Appendix C.5). These are genuine structural correspondences, not renamed versions of a single result. The D-PUnC proposal in Section 5 is presented as a new definition followed by a claimed theorem; its novelty claim depends on the validity of Theorem 5.8, not on a self-citation chain. The only self-citation is the use of partition trees from Zuidberg Dos Martires [2024] as expositional scaffolding, which is not load-bearing and could be replaced by standard variable-tree definitions. I did find a serious non-circular correctness gap: Definition 5.7 does not require smoothness, although Section 5.1 notes that smoothness is usually assumed (Definition 5.5), and Appendix E.1 pushes sums through sum units as if all inputs had the same scope. For non-smooth sum units this step can introduce extra domain-size factors and break normalization, so Theorem 5.8 is unproven as stated. That is a mathematical flaw, not a circularity: the failure is an invalid inference step, not an output being equivalent to an input by construction. Overall, the paper is self-contained against external circuit classes and does not fit parameters or rename known results, so it receives a low circularity score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The framework is definitional: no numbers are fitted to data, and the central claim rests on standard quantum-information theorems plus the unstated smoothness of D-PUnC sum units. The strict generalization claims are not supported by separation examples.

assumptions (6)
  • standard math Kraus theorem: every quantum operation can be written as Phi(E)=sum_j K_j E K_j^* with sum_j K_j^* K_j <= 1.
    Invoked as Theorem 2.6 from Nielsen and Chuang to represent internal computation units; however the paper's unital maps need not satisfy the stated inequality.
  • standard math The Born rule p(i)=Tr[rho E(i)] defines a probability distribution for a density matrix and a POVM.
    Used in Proposition 2.5 and Theorem 3.8 to turn PSD circuit outputs into probabilities.
  • domain assumption Leaves of a PUnC form a POVM for each variable and each internal operation is unital.
    These are the defining constraints of positive unital circuits (Definitions 3.4 and 3.6); without them normalization fails.
  • domain assumption All sum units in D-PUnCs are smooth, meaning their inputs share the same scope.
    Not included in Definition 5.7, but required for the normalization proof of Theorem 5.8 when sums are pushed to the leaves.
  • standard math Spectral theorem: any PSD matrix decomposes as sum_j V_j V_j^*.
    Used in Section 4.2 to relate pure states and PSD circuits.
  • standard math Visick's embedding: the Hadamard product is a principal submatrix of the Kronecker product.
    Used in Section 4.1 to rewrite Hadamard-product units as Kronecker units.

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Cite this review

Pith. "Pith review of A Quantum Information Theoretic Approach to Tractable Probabilistic Models." pith.science (2026). https://pith.science/paper/QRY36NQI

@misc{pith2026250601824,
  author       = {Pith},
  title        = {Pith review of: A Quantum Information Theoretic Approach to Tractable Probabilistic Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QRY36NQI}},
  note         = {Machine review of arXiv:2506.01824}
}
read the original abstract

By recursively nesting sums and products, probabilistic circuits have emerged in recent years as an attractive class of generative models as they enjoy, for instance, polytime marginalization of random variables. In this work we study these machine learning models using the framework of quantum information theory, leading to the introduction of positive unital circuits (PUnCs), which generalize circuit evaluations over positive real-valued probabilities to circuit evaluations over positive semi-definite matrices. As a consequence, PUnCs strictly generalize probabilistic circuits as well as recently introduced circuit classes such as PSD circuits.

Figures

Figures reproduced from arXiv: 2506.01824 by the authors.

Figure 1
Figure 1. Note, the concept of a partition tree, and hence [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Graphical representation of a (non-structured) [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Graphical representation of a structured decompos [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Graphical representation of operator mixing as described in Equation 72. The operators [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]

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Works this paper leans on

51 extracted references · 46 canonical work pages

  1. [1]

    Probabilistic generating circuits-demystified

    Sanyam Agarwal and Markus Bl \"a ser. Probabilistic generating circuits-demystified. In Forty-first International Conference on Machine Learning, 2024

    work page 2024

  2. [2]

    Negative probabilities in probabilistic logic programs

    David Buchman and David Poole. Negative probabilities in probabilistic logic programs. International Journal of Approximate Reasoning, 83: 0 43--59, 2017 a

    work page 2017

  3. [3]

    Why rules are complex: Real-valued probabilistic logic programs are not fully expressive

    David Buchman and David Poole. Why rules are complex: Real-valued probabilistic logic programs are not fully expressive. In UAI, 2017 b

    work page 2017

  4. [4]

    eckart-young

    J Douglas Carroll and Jih-Jie Chang. Analysis of individual differences in multidimensional scaling via an n-way generalization of “eckart-young” decomposition. Psychometrika, 35 0 (3): 0 283--319, 1970

    work page 1970

  5. [5]

    Tree tensor networks for generative modeling

    Song Cheng, Lei Wang, Tao Xiang, and Pan Zhang. Tree tensor networks for generative modeling. Physical Review B, 99 0 (15): 0 155131, 2019

    work page 2019

  6. [6]

    Probabilistic circuits: A unifying framework for tractable probabilistic models

    Y Choi, Antonio Vergari, and Guy Van den Broeck. Probabilistic circuits: A unifying framework for tractable probabilistic models. UCLA. URL: http://starai. cs. ucla. edu/papers/ProbCirc20. pdf, page 6, 2020

    2020

  7. [7]

    Monarch: Expressive structured matrices for efficient and accurate training

    Tri Dao, Beidi Chen, Nimit S Sohoni, Arjun Desai, Michael Poli, Jessica Grogan, Alexander Liu, Aniruddh Rao, Atri Rudra, and Christopher R \'e . Monarch: Expressive structured matrices for efficient and accurate training. In International Conference on Machine Learning, pages 4690--4721. PMLR, 2022

    work page 2022

  8. [8]

    Decomposable negation normal form

    Adnan Darwiche. Decomposable negation normal form. Journal of the ACM (JACM), 48 0 (4): 0 608--647, 2001 a

    work page 2001

Show all 51 references
  1. [9]

    On the tractable counting of theory models and its application to truth maintenance and belief revision

    Adnan Darwiche. On the tractable counting of theory models and its application to truth maintenance and belief revision. Journal of Applied Non-Classical Logics, 11 0 (1-2): 0 11--34, 2001 b

    2001

  2. [10]

    A differential approach to inference in bayesian networks

    Adnan Darwiche. A differential approach to inference in bayesian networks. Journal of the ACM, 50 0 (3): 0 280--305, 2003

    2003

  3. [11]

    Sdd: A new canonical representation of propositional knowledge bases

    Adnan Darwiche. Sdd: A new canonical representation of propositional knowledge bases. In IJCAI, 2011

    2011

  4. [12]

    A knowledge compilation map

    Adnan Darwiche and Pierre Marquis. A knowledge compilation map. Journal of Artificial Intelligence Research, 17: 0 229--264, 2002

    2002

  5. [13]

    A compilation of succinctness results for arithmetic circuits

    Alexis de Colnet and Stefan Mengel. A compilation of succinctness results for arithmetic circuits. In KR, 2021

    2021

  6. [14]

    Enhancing generative models via quantum correlations

    Xun Gao, Eric R Anschuetz, Sheng-Tao Wang, J Ignacio Cirac, and Mikhail D Lukin. Enhancing generative models via quantum correlations. Physical Review X, 12 0 (2): 0 021037, 2022

    2022

  7. [15]

    Expressive power of tensor-network factorizations for probabilistic modeling

    Ivan Glasser, Ryan Sweke, Nicola Pancotti, Jens Eisert, and Ignacio Cirac. Expressive power of tensor-network factorizations for probabilistic modeling. In Advances in neural information processing systems, 2019

    2019

  8. [16]

    A fast quantum mechanical algorithm for database search

    Lov K Grover. A fast quantum mechanical algorithm for database search. In Proceedings of the twenty-eighth annual ACM symposium on Theory of computing, pages 212--219, 1996

    1996

  9. [17]

    Unsupervised generative modeling using matrix product states

    Zhao-Yu Han, Jun Wang, Heng Fan, Lei Wang, and Pan Zhang. Unsupervised generative modeling using matrix product states. Physical Review X, 8 0 (3): 0 031012, 2018

    2018

  10. [18]

    On inference and learning with probabilistic generating circuits

    Juha Harviainen, Vaidyanathan Peruvemba Ramaswamy, and Mikko Koivisto. On inference and learning with probabilistic generating circuits. In UAI, pages 829--838. PMLR, 2023

    2023

  11. [19]

    On tractable computation of expected predictions

    Pasha Khosravi, YooJung Choi, Yitao Liang, Antonio Vergari, and Guy Van den Broeck. On tractable computation of expected predictions. Advances in Neural Information Processing Systems, 32, 2019

    2019

  12. [20]

    Algebraic model counting

    Angelika Kimmig, Guy Van den Broeck, and Luc De Raedt. Algebraic model counting. Journal of Applied Logic, 22: 0 46--62, 2017

    2017

  13. [21]

    States, effects, and operations fundamental notions of quantum theory

    K Kraus. States, effects, and operations fundamental notions of quantum theory. Lecture Notes in Physics, 190, 1983

    1983

  14. [22]

    Complex markov logic networks: Expressivity and liftability

    Ondrej Kuzelka. Complex markov logic networks: Expressivity and liftability. In Conference on Uncertainty in Artificial Intelligence, pages 729--738. PMLR, 2020

    2020

  15. [23]

    a mpfungsproblem in der wellenmechanik. Zeitschrift f \

    Lev Landau. Das d \"a mpfungsproblem in der wellenmechanik. Zeitschrift f \"u r Physik , 45 0 (5): 0 430--441, 1927

    1927

  16. [24]

    On faster marginalization with squared circuits via orthonormalization

    Lorenzo Loconte and Antonio Vergari. On faster marginalization with squared circuits via orthonormalization. arXiv preprint arXiv:2412.07883, 2024

    2024 arXiv

  17. [25]

    Subtractive mixture models via squaring: Representation and learning

    Lorenzo Loconte, Aleksanteri M Sladek, Stefan Mengel, Martin Trapp, Arno Solin, Nicolas Gillis, and Antonio Vergari. Subtractive mixture models via squaring: Representation and learning. In ICLR, 2024

    2024

  18. [26]

    What is the relationship between tensor factorizations and circuits (and how can we exploit it)? Transactions on Machine Learning Research, 2025 a

    Lorenzo Loconte, Antonio Mari, Gennaro Gala, Robert Peharz, Cassio de Campos, Erik Quaeghebeur, Gennaro Vessio, and Antonio Vergari. What is the relationship between tensor factorizations and circuits (and how can we exploit it)? Transactions on Machine Learning Research, 2025...

    2025

  19. [27]

    Sum of squares circuits

    Lorenzo Loconte, Stefan Mengel, and Antonio Vergari. Sum of squares circuits. In AAAI, 2025 b

    2025

  20. [28]

    On the expressive efficiency of sum product networks

    James Martens and Venkatesh Medabalimi. On the expressive efficiency of sum product networks. arXiv preprint arXiv:1411.7717, 2014

    2014 arXiv

  21. [29]

    Quantum computation and quantum information

    Michael A Nielsen and Isaac L Chuang. Quantum computation and quantum information. Phys. Today, 54 0 (2): 0 60, 2001

    2001

  22. [30]

    A practical introduction to tensor networks: Matrix product states and projected entangled pair states

    Rom \'a n Or \'u s. A practical introduction to tensor networks: Matrix product states and projected entangled pair states. Annals of physics, 349: 0 117--158, 2014

    2014

  23. [31]

    On theoretical properties of sum-product networks

    Robert Peharz, Sebastian Tschiatschek, Franz Pernkopf, and Pedro Domingos. On theoretical properties of sum-product networks. In Artificial Intelligence and Statistics, 2015

    2015

  24. [32]

    Random sum-product networks: A simple and effective approach to probabilistic deep learning

    Robert Peharz, Antonio Vergari, Karl Stelzner, Alejandro Molina, Xiaoting Shao, Martin Trapp, Kristian Kersting, and Zoubin Ghahramani. Random sum-product networks: A simple and effective approach to probabilistic deep learning. In UAI, 2019

    2019

  25. [33]

    Einsum networks: Fast and scalable learning of tractable probabilistic circuits

    Robert Peharz, Steven Lang, Antonio Vergari, Karl Stelzner, Alejandro Molina, Martin Trapp, Guy Van den Broeck, Kristian Kersting, and Zoubin Ghahramani. Einsum networks: Fast and scalable learning of tractable probabilistic circuits. In ICML, 2020

    2020

  26. [34]

    New compilation languages based on structured decomposability

    Knot Pipatsrisawat and Adnan Darwiche. New compilation languages based on structured decomposability. In AAAI, volume 8, pages 517--522, 2008

    2008

  27. [35]

    Sum-product networks: A new deep architecture

    Hoifung Poon and Pedro Domingos. Sum-product networks: A new deep architecture. In IEEE International Conference on Computer Vision Workshops. IEEE, 2011

    2011

  28. [36]

    Quantum algorithms for weighted constrained sampling and weighted model counting

    Fabrizio Riguzzi. Quantum algorithms for weighted constrained sampling and weighted model counting. Quantum Machine Intelligence, 6 0 (2): 0 1--24, 2024

    2024

  29. [37]

    Hyperspns: Compact and expressive probabilistic circuits

    Andy Shih, Dorsa Sadigh, and Stefano Ermon. Hyperspns: Compact and expressive probabilistic circuits. NeurIPS, 2021

    2021

  30. [38]

    Algorithms for quantum computation: discrete logarithms and factoring

    Peter W Shor. Algorithms for quantum computation: discrete logarithms and factoring. In Proceedings 35th annual symposium on foundations of computer science, pages 124--134. Ieee, 1994

    1994

  31. [39]

    Arithmetic circuits: A survey of recent results and open questions

    Amir Shpilka and Amir Yehudayoff. Arithmetic circuits: A survey of recent results and open questions. Foundations and Trends in Theoretical Computer Science , 5 0 (3--4): 0 207--388, 2010

    2010

  32. [40]

    Encoding negative dependencies in probabilistic circuits

    Aleksanteri Mikulus Sladek, Martin Trapp, and Arno Solin. Encoding negative dependencies in probabilistic circuits. In Workshop on Tractable Probabilistic Modeling, 2023

    2023

  33. [41]

    Supervised learning with tensor networks

    Edwin Stoudenmire and David J Schwab. Supervised learning with tensor networks. Advances in neural information processing systems, 2016

    2016

  34. [42]

    Negation can be exponentially powerful

    Leslie G Valiant. Negation can be exponentially powerful. In Proceedings of the eleventh annual ACM symposium on theory of computing, pages 189--196, 1979

    1979

  35. [43]

    A compositional atlas of tractable circuit operations for probabilistic inference

    Antonio Vergari, YooJung Choi, Anji Liu, Stefano Teso, and Guy Van den Broeck. A compositional atlas of tractable circuit operations for probabilistic inference. In NeurIPS, 2021

    2021

  36. [44]

    A quantitative version of the observation that the hadamard product is a principal submatrix of the kronecker product

    George Visick. A quantitative version of the observation that the hadamard product is a principal submatrix of the kronecker product. Linear Algebra and Its Applications, 304 0 (1-3): 0 45--68, 2000

    2000

  37. [45]

    Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik

    John von Neumann. Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik . Nachrichten von der Gesellschaft der Wissenschaften zu G \"o ttingen, Mathematisch-Physikalische Klasse , 1927: 0 245--272, 1927

    1927

  38. [46]

    On the relationship between monotone and squared probabilistic circuits

    Benjie Wang and Guy Van den Broeck. On the relationship between monotone and squared probabilistic circuits. In AAAI, 2025

    2025

  39. [47]

    Density matrix formulation for quantum renormalization groups

    Steven R White. Density matrix formulation for quantum renormalization groups. Physical review letters, 69 0 (19): 0 2863, 1992

    1992

  40. [48]

    Quantum measurement and control

    Howard M Wiseman and Gerard J Milburn. Quantum measurement and control. Cambridge university press, 2009

    2009

  41. [49]

    On the relationship between probabilistic circuits and determinantal point processes

    Honghua Zhang, Steven Holtzen, and Guy Broeck. On the relationship between probabilistic circuits and determinantal point processes. In UAI, pages 1188--1197. PMLR, 2020

    2020

  42. [50]

    Scaling up probabilistic circuits via monarch matrices

    Honghua Zhang, Benjie Wang, Meihua Dang, Nanyun Peng, Stefano Ermon, and Guy Van den Broeck. Scaling up probabilistic circuits via monarch matrices. In Connecting Low-Rank Representations in AI, 2025

    2025

  43. [51]

    Probabilistic neural circuits

    Pedro Zuidberg Dos Martires. Probabilistic neural circuits. In AAAI, 2024

    2024

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