Explicit Solution Equation for Every Combinatorial Problem via Tensor Networks: MeLoCoToN

Alejandro Mata Ali

REVIEW 4 major objections 5 minor 1 cited by

Explicit Solution Equation for Every Combinatorial Problem via Tensor Networks: MeLoCoToN

T0 review · 4 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Every finite combinatorial problem has an exact explicit equation, built by tensorizing its logic circuit and reading off variables with nested Heaviside steps.

desk verdict A genuinely useful catalog of tensor-network encodings, but the optimization readout's unspecified tau sinks the 'explicit equation' theorem. read the letter →

arxiv 2502.05981 v1 pith:5YHBIX3Q submitted 2025-02-09 cs.ET quant-ph

Alejandro Mata Ali This is my paper · ORCID

classification cs.ETquant-ph

keywords tensornetworkscombinatorialoptimizationconstraintsatisfactioninversionproblemsexplicitsolutionequationimaginarytimeevolutionQUBOHeavisidestepfunction

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

The paper's central claim is that every finite combinatorial problem—whether it asks to invert a function, satisfy constraints, or optimize a cost—has an exact explicit equation that returns its solution or solutions. The construction starts from a classical logic circuit for the problem, translates each gate into a tensor, and contracts the resulting tensor network while summing over all variables except one at a time. Each variable's value is then given by the sign of a scalar, expressed as a nested Heaviside step function, Eq. (2.41). The paper stresses that this equation need not be computable in a reasonable time and is not claimed to beat state-of-the-art complexity, but it does give every problem a closed-form-like analytic handle. A side consequence is the conditional statement that if some physical device could contract these networks in polynomial time, every NP-hard problem would be solvable in polynomial time.

What carries the argument

The load-bearing object is the Tensor Logical Circuit (TLC) together with the Half Partial Trace readout. A TLC is a tensor network obtained from the classical logical circuit of the problem by replacing each operator with a sparse tensor whose nonzero entries enforce the input-output relation and multiply the state's amplitude; equality of indexes encodes the circuit wiring. The readout contracts the TLC with Plus Vectors (all-ones vectors) on all variables except the one being determined, and with a Minus Vector $(-1,1)$ on that variable, producing a scalar $\Omega_n$ whose sign decides the bit through the Heaviside function. Earlier determined values enter as projection vectors, which makes the expression for the $n$-th variable a nested composition $x_n = H(\Omega_n(H(\Omega_0), H(\Omega_1(H(\Omega_0))),\dots))$. For optimization, the amplitude factor $e^{-\tau C(\vec x)}$ is the engine that makes low-cost states dominant, with the intended limit $\tau\to\infty$ isolating the optimum.

What would settle it

Run the Half Partial Trace iteration, with increasing values of $\tau$, on a small optimization instance whose optimum is known (for example a 4-variable QUBO with a unique minimizer) and record whether the sign of each $\Omega_n$ eventually matches the optimal assignment for every instance; any instance for which no finite $\tau$ yields the optimum would falsify the optimization version of Theorem 1.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 1: given any combinatorial problem—inversion, constraint satisfaction, or optimization—there is an exact explicit equation for its solution(s). The equation is produced by (1) rewriting the problem in chosen variables, (2) building a logical circuit (LSTC for inversion, LSVC for constraints, LSMC for optimization) whose operators carry only the necessary internal signals, (3) tensorizing it into a Tensor Logical Circuit, and (4) contracting with the 'Half Partial Trace': impose the known output, or for optimization apply imaginary-time weight $e^{-\tau C(\vec x)}$, then sum over all variables except one using Plus Vectors and read that variable from a Minus Vector $(-1,1)$. Each decision is a Heaviside step $x_n = H(\Omega_n(...))$, and because earlier decisions are fed back as projection vectors, the full solution is a nesting of Heaviside functions inside tensor-network contractions. The paper further claims (Theorem 2) that this equation can be written down in time polynomial in the problem's formulation, and (Theorem 3) that infinitely many equivalent equations exist.

Load-bearing premise

For optimization problems the readout assumes that a single finite damping parameter $\tau$ can be made large enough that each partial-trace scalar $\Omega_n$ keeps the sign of an optimal assignment; the paper states this informally in Section 2.5.2 and gives no proof or constructive rule for choosing $\tau$.

Editorial extensions

If this is right

Every finite well-formulated combinatorial problem acquires a closed-form-like equation, so problems without a known analytic solution still have an explicit mathematical expression for their answer.
The same recipe covers inversion, constraint satisfaction, and optimization uniformly, suggesting a common analytic language for problems usually treated by separate algorithms.
Because the equation's construction time is polynomial in the problem formulation, any future method that contracts these tensor networks in polynomial time would imply polynomial-time algorithms for all NP-hard problems.
The framework yields concrete equations for many named problems (QUBO, TSP, knapsack, integer programming, k-colouring, maximum flow, and others), which can be studied analytically or approximated by tensor-network compression.
The equations also give a new target for approximation: compressing the TLC with Matrix Product States or removing constraint layers can turn the exact form into an approximate solver.

Reading between the lines

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

Editorial inference: if the claimed construction is correct, it reframes the P versus NP question as a question about the contraction cost of a specific family of tensor networks, giving complexity theorists a concrete combinatorial object to bound.
Editorial inference: a natural testable next step is to search for polynomial-time contractible subfamilies (for example, low-treewidth or chain-structured TLCs) where the Half Partial Trace readout provably returns the optimum, bypassing the unresolved general choice of $\tau$.
Editorial inference: the paper's explicit equations could be used to generate certificates, because evaluating the nested Heaviside composition at a candidate solution gives a direct check, though checking sign consistency would itself cost as much as contraction.
Editorial inference: since the TLC is a positive tensor network, Monte-Carlo or approximate contraction methods may be adapted to sample from its distribution, connecting the exact equation to randomized algorithms.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: the catalog of explicit tensor networks for dozens of combinatorial problems is a real contribution, but the headline theorem is not established for the optimization case because the readout equation depends on an unspecified finite tau and fails on concrete small instances.

What's good: the circuit-to-tensor translation in Section 2.4 is coherent, and Input-Output Indexing is a clean way to turn logical circuits into sparse tensors. The paper gives explicit tensor definitions for QUBO, T-QUDO, linear systems, k-colouring, partition, shortest path, TSP, knapsack, ILP, and more. That is a useful reference even if the networks are exponentially large. The inversion and CSP readouts are count-based, so they survive degeneracy, and the projection-vector handling of multiple solutions is sensible. The paper is also honest in the abstract and conclusions that the equations are not efficient.

The soft spots: Theorem 1 is asserted, not proved. For optimization, the Half Partial Trace readout in Eq. (2.41) requires a 'sufficiently large finite value of tau' that is never specified. The stress-test example is correct: two binary variables with costs C(10)=0, C(11)=100, C(00)=0.01, C(01)=0.02, at tau=1 the amplitude sum for x0=0 exceeds that for x0=1, so the equation returns the suboptimal state 00. This is not a pathological corner case; the required tau depends on the cost landscape, which is exactly what the equation is supposed to return. The paper lists 'determination of the minimum tau value' as future work, which concedes the point. For optimization, the claimed exact explicit equation is either tautological (the tensor network encodes the truth table) or under-specified (tau as an existential parameter). The reader's circularity concern is fair: the tensor elements are defined directly from the problem's logical relations, so the contraction is a restatement of the problem.

The paper is also incomplete: several subsections are missing because they refer to unpublished work, including prime factorization and SHA-3 inversion. That makes a full evaluation harder.

Bottom line: readers who want a catalogue of tensor-network formulations will find value here; the theorem as stated does not hold for optimization. I would send it to a serious referee but expect major revision: either fix the tau issue or weaken the theorem to the inversion and CSP cases. I would cite the catalog, not the theorem.

Referee Report

4 major / 5 minor

Summary. The paper proposes a method, MeLoCoToN, which associates to any combinatorial problem—inversion, constraint satisfaction, or optimization—a tensor network whose contraction yields scalars Omega_n, and then defines the solution by nested Heaviside equations x_n = H(Omega_n(...)). The construction has four steps: choose variables, build a classical logical circuit (LSTC, LSVC, or LSMC), tensorize the circuit by Input-Output Indexing (Eq. 2.19), and read out variables iteratively via the Half Partial Trace. The paper presents tensor networks for a large list of problems, including QUBO/HOBO, integer sum, linear systems, k-colouring, TSP, knapsack, and several graph problems, and it describes approximation techniques under the name Motion Onion. It further claims that if a physical system could contract these tensor networks efficiently, every NP-hard problem would be solvable in polynomial time.

Significance. The circuit-to-tensor translation is coherent for finite circuits, and several of the example tensor definitions appear correct; the paper is also honest that no computational advantage over existing algorithms is claimed. If Theorem 1 were rigorously established, the work would offer a unified exact-formulation perspective on combinatorial problems and a conditional complexity statement. However, the central claim is weakened by three issues: the optimization readout depends on an unspecified sufficiently large tau or an unjustified limit interchange; Theorems 1-3 are asserted without formal proof; and the construction is essentially a transcription of the problem's truth table into tensor form, so the sense in which the resulting equation is 'explicit' needs to be made precise. These issues are load-bearing because the abstract and conclusions assert an exact explicit equation for every combinatorial problem.

major comments (4)

[Section 2.5.2, Eqs. (2.36)-(2.41)] The optimization readout is not justified, and the displayed limit is incorrect in the degenerate case. Eq. (2.36) claims that the normalized state tends to |X>; this holds only for a unique minimizer. With multiple optima, the limit is a uniform superposition over all minimizers. More importantly, Eq. (2.41) uses finite-tau unnormalized scalars Omega_n, and the paper only asserts that 'for a sufficiently large finite value of tau' the correct variable is selected, without a proof or a rule for choosing tau. This is load-bearing: for fixed tau, greedy Heaviside decisions can return suboptimal solutions. For example, with two binary variables and costs C(10)=0, C(11)=100, C(00)=0.01, C(01)=0.02, at tau=1 Eq. (2.37) gives Omega_0 = 1 + e^{-100} - e^{-0.01} - e^{-0.02} < 0, so x_0=0, and then x_1=0, returning x=(0,0) with cost 0.01 instead of the optimum (1,0) with cost 0. The Conclusions list 'determination of the minimum tau value' as future work, so the exactness claim for optimization is not established.
[Section 2.6, Theorem 1] The main theorem is stated without a formal proof. The preceding text gives a construction, but no correctness statement, for example by induction on circuit size, shows that the contracted TLC equals the problem's indicator or cost tensor, nor that the nested Heaviside readout returns a solution for every input instance. The assertion in Section 2.3.1 that any known function can be implemented as an LSTC is also unproved. For finite circuits this is a standard fact, but the theorem's quantifier 'every combinatorial problem' requires a precise statement about the input representation and about the size of the circuit relative to the problem formulation.
[Section 2.4, Eq. (2.19)] The construction is essentially a transcription of the problem's defining relations into tensor elements via Input-Output Indexing. Because the tensor elements are defined directly from the same logical relations that define the problem, the contraction of the TLC is exactly the problem's truth table in tensor form. Correctness is therefore inherited by construction, and the claimed 'solution equation' is close to a notational restatement of exhaustive enumeration. The paper should define what counts as an 'explicit equation' and state why this transcription is not merely a disguised enumeration of all configurations. Without such a definition, the abstract's claim that the method proves the existence of an exact explicit equation for every combinatorial problem is not a substantive mathematical result.
[Section 8.1, Eq. (8.3); Sections 7 and 9] Several constrained formulations use a penalty weight lambda that is only described as 'large enough', with no finite instance-dependent bound. For example, Eq. (8.3) minimizes sum_i (C_{i,x_i} + lambda V_{x_i}) and states that lambda must be large enough to impose the maximum number of tasks performed, but no value or bound is given. Since the paper claims exact solutions, every free parameter used to enforce constraints needs a concrete, instance-dependent value that guarantees feasibility without changing the optimum. As written, the exactness claim for constrained optimization inherits the same unsupported-parameter problem as the tau readout.

minor comments (5)

[Front matter and Section 2] The manuscript contains numerous typographical errors, including 'sintetized', 'unassumingly expensive', and inconsistent capitalization of 'Half Partial Trace' and 'Humbucker'. A careful proofreading pass is needed.
[Sections 5.1.1, 5.1.2, 10.2, 10.3] Several subsections are placeholders stating 'Subsubsection not available due to paper pending publication', and references such as '[pending to publish]' appear throughout. These must be either filled in or removed before the paper can be considered a complete manuscript.
[Section 2.6, Theorem 2] The statement that the equation 'can be obtained in a polynomial time with respect to the time needed to formulate it' is not a well-defined complexity measure. The paper should specify the input size and the computational model, and prove the claim with respect to that measure.
[Appendix D.1] In the definition of the Plus Vector with local imaginary time evolution, the notation '+_i = e^{-tau C_i}' uses C_i without defining it; the symbol should be connected to the problem's cost function or removed.
[Equations (4.3)-(4.4) and (5.1)-(5.4)] The same tensor names, such as delta and DOT, are reused for tensors of different arities in different problems. A summary table of index conventions for each problem would improve readability and reduce ambiguity.

Circularity Check

2 steps flagged · score 8.0 of 10

The solution equation is a truth-table lookup: IOI tensors encode the problem's own relations (Eq. 2.19) and Eq. 2.41 reads them back; optimization additionally relies on an unspecified 'sufficiently large' τ left as future work.

self definitional [Section 2.4, Eq. (2.19); Section 2.6, Eq. (2.41)]
"if we have an operator U with 3 inputs x, y, z and 2 outputs µ, ν, calculated as µ = f (x, y, z), ν = g(x, y, z), which multiplies the amplitude of the state by h(x, y, z), then its associated tensor U has as non-zero elements those that satisfy µ = f (x, y, z), ν = g(x, y, z), Ux,y,z,µ,ν = h(x, y, z)."

IOI fixes each tensor's nonzero support to exactly the operator's input-output relation. Contracting the TLC therefore produces a tensor whose nonzero entries sit precisely on the problem's solution set; Section 2.5.1 states that 'there is only one non-zero element in the tensor... whose indexes give us the solution'. Equation (2.41) is then a Heaviside-encoded Half Partial Trace that locates that pre-planted nonzero support. The proof that the equation returns the solution is therefore nothing more than reading back the truth table inserted at Eq. (2.19), so the theorem's content is definitional rather than a derived prediction.
other [Section 2.5.2 and Conclusions]
"For a sufficiently large finite value of τ, the peak amplitude in the optimal combination will be large enough so that, when summing over the other variables values to obtain the vector of amplitudes of the variable we want to determine, this amplitude will be greater than the sum of all the suboptimals."

The finite-τ readout guarantee is asserted, not derived: Eq. (2.36) proves only the τ→∞ renormalized limit, not the unnormalized greedy sign of Ω_n in Eq. (2.41). The required size of τ depends on the gap between the optimal cost and the aggregate suboptimal amplitude, i.e., on the very solution the equation is supposed to produce. The paper lists 'the determination of the minimum τ value' as future work, so the optimization branch is either non-explicit (τ unspecified) or, for any concrete finite τ, can select a suboptimal state on some instances. This is a self-referential guarantee rather than an exactness proof.

full rationale

The paper is self-contained and does not fit parameters to external benchmarks, so this is not a data-fitting circularity. Its central theorem, however, is a definitional construction: Eq. (2.19) places nonzero tensor elements exactly at the problem's valid input-output/cost-value pairs, and Eq. (2.41) is the Half Partial Trace that locates those elements. The 'explicit equation' is therefore a Heaviside-encoded lookup in the truth table that was inserted by construction; the derivation of the solution from the equation is equivalent to undoing the definition of the TLC. Self-citations [37]–[39] are used for refinements and heuristics (e.g., Humbucker) and are not load-bearing for Theorem 1. Separately, the optimization branch requires a 'sufficiently large' finite τ that is asserted without proof; Eq. (2.36) only treats the τ→∞ normalized limit, and the paper defers the minimum-τ problem to future work. This is a correctness gap as well as a self-referential condition: for any fixed τ the readout can fail, while an unspecified 'sufficiently large' τ cannot be instantiated from the problem data. Overall, the central result is forced by construction rather than independently derived, giving a circularity score of 8.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central construction depends on: (1) exact finite logical-circuit specification of each problem, (2) the multiplicative exponential amplitude model, and (3) the unproved large-tau greedy readout for optimization. The first is a domain restriction that the abstract overstates; the second is a design choice; the third is the least-supported load-bearing premise. No new physical entities are introduced.

free parameters (2)

tau (imaginary time evolution parameter) = not specified; limit tau -> infinity proposed
Used throughout as the damping amplitude in e^{-tau C(x)}; exactness of the optimization readout depends on choosing it sufficiently large or taking the limit, but no constructive rule is given.
lambda (Motion Onion soft-constraint weight) = not specified
Introduced in Eq. (3.3) to weight the soft uniformity constraint in the heuristic Motion Onion section; not part of the central theorem but a hand-chosen parameter.

assumptions (4)

domain assumption Every considered problem is exactly specified by a finite classical logical circuit (LSTC, LSVC, or LSMC).
Used throughout Section 2.3 to justify tensorization; the FAQ in Appendix A narrows 'every combinatorial problem' to well-formulated problems with exact finite data, which is a restriction rather than a theorem.
ad hoc to paper Amplitude changes in the LSMC are multiplicative and take the exponential form e^{-tau C(x)}.
Section 2.3.3 imposes this exponential form so that products factor over cost terms; this design choice is load-bearing for the optimization construction.
ad hoc to paper There exists a sufficiently large finite tau, or an admissible tau-to-infinity limit, such that greedy Half Partial Trace decisions recover a valid or optimal solution.
Section 2.5.2 asserts this informally ('For a sufficiently large finite value of tau') but provides no proof, no bound, and no algorithm to select tau. The optimization theorem depends on this premise.
standard math Tensor contraction, Kronecker deltas, and Heaviside step functions behave in the standard way.
Uncontroversial background used throughout the tensor definitions and the readout formulas.

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Cite this review

Pith. "Pith review of Explicit Solution Equation for Every Combinatorial Problem via Tensor Networks: MeLoCoToN." pith.science (2026). https://pith.science/paper/5YHBIX3Q

@misc{pith2026250205981,
  author       = {Pith},
  title        = {Pith review of: Explicit Solution Equation for Every Combinatorial Problem via Tensor Networks: MeLoCoToN},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5YHBIX3Q}},
  note         = {Machine review of arXiv:2502.05981}
}

read the original abstract

In this paper we show that every combinatorial problem has an exact explicit equation that returns its solution. We present a method to obtain an equation that solves exactly any combinatorial problem, both inversion, constraint satisfaction and optimization, by obtaining its equivalent tensor network. This formulation only requires a basic knowledge of classical logical operators, at a first year level of any computer science degree. These equations are not necessarily computable in a reasonable time, nor do they allow to surpass the state of the art in computational complexity, but they allow to have a new perspective for the mathematical analysis of these problems. These equations computation can be approximated by different methods such as Matrix Product State compression. We also present the equations for numerous combinatorial problems. This work proves that, if there is a physical system capable of contracting in polynomial time the tensor networks presented, every NP-Hard problem can be solved in polynomial time.

Figures

Figures reproduced from arXiv: 2502.05981 by the authors.

**Figure 1.** General scheme of ideas presented. There are three general types of combinatorial problems that we can define. The first are inversion problems, which consist in having a function γ which associates one output combination to each input combination, and given a known output Y⃗ , we search for the input X⃗ that generates it, Y⃗ = γ(X⃗ ). An example would be the factorization of prime numbers. The second are the constr… view at source ↗

**Figure 2.** LSTC to add two numbers a and b to obtain a number c. cADDb cADDb . . . cADDb = cADD w w [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** LSTC to add two numbers a and b to obtain a number c if w = 1. binary sum is performed in an ADDb operator, we can make them return both the modular sum of the bits and the carry for the sum of the following bits. This carry information is sent in what we call the internal signal. The ADDb operator has three inputs, which are the three bits to be added, and two outputs. The first output is the function f(x, y, z) = … view at source ↗

Figures from the paper (42 more)

**Figure 4.** Figure 4: LSTC that performs the multiplication of two numbers [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: LSVC to determine if a number has only one bit on [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: LSMC that multiplies the amplitude of an input [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

**Figure 7.** Figure 7: LSMC that multiplies the amplitude of an input [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

**Figure 8.** Figure 8: LSMC that multiplies the amplitude of an input [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

**Figure 9.** Figure 9: TLC of Fig. 2 and its index correspondence for the tensors. [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]

**Figure 10.** Figure 10: TLC of Fig. 5 for the Single One Input problem. [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]

**Figure 11.** Figure 11: TLC of Fig. 6 and its index correspondence for the tensors. [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]

**Figure 12.** Figure 12: TLC of Fig. 7 and its index correspondence for the tensors. [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]

**Figure 13.** Figure 13: TLC of Fig. 8 and its index correspondence for the tensors. [PITH_FULL_IMAGE:figures/full_fig_p017_13.png]

**Figure 14.** Figure 14: Iterative method for the determination of the solution variables in an inversion problem, for a chain-type [PITH_FULL_IMAGE:figures/full_fig_p018_14.png]

**Figure 15.** Figure 15: Iterative method for the determination of the solution variables in an constraint satisfaction and optimization [PITH_FULL_IMAGE:figures/full_fig_p019_15.png]

**Figure 16.** Figure 16: Tensor Network for solving the minimization problem in the a) first iteration, b) second iteration, c) third [PITH_FULL_IMAGE:figures/full_fig_p022_16.png]

**Figure 17.** Figure 17: LSMC for the QUBO/QUDO/T-QUDO problem with 6 variables. a) One operator per variable, b) One [PITH_FULL_IMAGE:figures/full_fig_p024_17.png]

**Figure 18.** Figure 18: Tensor Network for the QUBO/QUDO/T-QUDO problem with 6 variables to determine the first variable [PITH_FULL_IMAGE:figures/full_fig_p025_18.png]

**Figure 19.** Figure 19: Tensor network for determine the first variable in the QUBO/QUDO/T-QUDO problem in a linear chain [PITH_FULL_IMAGE:figures/full_fig_p026_19.png]

**Figure 20.** Figure 20: Tensor network for the QUBO/QUDO/T-QUDO problem with 6 variables to determine, a) the second [PITH_FULL_IMAGE:figures/full_fig_p027_20.png]

**Figure 21.** Figure 21: Tensor network for the T-HODO problem with 6 variables and [PITH_FULL_IMAGE:figures/full_fig_p028_21.png]

**Figure 22.** Figure 22: Tensor network for the Integer sum total function problem with a) lineal argument b) non-linear argument. c) [PITH_FULL_IMAGE:figures/full_fig_p029_22.png]

**Figure 23.** Figure 23: MeLoCoToN for solving systems of linear equations with a [PITH_FULL_IMAGE:figures/full_fig_p031_23.png]

**Figure 24.** Figure 24: Indexes names for the tensor of the linear solver. [PITH_FULL_IMAGE:figures/full_fig_p032_24.png]

**Figure 25.** Figure 25: a) Original graph. b) LSTC of one step of the Closure finding problem. c) Operator signals. d) Tensors of [PITH_FULL_IMAGE:figures/full_fig_p032_25.png]

**Figure 26.** Figure 26: a) Operator decomposition for vertex and edge operators. b) Tensor decomposition for vertex and edge [PITH_FULL_IMAGE:figures/full_fig_p033_26.png]

**Figure 27.** Figure 27: a) k-colouring graph with k = 3 and 10 vertices, b) Tensor Network for the problem to determine the colour of the first vertex. To solve this problem we have to make a tensor network with the same shape of the graph, so that each tensor substitutes a vertex and each i…

**Figure 28.** Figure 28: Simplified k-colouring tensor network. If now the problem to be solved is a k-colouring in which we want to paint the graph with the minimum number of colors possible, we move on to an optimization problem. In this case, we only have to make the cost be of the type C(…

**Figure 29.** Figure 29: Tensor Network for the a) Shortest Path Cost Problem, b) Shortest Path Problem, and its notation. [PITH_FULL_IMAGE:figures/full_fig_p038_29.png]

**Figure 30.** Figure 30: Tensor network for the TSP with 6 variables. [PITH_FULL_IMAGE:figures/full_fig_p039_30.png]

**Figure 31.** Figure 31: Graph of the TSP in square grid and its associated tensor network. [PITH_FULL_IMAGE:figures/full_fig_p040_31.png]

**Figure 32.** Figure 32: Tensor network for the Vehicle routing problem with two vehicles and 6 vertexes. In this case, each vehicle [PITH_FULL_IMAGE:figures/full_fig_p041_32.png]

**Figure 33.** Figure 33: Tensor Network that solves the Chinese postman problem with 5 edges and 6 time steps. [PITH_FULL_IMAGE:figures/full_fig_p042_33.png]

**Figure 34.** Figure 34: a) Directed graph. b) Tensor Network for its Minimal Cost Closure Problem. c) Tensor Train decomposition [PITH_FULL_IMAGE:figures/full_fig_p043_34.png]

**Figure 35.** Figure 35: a) Maximum Flow graph, b) Tensor Network to determine the upper edge flow after determine the flow of [PITH_FULL_IMAGE:figures/full_fig_p045_35.png]

**Figure 36.** Figure 36: Decomposition of the vertex tensors. 7.8 Maximum Independent Set Problem This problem consists in, given a graph G of V vertices and E edges, finding the largest possible independent set. An independent set is a set of vertices of the graph such that no two vertices o…

**Figure 37.** Figure 37: a) Vertex Cover Solution, b) Tensor Network for the Minimum Vertex Cover Problem. [PITH_FULL_IMAGE:figures/full_fig_p046_37.png]

**Figure 38.** Figure 38: Tensor Network for the Assignment Problem with 6 agents and 4 tasks. [PITH_FULL_IMAGE:figures/full_fig_p047_38.png]

**Figure 39.** Figure 39: Tensor Network for determining the first job feature for first agent. [PITH_FULL_IMAGE:figures/full_fig_p048_39.png]

**Figure 40.** Figure 40: Cutting stock problem tensor network. To solve this problem, we will use the tensor network of [PITH_FULL_IMAGE:figures/full_fig_p050_40.png]

**Figure 41.** Figure 41: Tensor network to solve the Integer Quadratic Programming. [PITH_FULL_IMAGE:figures/full_fig_p052_41.png]

**Figure 42.** Figure 42: Task Scheduling Problem Tensor Network with 6 machines and 3 rules. [PITH_FULL_IMAGE:figures/full_fig_p053_42.png]

**Figure 43.** Figure 43: Representation of standard tensor network vs. its equivalent in simplified notation. a) Normal nodes. b) [PITH_FULL_IMAGE:figures/full_fig_p064_43.png]

**Figure 44.** Figure 44: Kronecker delta decomposition in tensor train. [PITH_FULL_IMAGE:figures/full_fig_p065_44.png]

**Figure 45.** Figure 45: Pass tensor decomposition. D.4 Counting layer This layer is responsible for counting how many times the value a has appeared in a set of variables, and removing the state if it has appeared more than Na times. It is a Matrix Product Operator (MPO) layer, and its non-z…

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Reference graph

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