Q38: What is the Quantum Max-Cut optimum at fixed total spin?

Working mathematical formulation

For a positive-weight graph $G$ , let $H_G=\sum_e w_eP_s^e$ and $\operatorname{QMC}_j(G)=\lambda_{\max}\!\left(H_G\big|_{J=j}\right).$ Obtain graph-sensitive upper and lower bounds, especially for sparse graphs. Compare with an explicitly specified localized ansatz, such as products of bounded-size singlet clusters or mixtures of dimer coverings, and bound the energy lost by that restriction. Normalize weights and state the additive or multiplicative approximation target.

Why this is interesting

Total singlet weight is fixed at a given spin, but a graph weights its locations differently. Delocalization may improve the objective without changing the total amount of antisymmetry.

Explicit variants

Incompatible singlet constraints
Overlapping pairs cannot both be perfect singlets. On three qubits, $p_s(12)+p_s(23)\le3/2$ , whereas disjoint singlets can be prepared independently. For $H_G=\sum_e w_eP_s^e$ , prove graph-dependent bounds on the energy lost by localized singlet ansätze, and identify families where correlations between their blocks are necessary for a chosen approximation ratio. Distinguish frustration from computational hardness by comparing efficiently solvable families with hard families. Which property of the graph and its accessible multiplicity states controls approximation or preparation cost beyond the mere presence of overlapping edges?

A place to start

Solve small graphs sector by sector and compare localized preparations with unrestricted multiplicity states. For fixed spin deficit $r=n/2-j$ , the multiplicity dimension is polynomial in $n$ ; use this tractable regime as a baseline.

Related work

These works provide context and established results. They need not be the origin of this exact formulation.

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