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An accurate calculation of the properties of quantum many-body systems is oneof the most important yet intricate challenges of modern physics and computerscience. In recent years, the tensor network ansatz has established itself asone of the most promising approaches enabling striking efficiency of simulatingstatic properties of one-dimensional systems and abounding numericalapplications in condensed matter theory. In higher dimensions, however, aconnection to the field of computational complexity theory has shown that theaccurate normalization of the two-dimensional tensor networks called projectedentangled pair states (PEPS) is #P-complete. Therefore, an efficient algorithmfor PEPS contraction would allow to solve exceedingly difficult combinatorialcounting problems, which is considered highly unlikely. Due to the importanceof understanding two- and three-dimensional systems the question currentlyremains: Are the known constructions typical of states relevant for quantummany-body systems? In this work, we show that an accurate evaluation ofnormalization or expectation values of PEPS is as hard to compute for typicalinstances as for special configurations of highest computational hardness. Wediscuss the structural property of average-case hardness in relation to thecurrent research on efficient algorithms attempting tensor network contraction,hinting at a wealth of possible further insights into the average-case hardnessof important problems in quantum many-body theory.

Contracting projected entangled pair states is average-case hard