Abstract

An accurate calculation of the properties of quantum many-body systems is one of the most important yet intricate challenges of modern physics and computer science. In recent years, the tensor network ansatz has established itself as one of the most promising approaches enabling striking efficiency of simulating static properties of one-dimensional systems and abounding numerical applications in condensed matter theory. In higher dimensions, however, a connection to the field of computational complexity theory has shown that the accurate normalization of the two-dimensional tensor networks called projected entangled pair states (PEPS) is #P-complete. Therefore, an efficient algorithm for PEPS contraction would allow to solve exceedingly difficult combinatorial counting problems, which is considered highly unlikely. Due to the importance of understanding two- and three-dimensional systems the question currently remains: Are the known constructions typical of states relevant for quantum many-body systems? In this work, we show that an accurate evaluation of normalization or expectation values of PEPS is as hard to compute for typical instances as for special configurations of highest computational hardness. We discuss the structural property of average-case hardness in relation to the current research on efficient algorithms attempting tensor network contraction, hinting at a wealth of possible further insights into the average-case hardness of important problems in quantum many-body theory.

Highlights

  • Determining the properties of quantum many-body systems is of paramount importance in our efforts to understand conductance and thermodynamics of solid-state materials [1,2], designing new sensors and devising novel quantum technologies [3], inferring nuclear processes in stars or the early universe [4,5]

  • Our worst-toaverage case reduction works just as well in this special case, by choosing (Q[v])v = (Q)v, where Q is drawn from the Gaussian distribution NC (0, σ )D4d

  • The same argument and statement of the main theorem goes through. This leaves us with two mutually exclusive options: If the translationinvariant problem is hard for a complexity class C, it follows that the problem is C-hard on average in the sense of our main theorem

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Summary

INTRODUCTION

Determining the properties of quantum many-body systems is of paramount importance in our efforts to understand conductance and thermodynamics of solid-state materials [1,2], designing new sensors and devising novel quantum technologies [3], inferring nuclear processes in stars or the early universe [4,5]. In particular a similar statement would hold if the local tensors are drawn from a uniform distribution supported on a bounded region, i.e., the overall “shape” is not crucial as long as the distribution is not infinitely peaked or has unusually broad tails This rules out the possibility that the computational hardness could be hidden in particular instances that are intractable, as it says that one could use the algorithm O to construct an algorithm O that is efficient for all inputs. Our average-case hardness result, suggests that these approaches could break down even for relevant PEPS instances as otherwise difficult computational problems would admit (quasi-) polynomial algorithms. A worst-to-average case reduction as described in this paper works just as well for translation-invariant systems but we are unaware of a hardness result in the worst case for such systems

Translation invariance
Evaluation precision
Expectation values
Implications on practical tensor network algorithms
Projected entangled pair states
PEPS evaluation
Proof idea
Proof of Theorem 2
Proof of Theorem 3
Proof of Theorem 4
EXPONENTIAL DEPENDENCE ON PEPS DATA
OUTLOOK
Full Text

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