Statistical Mechanics of Classical and Quantum Computational Complexity

  • C. R. Laumann
  • R. Moessner
  • A. Scardicchio
  • S. L. Sondhi
Part of the Lecture Notes in Physics book series (LNP, volume 843)


The quest for quantum computers is motivated by their potential for solving problems that defy existing, classical, computers. The theory of computational complexity, one of the crown jewels of computer science, provides a rigorous framework for classifying the hardness of problems according to the computational resources, most notably time, needed to solve them. Its extension to quantum computers allows the relative power of quantum computers to be analyzed. This framework identifies families of problems which are likely hard for classical computers (“NP-complete”) and those which are likely hard for quantum computers (“QMA-complete”) by indirect methods. That is, they identify problems of comparable worst-case difficulty without directly determining the individual hardness of any given instance. Statistical mechanical methods can be used to complement this classification by directly extracting information about particular families of instances—typically those that involve optimization—by studying random ensembles of them. These pose unusual and interesting (quantum) statistical mechanical questions and the results shed light on the difficulty of problems for large classes of algorithms as well as providing a window on the contrast between typical and worst case complexity. In these lecture notes we present an introduction to this set of ideas with older work on classical satisfiability and recent work on quantum satisfiability as primary examples. We also touch on the connection of computational hardness with the physical notion of glassiness.


Quantum Computer Ground State Energy Complexity Theory Interaction Graph Satisfying Assignment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We very gratefully acknowledge collaborations with Andreas Läuchli, in particular on the work reported in Ref. [34]. Chris Laumann was partially supported by a travel award of ICAM-I2CAM under NSF grant DMR-0844115.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • C. R. Laumann
    • 1
  • R. Moessner
    • 2
  • A. Scardicchio
    • 3
  • S. L. Sondhi
    • 1
  1. 1.Department of PhysicsPrinceton UniversityPrincetonUSA
  2. 2.Max-Planck-Institut für Physik komplexer SystemeDresdenGermany
  3. 3.Abdus Salam International Centre for Theoretical PhysicsTriesteItaly

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