Guest column

Abstract

The classical PCP theorem is arguably the most important achievement of classical complexity theory in the past quarter century. In recent years, researchers in quantum computational complexity have tried to identify approaches and develop tools that address the question: does a quantum version of the PCP theorem hold? The story of this study starts with classical complexity and takes unexpected turns providing fascinating vistas on the foundations of quantum mechanics and multipartite entanglement, topology and the so-called phenomenon of topological order, quantum error correction, information theory, and much more; it raises questions that touch upon some of the most fundamental issues at the heart of our understanding of quantum mechanics. At this point, the jury is still out as to whether or not such a theorem holds. This survey aims to provide a snapshot of the status in this ongoing story, tailored to a general theory-of-CS audience.

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“…Indeed, it could be the case that the local Hamiltonian problem remains QMA-hard if we have the promise b − a ≥ cm for some constant 0 < c < 1. This is (one formulation of) the quantum PCP conjecture; see a recent survey of Aharonov et al [10] for much more on this conjecture and its implications. Classically, one version of the famous PCP Theorem states that there exist constraint satisfaction problems for which it is hard to distinguish between there existing an assignment to the variables that satisfies all of the constraints, and there being no assignment that satisfies more than a constant fraction of them; the quantum PCP conjecture would be a direct quantization of this result.…”

“…10 Høyer et al [99] showed that this quantity is indeed a valid lower bound: every quantum algorithm that computes F with error probability ≤ ε needs to make at least…”

Divide and conquer method for proving gaps of frustration free Hamiltonians

“…The low-temperature behavior of the model (and in particular of its ground states) relies on whether the spectral gap is lower bounded by a constant which is independent on the number of particles (a situation usually referred to as gapped), or on the contrary the spectral gap tends to zero as we take the number of particle to infinity (the gapless 1 case). * angelo@math.ku.dk 1 We are using the terminology as it is frequently used in the quantum information community. In other contexts, one could only be interested in the thermodynamic limit, and the situation we have denoted as gapless does not necessarily imply that there is a continuous spectrum above the groundstate energy in such limit.…”

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“…Is there a compelling formulation of a quantum PCP conjecture [21] in terms of QMA T , e.g., QMA equals QMA T with such a noise T that the expected number of restored witness qubits is constant? Open Question 4.…”