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QMAcomplete problems
, 2012
"... In this paper we give an overview of the quantum computational complexity class QMA and a description of known QMAcomplete problems to date 1. Such problems are believed to be difficult to solve, even with a quantum computer, but have the property that if a purported solution to the problem is give ..."
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In this paper we give an overview of the quantum computational complexity class QMA and a description of known QMAcomplete problems to date 1. Such problems are believed to be difficult to solve, even with a quantum computer, but have the property that if a purported solution to the problem is given, a quantum computer would easily be able to verify whether it is correct. An attempt has been made to make this paper as selfcontained as possible so that it can be accessible to computer scientists, physicists, mathematicians, and quantum chemists. Problems of interest to all of these professions can be found here.
The quantum PCP conjecture
 ACM SIGACT News
, 2013
"... 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 t ..."
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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, the global nature of entanglement and its topological properties, 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 theoryofCS audience. 1
1 QUANTUM COMPUTING AND THE ENTANGLEMENT FRONTIER
, 2014
"... Quantum information science explores the frontier of highly complex quantum states, the “entanglement frontier. ” This study is motivated by the observation (widely believed but unproven) that classical systems cannot simulate highly entangled quantum systems efficiently, and we hope to hasten the d ..."
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Quantum information science explores the frontier of highly complex quantum states, the “entanglement frontier. ” This study is motivated by the observation (widely believed but unproven) that classical systems cannot simulate highly entangled quantum systems efficiently, and we hope to hasten the day when well controlled quantum systems can perform tasks surpassing what can be done in the classical world. One way to achieve such “quantum supremacy ” would be to run an algorithm on a quantum computer which solves a problem with a superpolynomial speedup relative to classical computers, but there may be other ways that can be achieved sooner, such as simulating exotic quantum states of strongly correlated matter. To operate a large scale quantum computer reliably we will need to overcome the debilitating effects of decoherence, which might be done using “standard ” quantum hardware protected by quantum errorcorrecting codes, or by exploiting the nonabelian quantum statistics of anyons realized in solid state systems, or by combining both methods. Only by challenging the entanglement frontier will we learn whether Nature provides extravagant resources far beyond what the classical world would allow. Rapporteur talk at the 25th Solvay Conference on Physics
The commuting local Hamiltonian on locallyexpanding graphs is in NP
, 2013
"... The local Hamiltonian problem is famously complete for the class QMA, the quantum analogue of NP [30]. The complexity of its semiclassical version, in which the terms of the Hamiltonian are required to commute (the CLH problem), has attracted considerable attention recently [4, 18, 27, 38, 28] due ..."
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The local Hamiltonian problem is famously complete for the class QMA, the quantum analogue of NP [30]. The complexity of its semiclassical version, in which the terms of the Hamiltonian are required to commute (the CLH problem), has attracted considerable attention recently [4, 18, 27, 38, 28] due to its intriguing nature, as well as in relation to growing interest in the qPCP conjecture [2, 3]. We show here that if the underlying bipartite interaction graph of the CLH instance is a good locallyexpanding graph, namely, the expansion of any constantsize set is εclose to optimal, then approximating its ground energy to within additive factor O(ε) lies in NP. The proof holds for klocal Hamiltonians for any constant k and any constant dimensionality of particles d. We also show that the approximation problem of CLH on such good local expanders is NPhard. This implies that too good local expansion of the interaction graph constitutes an obstacle against quantum hardness of the approximation problem, though it retains its classical hardness. The result highlights new difficulties in trying to mimic classical proofs (in particular Dinur’s PCP proof [21]) in an attempt to prove the quantum PCP conjecture. A related result was discovered recently independently by Brandão and Harrow [16], for 2local general Hamiltonians, bounding the quantum hardness of the approximation problem on good expanders, though no NPhardness is known in that case.
cb Licensed under a Creative Commons Attribution License (CCBY)
, 2011
"... Abstract: We study the complexity of a class of problems involving satisfying constraints which remain the same under translations in one or more spatial directions. In this paper, we show hardness of a classical tiling problem on an N × N 2dimensional grid and a quantum problem involving finding t ..."
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Abstract: We study the complexity of a class of problems involving satisfying constraints which remain the same under translations in one or more spatial directions. In this paper, we show hardness of a classical tiling problem on an N × N 2dimensional grid and a quantum problem involving finding the ground state energy of a 1dimensional quantum system of N particles. In both cases, the only input is N, provided in binary. We show that the classical problem is NEXPcomplete and the quantum problem is QMAEXPcomplete. Thus, an algorithm for these problems which runs in time polynomial in N (exponential in the input size) would imply that EXP = NEXP or BQEXP = QMAEXP, respectively. Although tiling in general is already known to be NEXPcomplete, the usual approach is to require that either the set of tiles and their constraints or some varying boundary conditions be given as part of the input. In the problem considered here, these are fixed, constantsized parameters of the problem. Instead, the problem instance is encoded solely in the size of the system. ACM Classification: F.1.3