Computational Difficulty of Computing the Density of States

Computational Difficulty of Computing the Density of States...

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Unformatted text preview: arXiv:1010.3060v1 [quant-ph] 15 Oct 2010 Computational Difficulty of Computing the Density of States Brielin Brown, 1, 2 Steven T. Flammia, 2 and Norbert Schuch 3 1 University of Virginia, Departments of Physics and Computer Science, Charlottesville, Virginia 22904, USA 2 Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada 3 California Institute of Technology, Institute for Quantum Information, MC 305-16, Pasadena, California 91125, USA We study the computational difficulty of computing the ground state degeneracy and the density of states for local Hamiltonians. We show that the difficulty of both problems is exactly captured by a class which we call #BQP , which is the counting version of the quantum complexity class QMA . We show that #BQP is not harder than its classical counting counterpart #P , which in turn implies that computing the ground state degeneracy or the density of states for classical Hamiltonians is just as hard as it is for quantum Hamiltonians. Understanding the physical properties of correlated quantum many-body systems is a problem of central im- portance in condensed matter physics. The density of states, defined as the number of energy eigenstates per energy interval, plays a particularly crucial role in this endeavor. It is a key ingredient when deriving many thermodynamic properties from microscopic models, in- cluding specific heat capacity, thermal conductivity, band structure, and (near the Fermi energy) most electronic properties of metals. Computing the density of states can be a daunting task however, as it in principle involves diagonalizing a Hamiltonian acting on an exponentially large space, though other more efficient approaches which might take advantage of the structure of a given problem are not a priori ruled out. In this work, we precisely quantify the difficulty of com- puting the density of states by using the powerful tools of quantum complexity theory. Quantum complexity aims at generalizing the well-established field of classical com- plexity theory to assess the difficulty of tasks related to quantum mechanical problems, concerning both the clas- sical difficulty of simulating quantum systems as well as the fundamental limits to the power of quantum com- puters. In particular, quantum complexity theory has managed to explain the difficulty of computing ground state properties of quantum spin systems in various set- tings, such as two-dimensional lattices [ 1 ] and even one- dimensional chains [ 2 ], as well as fermionic systems [ 3 ]. We will determine the computational difficulty of two problems: First, computing the density of states of a local Hamiltonian, and second, counting the ground state de- generacy of a local gapped Hamiltonian. To this end, we will introduce the quantum counting class #BQP (sharp BQP), which constitutes the natural counting version of the class QMA (Quantum Merlin Arthur) which itself captures the difficulty of computing the ground state en- ergy of a local Hamiltonian [...
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Computational Difficulty of Computing the Density of States...

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