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Complexity classification of local Hamiltonian problems
, 2013
"... The calculation of groundstate energies of physical systems can be formalised as the klocal Hamiltonian problem, which is the natural quantum analogue of classical constraint satisfaction problems. One way of making the problem more physically meaningful is to restrict the Hamiltonian in question ..."
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The calculation of groundstate energies of physical systems can be formalised as the klocal Hamiltonian problem, which is the natural quantum analogue of classical constraint satisfaction problems. One way of making the problem more physically meaningful is to restrict the Hamiltonian in question by picking its terms from a fixed set S. Examples of such special cases are the Heisenberg and Ising models from condensedmatter physics. In this work we characterise the complexity of this problem for all 2local qubit Hamiltonians. Depending on the subset S, the problem falls into one of the following categories: in P; NPcomplete; polynomialtime equivalent to the Ising model with transverse magnetic fields; or QMAcomplete. The third of these classes contains NP and is contained within StoqMA. The characterisation holds even if S does not contain any 1local terms; for example, we prove for the first time QMAcompleteness of the Heisenberg and XY interactions in this setting. If S is assumed to contain all 1local terms, which is the setting considered by previous work, we have a characterisation that goes beyond 2local interactions: for any constant k, all klocal qubit Hamiltonians whose terms are picked from a fixed set S correspond to problems either in P; polynomialtime equivalent to the Ising model with transverse magnetic fields; or QMAcomplete. These results are a quantum analogue of Schaefer’s dichotomy theorem for boolean constraint satisfaction problems. 1
The complexity of the consistency and Nrepresentability problems for quantum states
"... QMA (Quantum MerlinArthur) is the quantum analogue of the class NP. There are a few QMAcomplete problems, most of which are variants of the “Local Hamiltonian” problem introduced by Kitaev. In this dissertation we show some new QMAcomplete problems which are very different from those known previo ..."
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QMA (Quantum MerlinArthur) is the quantum analogue of the class NP. There are a few QMAcomplete problems, most of which are variants of the “Local Hamiltonian” problem introduced by Kitaev. In this dissertation we show some new QMAcomplete problems which are very different from those known previously, and have applications in quantum chemistry. The first one is “Consistency of Local Density Matrices”: given a collection of density matrices describing different subsets of an nqubit system (where each subset has constant size), decide whether these are consistent with some global state of all n qubits. This problem was first suggested by Aharonov. We show that it is QMAcomplete, via an oracle reduction from Local Hamiltonian. Our reduction is based on algorithms for convex optimization with a membership oracle, due to Yudin and Nemirovskii. Next we show that two problems from quantum chemistry, “Fermionic Local Hamiltonian” and “Nrepresentability, ” are QMAcomplete. These problems involve systems of fermions, rather than qubits; they arise in calculating the ground state energies of molecular systems. Nrepresentability is particularly interesting, as it is a key component
The Local Consistency Problem for Stoquastic and 1D Quantum Systems. ArXiv eprints
, 2007
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Approximation, proof systems, and correlations in a quantum world
, 2013
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The complexity of antiferromagnetic interactions and 2D lattices
, 2015
"... Estimation of the minimum eigenvalue of a quantum Hamiltonian can be formalised as the Local Hamiltonian problem. We study the natural special case of the Local Hamiltonian problem where the same 2local interaction, with differing weights, is applied across each pair of qubits. First we consider an ..."
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Estimation of the minimum eigenvalue of a quantum Hamiltonian can be formalised as the Local Hamiltonian problem. We study the natural special case of the Local Hamiltonian problem where the same 2local interaction, with differing weights, is applied across each pair of qubits. First we consider antiferromagnetic/ferromagnetic interactions, where the weights of the terms in the Hamiltonian are restricted to all be of the same sign. We show that for symmetric 2local interactions with no 1local part, the problem is either QMAcomplete or in StoqMA. In particular the antiferromagnetic Heisenberg and antiferromagnetic XY interactions are shown to be QMAcomplete. We also prove StoqMAcompleteness of the antiferromagnetic transverse field Ising model. Second, we study the Local Hamiltonian problem under the restriction that the interaction terms can only be chosen to lie on a particular graph. We prove that nearly all of the QMAcomplete 2local interactions remain QMAcomplete when restricted to a 2D square lattice. Finally we consider both restrictions at the same time and discover that, with the exception of the antiferromagnetic Heisenberg interaction, all of the interactions which are QMAcomplete with positive coefficients remain QMAcomplete when restricted to a 2D triangular lattice. 1
Quantum Hamiltonian Complexity
, 2014
"... We survey the growing field of Quantum Hamiltonian Complexity, which includes the study of Quantum Constraint Satisfaction. In particular, our aim is to provide a computer scienceoriented introduction to the subject in order to help bridge the language barrier between computer scientists and physic ..."
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We survey the growing field of Quantum Hamiltonian Complexity, which includes the study of Quantum Constraint Satisfaction. In particular, our aim is to provide a computer scienceoriented introduction to the subject in order to help bridge the language barrier between computer scientists and physicists in the field. As such, we include the following in this paper: (1) The basic ideas, motivations, and history of the field, (2) a glossary of manybody physics terms explained in computerscience friendly language, (3) overviews of central ideas from manybody physics, such as Mean Field Theory and Tensor Networks, and (4) brief expositions of selected computer sciencebased results in the area. This paper is based largely on the discussions of the quantum reading group at UC Berkeley in Spring 2013. “Computers are physical objects, and computations are physical processes. What computers can or cannot compute is determined by the laws of physics alone... ” — David Deutsch 1
Tensor network nonzero testing
, 2014
"... Tensor networks are a central tool in condensed matter physics. In this paper, we initiate the study of tensor network nonzero testing (TNZ): Given a tensor network T, does T represent a nonzero vector? We show that TNZ is not in the PolynomialTime Hierarchy unless the hierarchy collapses. We nex ..."
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Tensor networks are a central tool in condensed matter physics. In this paper, we initiate the study of tensor network nonzero testing (TNZ): Given a tensor network T, does T represent a nonzero vector? We show that TNZ is not in the PolynomialTime Hierarchy unless the hierarchy collapses. We next show (among other results) that the special cases of TNZ on nonnegative and injective tensor networks are in NP. Using this, we make a simple observation: The commuting variant of the MAcomplete stoquastic kSAT problem on Ddimensional qudits is in NP for k ∈ O(log n) and D ∈ O(1). This reveals the first class of quantum Hamiltonians whose commuting variant is known to be in NP for all (1) logarithmic k, (2) constant D, and (3) for arbitrary interaction graphs. 1