Abstract: Computing the group of units in a field of algebraic numbers is one of the central tasks of computational algebraic number theory. It is believed to be hard classically, which is of interest for cryptography. In the quantum setting, efficient algorithms were previously known for fields of constant degree. We give a quantum algorithm that is polynomial in the degree of the field and the logarithm of its discriminant. This is achieved by combining three new results. The first is a classical algorithm for computing a basis for certain ideal lattices with doubly exponentially large generators. The second shows that a Gaussian-weighted superposition of lattice points, with an appropriate encoding, can be used to provide a unique representation of a real-valued lattice. The third is an extension of the hidden subgroup problem to continuous groups and a quantum algorithm for solving the HSP over the group ℝ^n.

ID: CaltechAUTHORS:20161010-172823440

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Abstract: Gapped phases of noninteracting fermions, with and without charge conservation and time-reversal symmetry, are classified using Bott periodicity. The symmetry and spatial dimension determines a general universality class, which corresponds to one of the 2 types of complex and 8 types of real Clifford algebras. The phases within a given class are further characterized by a topological invariant, an element of some Abelian group that can be 0, Z, or Z_2. The interface between two infinite phases with different topological numbers must carry some gapless mode. Topological properties of finite systems are described in terms of K-homology. This classification is robust with respect to disorder, provided electron states near the Fermi energy are absent or localized. In some cases (e.g., integer quantum Hall systems) the K-theoretic classification is stable to interactions, but a counterexample is also given.

No.: 1134
ID: CaltechAUTHORS:20100510-100944960

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Abstract: The basic building block of quantum computation is the qubit, a system with two (nearly) degenerate states that can be used to encode quantum information. Real systems typically have a full spectrum of excitations that are considered illegal from the point of view of a computation, and lead to decoherence if they couple too strongly into the qubit states during some process (see Fig. 4.1). The essential problem, then, is to preserve the quantum state of the qubit as long as possible to allow time for computations to take place.

No.: 89
ID: CaltechAUTHORS:20110207-134954371

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Abstract: The k-Local Hamiltonian problem is a natural complete problem for the complexity class QMA, the quantum analog of NP. It is similar in spirit to MAX- k -SAT, which is NP-complete for k ≥ 2. It was known that the problem is QMA-complete for any k ≥ 3. On the other hand 1-Local Hamiltonian is in P, and hence not believed to be QMA-complete. The complexity of the 2-Local Hamiltonian problem has long been outstanding. Here we settle the question and show that it is QMA-complete. We provide two independent proofs; our first proof uses a powerful technique for analyzing the sum of two Hamiltonians; this technique is based on perturbation theory and we believe that it might prove useful elsewhere. The second proof uses elementary linear algebra only. Using our techniques we also show that adiabatic computation with two-local interactions on qubits is equivalent to standard quantum computation.

No.: 3328
ID: CaltechAUTHORS:20191011-072647725

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