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 Rn.

The k-{\locHam} problem is a natural complete problem for the complexity class $\QMA$, the quantum analogue of $\NP$. It is similar in spirit to {\sc MAX-k-SAT}, which is $\NP$-complete for $k\geq 2$. It was known that the problem is $\QMA$-complete for any $k \geq 3$. On the other hand, 1-{\locHam} is in {\P} and hence not believed to be $\QMA$-complete. The complexity of the 2-{\locHam} 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 only elementary linear algebra. Our second 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. Using our techniques we also show that adiabatic computation with 2-local interactions on qubits is equivalent to standard quantum computation.

In this paper we consider quantum interactive proof systems, which are interactive proof systems in which the prover and verier may perform quantum computations and exchange quantum information. We prove that any polynomial-round quantum interactive proof system with two-sided bounded error can be parallelized to a quantum interactive proof system with exponentially small one-sided error in which the prover and verier exchange only 3 messages. This yields a simplied proof that PSPACE has 3-message quantum interactive proof systems. We also prove that any language having a quantum interactive proof system can be decided in deterministic exponential time, implying that single-prover quantum interactive proof systems are strictly less powerful than multiple-prover classical interactive proof systems unless EXP = NEXP. 1. INTRODUCTION Interactive proof systems were introduced by Babai [3] and Goldwasser, Micali, and Racko [17] in 1985. In the same year, Deutsch [10] gave the rst for...

Current formal models for quantum computation deal only with unitary gates operating on "pure quantum states". In these models it is difficult or impossible to deal formally with several central issues: measurements in the middle of the computation; decoherence and noise, using probabilistic subroutines, and more. It turns out, that the restriction to unitary gates and pure states is unnecessary. In this paper we generalize the formal model of quantum circuits to a model in which the state can be a general quantum state, namely a mixed state, or a "density matrix", and the gates can be general quantum operations, not necessarily unitary. The new model is shown to be equivalent in computational power to the standard one, and the problems mentioned above essentially disappear. The main result in this paper is a solution for the subroutine problem. The general function that a quantum circuit outputs is a probabilistic function. However, the question of using probabilistic functions as subroutines was not previously dealt with, the reason being that in the language of pure states, this simply can not be done. We define a natural notion of using general subroutines, and show that using general subroutines does not strengthen the model. As an example of the advantages of analyzing quantum complexity using density matrices, we prove a simple lower bound on depth of circuits that compute probabilistic functions. Finally, we deal with the question of inaccurate quantum computation with mixed states. Using the so called "trace metric" on density matrices, we show how to keep track of errors in the new model. Institutes of Physics and Computer science, The Hebrew University, Jerusalem, Israel y L.D.Landau Institute for Theoretical Physics,Moscow, Russia z Institute of Compu...