2,648 research outputs found
Perfect Computational Equivalence between Quantum Turing Machines and Finitely Generated Uniform Quantum Circuit Families
In order to establish the computational equivalence between quantum Turing
machines (QTMs) and quantum circuit families (QCFs) using Yao's quantum circuit
simulation of QTMs, we previously introduced the class of uniform QCFs based on
an infinite set of elementary gates, which has been shown to be computationally
equivalent to the polynomial-time QTMs (with appropriate restriction of
amplitudes) up to bounded error simulation. This result implies that the
complexity class BQP introduced by Bernstein and Vazirani for QTMs equals its
counterpart for uniform QCFs. However, the complexity classes ZQP and EQP for
QTMs do not appear to equal their counterparts for uniform QCFs. In this paper,
we introduce a subclass of uniform QCFs, the finitely generated uniform QCFs,
based on finite number of elementary gates and show that the class of finitely
generated uniform QCFs is perfectly equivalent to the class of polynomial-time
QTMs; they can exactly simulate each other. This naturally implies that BQP as
well as ZQP and EQP equal the corresponding complexity classes of the finitely
generated uniform QCFs.Comment: 11page
Uniform Diagonalization Theorem for Complexity Classes of Promise Problems including Randomized and Quantum Classes
Diagonalization in the spirit of Cantor's diagonal arguments is a widely used
tool in theoretical computer sciences to obtain structural results about
computational problems and complexity classes by indirect proofs. The Uniform
Diagonalization Theorem allows the construction of problems outside complexity
classes while still being reducible to a specific decision problem. This paper
provides a generalization of the Uniform Diagonalization Theorem by extending
it to promise problems and the complexity classes they form, e.g. randomized
and quantum complexity classes. The theorem requires from the underlying
computing model not only the decidability of its acceptance and rejection
behaviour but also of its promise-contradicting indifferent behaviour - a
property that we will introduce as "total decidability" of promise problems.
Implications of the Uniform Diagonalization Theorem are mainly of two kinds:
1. Existence of intermediate problems (e.g. between BQP and QMA) - also known
as Ladner's Theorem - and 2. Undecidability if a problem of a complexity class
is contained in a subclass (e.g. membership of a QMA-problem in BQP). Like the
original Uniform Diagonalization Theorem the extension applies besides BQP and
QMA to a large variety of complexity class pairs, including combinations from
deterministic, randomized and quantum classes.Comment: 15 page
Classically-Controlled Quantum Computation
Quantum computations usually take place under the control of the classical
world. We introduce a Classically-controlled Quantum Turing Machine (CQTM)
which is a Turing Machine (TM) with a quantum tape for acting on quantum data,
and a classical transition function for a formalized classical control. In
CQTM, unitary transformations and measurements are allowed. We show that any
classical TM is simulated by a CQTM without loss of efficiency. The gap between
classical and quantum computations, already pointed out in the framework of
measurement-based quantum computation is confirmed. To appreciate the
similarity of programming classical TM and CQTM, examples are given.Comment: 20 page
Polynomial time quantum computation with advice
Advice is supplementary information that enhances the computational power of
an underlying computation. This paper focuses on advice that is given in the
form of a pure quantum state and examines the influence of such advice on the
behaviors of an underlying polynomial-time quantum computation with
bounded-error probability.Comment: 9 page
An Introduction to Quantum Complexity Theory
We give a basic overview of computational complexity, query complexity, and
communication complexity, with quantum information incorporated into each of
these scenarios. The aim is to provide simple but clear definitions, and to
highlight the interplay between the three scenarios and currently-known quantum
algorithms.Comment: 28 pages, LaTeX, 11 figures within the text, to appear in "Collected
Papers on Quantum Computation and Quantum Information Theory", edited by C.
Macchiavello, G.M. Palma, and A. Zeilinger (World Scientific
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