297 research outputs found
Optimal simulation of three-qubit gates
In this paper, we study the optimal simulation of three-qubit unitary by
using two-qubit gates. First, we give a lower bound on the two-qubit gates cost
of simulating a multi-qubit gate. Secondly, we completely characterize the
two-qubit gate cost of simulating a three-qubit controlled controlled gate by
generalizing our result on the cost of Toffoli gate. The function of controlled
controlled gate is simply a three-qubit controlled unitary gate and can be
intuitively explained as follows: the gate will output the states of the two
control qubit directly, and apply the given one-qubit unitary on the target
qubit only if both the states of the control are . Previously, it is
only known that five two-qubit gates is sufficient for implementing such a gate
[Sleator and Weinfurter, Phys. Rev. Lett. 74, 4087 (1995)]. Our result shows
that if the determinant of is 1, four two-qubit gates is achievable
optimal. Otherwise, five is optimal. Thirdly, we show that five two-qubit gates
are necessary and sufficient for implementing the Fredkin gate(the controlled
swap gate), which settles the open problem introduced in [Smolin and
DiVincenzo, Phys. Rev. A, 53, 2855 (1996)]. The Fredkin gate is one of the most
important quantum logic gates because it is universal alone for classical
reversible computation, and thus with little help, universal for quantum
computation. Before our work, a five two-qubit gates decomposition of the
Fredkin gate was already known, and numerical evidence of showing five is
optimal is found.Comment: 16 Pages, comments welcom
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
A digital computer is generally believed to be an efficient universal
computing device; that is, it is believed able to simulate any physical
computing device with an increase in computation time of at most a polynomial
factor. This may not be true when quantum mechanics is taken into
consideration. This paper considers factoring integers and finding discrete
logarithms, two problems which are generally thought to be hard on a classical
computer and have been used as the basis of several proposed cryptosystems.
Efficient randomized algorithms are given for these two problems on a
hypothetical quantum computer. These algorithms take a number of steps
polynomial in the input size, e.g., the number of digits of the integer to be
factored.Comment: 28 pages, LaTeX. This is an expanded version of a paper that appeared
in the Proceedings of the 35th Annual Symposium on Foundations of Computer
Science, Santa Fe, NM, Nov. 20--22, 1994. Minor revisions made January, 199
Synthesis and Optimization of Reversible Circuits - A Survey
Reversible logic circuits have been historically motivated by theoretical
research in low-power electronics as well as practical improvement of
bit-manipulation transforms in cryptography and computer graphics. Recently,
reversible circuits have attracted interest as components of quantum
algorithms, as well as in photonic and nano-computing technologies where some
switching devices offer no signal gain. Research in generating reversible logic
distinguishes between circuit synthesis, post-synthesis optimization, and
technology mapping. In this survey, we review algorithmic paradigms ---
search-based, cycle-based, transformation-based, and BDD-based --- as well as
specific algorithms for reversible synthesis, both exact and heuristic. We
conclude the survey by outlining key open challenges in synthesis of reversible
and quantum logic, as well as most common misconceptions.Comment: 34 pages, 15 figures, 2 table
Three Quantum Algorithms to Solve 3-SAT
We propose three quantum algorithms to solve the 3-SAT NP-complete decision problem. The first algorithm builds, for any instance Á of 3-SAT, a quantum Fredkin
circuit that computes a superposition of all classical evaluations of Á in a given output
line. Similarly, the second and third algorithms compute the same superposition on a
given register of a quantum register machine, and as the energy of a given membrane in
a quantum P system, respectively.
Assuming that a specific non-unitary operator, built using the well known creation
and annihilation operators, can be realized as a quantum gate, as an instruction of the
quantum register machine, and as a rule of the quantum P system, respectively, we show
how to decide whether Á is a positive instance of 3-SAT. The construction relies also
upon the assumption that an external observer is able to distinguish, as the result of a
measurement, between a null and a non-null vector
The Classification of Reversible Bit Operations
We present a complete classification of all possible sets of classical reversible gates acting on bits, in terms of which reversible transformations they generate, assuming swaps and ancilla bits are available for free. Our classification can be seen as the reversible-computing analogue of Post\u27s lattice, a central result in mathematical logic from the 1940s. It is a step toward the ambitious goal of classifying all possible quantum gate sets acting on qubits.
Our theorem implies a linear-time algorithm (which we have implemented), that takes as input the truth tables of reversible gates G and H, and that decides whether G generates H. Previously, this problem was not even known to be decidable (though with effort, one can derive from abstract considerations an algorithm that takes triply-exponential time). The theorem also implies that any n-bit reversible circuit can be "compressed" to an equivalent circuit, over the same gates, that uses at most 2^{n}poly(n) gates and O(1) ancilla bits; these are the first upper bounds on these quantities known, and are close to optimal. Finally, the theorem implies that every non-degenerate reversible gate can implement either every reversible transformation, or every affine transformation, when restricted to an "encoded subspace."
Briefly, the theorem says that every set of reversible gates generates either all reversible transformations on n-bit strings (as the Toffoli gate does); no transformations; all transformations that preserve Hamming weight (as the Fredkin gate does); all transformations that preserve Hamming weight mod k for some k; all affine transformations (as the Controlled-NOT gate does); all affine transformations that preserve Hamming weight mod 2 or mod 4, inner products mod 2, or a combination thereof; or a previous class augmented by a NOT or NOTNOT gate. Prior to this work, it was not even known that every class was finitely generated. Ruling out the possibility of additional classes, not in the list, requires involved arguments about polynomials, lattices, and Diophantine equations
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