38 research outputs found
Holographic Algorithms Beyond Matchgates
Holographic algorithms introduced by Valiant are composed of two ingredients:
matchgates, which are gadgets realizing local constraint functions by weighted
planar perfect matchings, and holographic reductions, which show equivalences
among problems with different descriptions via certain basis transformations.
In this paper, we replace matchgates in the paradigm above by the affine type
and the product type constraint functions, which are known to be tractable in
general (not necessarily planar) graphs. More specifically, we present
polynomial-time algorithms to decide if a given counting problem has a
holographic reduction to another problem defined by the affine or product-type
functions. Our algorithms also find a holographic transformation when one
exists. We further present polynomial-time algorithms of the same decision and
search problems for symmetric functions, where the complexity is measured in
terms of the (exponentially more) succinct representations. The algorithm for
the symmetric case also shows that the recent dichotomy theorem for Holant
problems with symmetric constraints is efficiently decidable. Our proof
techniques are mainly algebraic, e.g., using stabilizers and orbits of group
actions.Comment: Inf. Comput., to appear. Author accepted manuscrip
Normal Factor Graphs and Holographic Transformations
This paper stands at the intersection of two distinct lines of research. One
line is "holographic algorithms," a powerful approach introduced by Valiant for
solving various counting problems in computer science; the other is "normal
factor graphs," an elegant framework proposed by Forney for representing codes
defined on graphs. We introduce the notion of holographic transformations for
normal factor graphs, and establish a very general theorem, called the
generalized Holant theorem, which relates a normal factor graph to its
holographic transformation. We show that the generalized Holant theorem on the
one hand underlies the principle of holographic algorithms, and on the other
hand reduces to a general duality theorem for normal factor graphs, a special
case of which was first proved by Forney. In the course of our development, we
formalize a new semantics for normal factor graphs, which highlights various
linear algebraic properties that potentially enable the use of normal factor
graphs as a linear algebraic tool.Comment: To appear IEEE Trans. Inform. Theor
Holographic Algorithms Beyond Matchgates
Holographic algorithms based on matchgates were introduced by Valiant. These algorithms run in polynomial-time and are intrinsically for planar problems. We introduce two new families of holographic algorithms, which work over general, i.e., not necessarily planar, graphs. The two underlying families of constraint functions are of the affine and product types. These play the role of Kasteleyn’s algorithm for counting planar perfect matchings. The new algorithms are obtained by transforming a problem to one of these two families by holographic reductions. We present a polynomial-time algorithm to decide if a given counting problem has a holographic algorithm using these constraint families. When the constraints are symmetric, we give a polynomial-time decision procedure in the size of the succinct presentation of symmetric constraint functions. This procedure shows that the recent dichotomy theorem for Holant problems with symmetric constraints is polynomial-time decidable
Matchgates and classical simulation of quantum circuits
Let G(A,B) denote the 2-qubit gate which acts as the 1-qubit SU(2) gates A
and B in the even and odd parity subspaces respectively, of two qubits. Using a
Clifford algebra formalism we show that arbitrary uniform families of circuits
of these gates, restricted to act only on nearest neighbour (n.n.) qubit lines,
can be classically efficiently simulated. This reproduces a result originally
proved by Valiant using his matchgate formalism, and subsequently related by
others to free fermionic physics. We further show that if the n.n. condition is
slightly relaxed, to allowing the same gates to act only on n.n. and next-n.n.
qubit lines, then the resulting circuits can efficiently perform universal
quantum computation. From this point of view, the gap between efficient
classical and quantum computational power is bridged by a very modest use of a
seemingly innocuous resource (qubit swapping). We also extend the simulation
result above in various ways. In particular, by exploiting properties of
Clifford operations in conjunction with the Jordan-Wigner representation of a
Clifford algebra, we show how one may generalise the simulation result above to
provide further classes of classically efficiently simulatable quantum
circuits, which we call Gaussian quantum circuits.Comment: 18 pages, 2 figure
Quantum Circuits and Spin(3n) Groups
All quantum gates with one and two qubits may be described by elements of
groups due to isomorphisms and . However, the group of -qubit gates for has bigger
dimension than . A quantum circuit with one- and two-qubit gates may
be used for construction of arbitrary unitary transformation .
Analogously, the ` circuits' are introduced in this work as products
of elements associated with one- and two-qubit gates with respect to the
above-mentioned isomorphisms.
The matrix tensor product implementation of the group together
with relevant models by usual quantum circuits with qubits are
investigated in such a framework. A certain resemblance with well-known sets of
non-universal quantum gates e.g., matchgates, noninteracting-fermion quantum
circuits) related with may be found in presented approach. Finally,
a possibility of the classical simulation of such circuits in polynomial time
is discussed.Comment: v1. REVTeX 4-1, 2 columns, 10 pages, no figures, v3. extended,
LaTeX2e, 1 col., 23+2 pages, v4. typos, accepted for publicatio
Clifford Gates in the Holant Framework
We show that the Clifford gates and stabilizer circuits in the quantum
computing literature, which admit efficient classical simulation, are
equivalent to affine signatures under a unitary condition. The latter is a
known class of tractable functions under the Holant framework.Comment: 14 page
P versus NP and geometry
I describe three geometric approaches to resolving variants of P v. NP,
present several results that illustrate the role of group actions in complexity
theory, and make a first step towards completely geometric definitions of
complexity classes.Comment: 20 pages, to appear in special issue of J. Symbolic. Comp. dedicated
to MEGA 200