arXiv subjects

Niel de Beaudrap

Publications and source records attributed to Niel de Beaudrap.

33 records · Page 2Linked to original sources

arXiv2015

On exact counting and quasi-quantum complexity

We present characterisations of "exact" gap-definable classes, in terms of indeterministic models of computation which slightly modify the standard model of quantum computation. This follows on work of Aaronson [arXiv:quant-ph/0412187], who shows that the counting class PP can be characterised in terms of bounded-error "quantum" algorithms which use invertible (and possibly non-unitary) transformations, or postselections on events of non-zero probability. Our work considers similar modifications of the quantum computational model, but in the setting of exact algorithms, and algorithms with zero error and constant success probability. We show that the gap-definable counting classes [J. Comput. Syst. Sci. 48 (1994), p.116] which bound exact and zero-error quantum algorithms can be characterised in terms of "quantum-like" algorithms involving nonunitary gates, and that postselection and nonunitarity have equivalent power for exact quantum computation only if these classes collapse.

cs.CC↗

arXiv2015

A linear time algorithm for quantum 2-SAT

The Boolean constraint satisfaction problem 3-SAT is arguably the canonical NP-complete problem. In contrast, 2-SAT can not only be decided in polynomial time, but in fact in deterministic linear time. In 2006, Bravyi proposed a physically motivated generalization of k-SAT to the quantum setting, defining the problem "quantum k-SAT". He showed that quantum 2-SAT is also solvable in polynomial time on a classical computer, in particular in deterministic time O(n^4), assuming unit-cost arithmetic over a field extension of the rational numbers, where n is number of variables. In this paper, we present an algorithm for quantum 2-SAT which runs in linear time, i.e. deterministic time O(n+m) for n and m the number of variables and clauses, respectively. Our approach exploits the transfer matrix techniques of Laumann et al. [QIC, 2010] used in the study of phase transitions for random quantum 2-SAT, and bears similarities with both the linear time 2-SAT algorithms of Even, Itai, and Shamir (based on backtracking) [SICOMP, 1976] and Aspvall, Plass, and Tarjan (based on strongly connected components) [IPL, 1979].

quant-ph↗

arXiv2014

On computation with 'probabilities' modulo k

We propose a framework to study models of computation of indeterministic data, represented by abstract "distributions". In these distributions, probabilities are replaced by "amplitudes" drawn from a fixed semi-ring $S$, of which the non-negative reals, the complex numbers, finite fields $\mathbb F_{p^r}$, and cyclic rings $\mathbb Z_k$ are examples. Varying $S$ yields different models of computation, which we may investigate to better understand the (likely) difference in power between randomised and quantum computation. The "modal quantum states" of Schumacher and Westmoreland [arXiv:1010.2929] are examples of such distributions, for $S$ a finite field. For $S = \mathbb F_2$, Willcock and Sabry [arXiv:1102.3587] show that UNIQUE-SAT is solvable by polynomial-time uniform circuit families consisting of invertible gates. We characterize the decision problems solvable by polynomial uniform circuit families, using either invertible or "unitary" transformations over cyclic rings $S = \mathbb Z_k$, or (in the case that $k$ is a prime power) finite fields $S = \mathbb F_k$. In particular, for $k$ a prime power, these are precisely the problems in the class $\mathsf{Mod}_k\mathsf P$.

cs.CC↗

arXiv2014

Difficult instances of the counting problem for 2-quantum-SAT are very atypical

The problem 2-quantum-satisfiability (2-QSAT) is the generalisation of the 2-CNF-SAT problem to quantum bits, and is equivalent to determining whether or not a spin-1/2 Hamiltonian with two-body terms is frustration-free. Similarly to the classical problem 2-SAT, the counting problem #2-QSAT of determining the size (i.e. the dimension) of the set of satisfying states is #P-complete. However, if we consider random instances of #2-QSAT in which constraints are sampled from the Haar measure, intractible instances have measure zero. An apparent reason for this is that almost all two-qubit constraints are entangled, which more readily give rise to long-range constraints. We investigate under which conditions product constraints also give rise to efficiently solvable families of #2-QSAT instances. We consider #2-QSAT involving only discrete distributions over tensor product operators, which interpolates between classical #2-SAT and #2-QSAT involving arbitrary product constraints. We find that such instances of #2-QSAT, defined on Erdos--Renyi graphs or bond-percolated lattices, are asymptotically almost surely efficiently solvable except to the extent that they are biased to resemble monotone instances of #2-SAT.

quant-ph↗

arXiv2013

On the complexity of solving linear congruences and computing nullspaces modulo a constant

We consider the problems of determining the feasibility of a linear congruence, producing a solution to a linear congruence, and finding a spanning set for the nullspace of an integer matrix, where each problem is considered modulo an arbitrary constant k>1. These problems are known to be complete for the logspace modular counting classes {Mod_k L} = {coMod_k L} in special case that k is prime (Buntrock et al, 1992). By considering variants of standard logspace function classes --- related to #L and functions computable by UL machines, but which only characterize the number of accepting paths modulo k --- we show that these problems of linear algebra are also complete for {coMod_k L} for any constant k>1. Our results are obtained by defining a class of functions FUL_k which are low for {Mod_k L} and {coMod_k L} for k>1, using ideas similar to those used in the case of k prime in (Buntrock et al, 1992) to show closure of Mod_k L under NC^1 reductions (including {Mod_k L} oracle reductions). In addition to the results above, we briefly consider the relationship of the class FUL_k for arbitrary moduli k to the class {F.coMod_k L} of functions whose output symbols are verifiable by {coMod_k L} algorithms; and consider what consequences such a comparison may have for oracle closure results of the form {Mod_k L}^{Mod_k L} = {Mod_k L} for composite k.

cs.CC↗

arXiv2013

Interpreting the von Neumann entropy of graph Laplacians, and coentropic graphs

For any graph, we define a rank-1 operator on a bipartite tensor product space, with components associated to the set of vertices and edges respectively. We show that the partial traces of the operator are the Laplacian and the edge-Laplacian. This provides an interpretation of the von Neumann entropy of the (normalized)\ Laplacian as the amount of quantum entanglement between two systems corresponding to vertices and edges. In this framework, cospectral graphs correspond exactly to local unitarily equivalent pure states. Finally, we introduce the notion of coentropic graphs, that is, graphs with equal von Neumann entropy. The smallest coentropic (but not cospectral) graphs that we are able to construct have 8 vertices. The number of equivalence classes of coentropic graphs with n vertices and m edges is a lower bound to the number of (pure) bipartite entanglement classes with subsystems of corresponding dimension.

math.CO↗

arXiv2012

A linearized stabilizer formalism for systems of finite dimension

The stabilizer formalism is a scheme, generalizing well-known techniques developed by Gottesman [quant-ph/9705052] in the case of qubits, to efficiently simulate a class of transformations ("stabilizer circuits", which include the quantum Fourier transform and highly entangling operations) on standard basis states of d-dimensional qudits. To determine the state of a simulated system, existing treatments involve the computation of cumulative phase factors which involve quadratic dependencies. We present a simple formalism in which Pauli operators are represented using displacement operators in discrete phase space, expressing the evolution of the state via linear transformations modulo D <= 2d. We thus obtain a simple proof that simulating stabilizer circuits on n qudits, involving any constant number of measurement rounds, is complete for the complexity class coMod_{d}L and may be simulated by O(log(n)^2)-depth boolean circuits for any constant d >= 2.

quant-ph↗

arXiv2010

Ground states of unfrustrated spin Hamiltonians satisfy an area law

We show that ground states of unfrustrated quantum spin-1/2 systems on general lattices satisfy an entanglement area law, provided that the Hamiltonian can be decomposed into nearest-neighbor interaction terms which have entangled excited states. The ground state manifold can be efficiently described as the image of a low-dimensional subspace of low Schmidt measure, under an efficiently contractible tree-tensor network. This structure gives rise to the possibility of efficiently simulating the complete ground space (which is in general degenerate). We briefly discuss "non-generic" cases, including highly degenerate interactions with product eigenbases, using a relationship to percolation theory. We finally assess the possibility of using such tree tensor networks to simulate almost frustration-free spin models.

quant-ph↗

arXiv2010

On restricted unitary Cayley graphs and symplectic transformations modulo n

We present some observations on a restricted variant of unitary Cayley graphs modulo n, and the implications for a decomposition of elements of symplectic operators over the integers modulo n. We define quadratic unitary Cayley graphs G_n, whose vertex set is the ring Z_n, and where residues a, b modulo n are adjacent if and only if their difference is a quadratic residue. By bounding the diameter of such graphs, we show an upper bound on the number of elementary operations (symplectic scalar multiplications, symplectic row swaps, and row additions or subtractions) required to decompose a symplectic matrix over Z_n. We also characterize the conditions on n for G_n to be a perfect graph.

math.CO↗

arXiv2009

Unitary-circuit semantics for measurement-based computations

One-way measurement based quantum computations (1WQC) may describe unitary transformations, via a composition of CPTP maps which are not all unitary themselves. This motivates the following decision problems: Is it possible to determine whether a ``quantum-to-quantum'' 1WQC procedure (having non-trivial input and output subsystems) performs a unitary transformation? Is it possible to describe precisely how such computations transform quantum states, by translation to a quantum circuit of comparable complexity? In this article, we present an efficient algorithm for transforming certain families of measurement-based computations into a reasonable unitary circuit model, in particular without employing the principle of deferred measurement.

quant-ph↗

arXiv2008

Finding flows in the one-way measurement model

The one-way measurement model is a framework for universal quantum computation, in which algorithms are partially described by a graph G of entanglement relations on a collection of qubits. A sufficient condition for an algorithm to perform a unitary embedding between two Hilbert spaces is for the graph G, together with input/output vertices I, O \subset V(G), to have a flow in the sense introduced by Danos and Kashefi [quant-ph/0506062]. For the special case of |I| = |O|, using a graph-theoretic characterization, I show that such flows are unique when they exist. This leads to an efficient algorithm for finding flows, by a reduction to solved problems in graph theory.

quant-ph↗

arXiv2008

Quadratic Form Expansions for Unitaries

We introduce techniques to analyze unitary operations in terms of quadratic form expansions, a form similar to a sum over paths in the computational basis when the phase contributed by each path is described by a quadratic form over $\mathbb R$. We show how to relate such a form to an entangled resource akin to that of the one-way measurement model of quantum computing. Using this, we describe various conditions under which it is possible to efficiently implement a unitary operation U, either when provided a quadratic form expansion for U as input, or by finding a quadratic form expansion for U from other input data.

quant-ph↗

arXiv2008

An extremal result for geometries in the one-way measurement model

We present an extremal result for the class of graphs G which (together with some specified sets of input and output vertices, I and O) have a certain "flow" property introduced by Danos and Kashefi for the one-way measurement model of quantum computation. The existence of a flow for a triple (G,I,O) allows a unitary embedding to be derived from any choice of measurement bases allowed in the one-way measurement model. We prove an upper bound on the number of edges that a graph G may have, in order for a triple (G,I,O) to have a flow for some $I, O \subseteq V(G)$, in terms of the number of vertices in G and O. This implies that finding a flow for a triple (G,I,O) when |I| = |O| = k (corresponding to unitary transformations in the measurement model) and |V(G)| = n can be performed in time O(k^2 n), improving the earlier known bound of O(km) given in [quant-ph/0611284], where m = |E(G)|.

quant-ph↗

arXiv2007

A complete algorithm to find flows in the one-way measurement model

This article is the complement to [quant-ph/0611284], which proves that flows (as introduced by [quant-ph/0506062]) can be found efficiently for patterns in the one-way measurement model which have non-empty input and output subsystems of the same size. This article presents a complete algorithm for finding flows, and a proof of its' correctness, without assuming any knowledge of graph-theoretic algorithms on the part of the reader. This article is a revised version of [quant-ph/0603072v2], where the results of [quant-ph/0611284] also first appeared.

quant-ph↗

arXiv2006

Phase map decompositions for unitaries

We propose a universal decomposition of unitary maps over a tensorial power of C^2, introducing the key concept of "phase maps", and investigate how this decomposition can be used to implement unitary maps directly in the measurement-based model for quantum computing. Specifically, we show how to extract from such a decomposition a matching entangled graph state (with inputs), and a set of measurements angles, when there is one. Next, we check whether the obtained graph state verifies a "flow" condition, which guarantees an execution order such that the dependent measurements and corrections of the pattern yield deterministic results. Using a graph theoretic characterization of flows, we can determine whether a flow can be constructed for a graph state in polynomial time. This approach yields an algorithmic procedure which, when it succeeds, may produce an efficient pattern for a given unitary.

quant-ph↗