Semidefinite programming characterization and spectral adversary method for quantum complexity with noncommuting unitary queries

Generalizing earlier work characterizing the quantum query complexity of computing a function of an unknown classical ``black box'' function drawn from some set of such black box functions, we investigate a more general quantum query model in which the goal is to compute functions of N by N ``black box'' unitary matrices drawn from a set of such matrices, a problem with applications to determining properties of quantum physical systems. We characterize the existence of an algorithm for such a query problem, with given error and number of queries, as equivalent to the feasibility of a certain set of semidefinite programming constraints, or equivalently the infeasibility of a dual of these constraints, which we construct. Relaxing the primal constraints to correspond to mere pairwise near-orthogonality of the final states of a quantum computer, conditional on black-box inputs having distinct function values, rather than bounded-error determinability of the function value via a single measurement on the output states, we obtain a relaxed primal program the feasibility of whose dual still implies the nonexistence of a quantum algorithm. We use this to obtain a generalization, to our not-necessarily-commutative setting, of the ``spectral adversary method'' for quantum query lower bounds.Comment: Dagstuhl Seminar Proceedings 06391, "Algorithms and Complexity for Continuous Problems," ed. S. Dahlke, K. Ritter, I. H. Sloan, J. F. Traub (2006), available electronically at http://drops.dagstuhl.de/portals/index.php?semnr=0639

Tensor Norms and the Classical Communication Complexity of Nonlocal Quantum Measurement

We initiate the study of quantifying nonlocalness of a bipartite measurement by the minimum amount of classical communication required to simulate the measurement. We derive general upper bounds, which are expressed in terms of certain tensor norms of the measurement operator. As applications, we show that (a) If the amount of communication is constant, quantum and classical communication protocols with unlimited amount of shared entanglement or shared randomness compute the same set of functions; (b) A local hidden variable model needs only a constant amount of communication to create, within an arbitrarily small statistical distance, a distribution resulted from local measurements of an entangled quantum state, as long as the number of measurement outcomes is constant.Comment: A preliminary version of this paper appears as part of an article in Proceedings of the the 37th ACM Symposium on Theory of Computing (STOC 2005), 460--467, 200

An Algorithmic Argument for Nonadaptive Query Complexity Lower Bounds on Advised Quantum Computation

This paper employs a powerful argument, called an algorithmic argument, to prove lower bounds of the quantum query complexity of a multiple-block ordered search problem in which, given a block number i, we are to find a location of a target keyword in an ordered list of the i-th block. Apart from much studied polynomial and adversary methods for quantum query complexity lower bounds, our argument shows that the multiple-block ordered search needs a large number of nonadaptive oracle queries on a black-box model of quantum computation that is also supplemented with advice. Our argument is also applied to the notions of computational complexity theory: quantum truth-table reducibility and quantum truth-table autoreducibility.Comment: 16 pages. An extended abstract will appear in the Proceedings of the 29th International Symposium on Mathematical Foundations of Computer Science, Lecture Notes in Computer Science, Springer-Verlag, Prague, August 22-27, 200

Quantum vs. Classical Read-once Branching Programs

The paper presents the first nontrivial upper and lower bounds for (non-oblivious) quantum read-once branching programs. It is shown that the computational power of quantum and classical read-once branching programs is incomparable in the following sense: (i) A simple, explicit boolean function on 2n input bits is presented that is computable by error-free quantum read-once branching programs of size O(n^3), while each classical randomized read-once branching program and each quantum OBDD for this function with bounded two-sided error requires size 2^{\Omega(n)}. (ii) Quantum branching programs reading each input variable exactly once are shown to require size 2^{\Omega(n)} for computing the set-disjointness function DISJ_n from communication complexity theory with two-sided error bounded by a constant smaller than 1/2-2\sqrt{3}/7. This function is trivially computable even by deterministic OBDDs of linear size. The technically most involved part is the proof of the lower bound in (ii). For this, a new model of quantum multi-partition communication protocols is introduced and a suitable extension of the information cost technique of Jain, Radhakrishnan, and Sen (2003) to this model is presented.Comment: 35 pages. Lower bound for disjointness: Error in application of info theory corrected and regularity of quantum read-once BPs (each variable at least once) added as additional assumption of the theorem. Some more informal explanations adde

A Survey on Continuous Time Computations

We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature

New Developments in Quantum Algorithms

In this survey, we describe two recent developments in quantum algorithms. The first new development is a quantum algorithm for evaluating a Boolean formula consisting of AND and OR gates of size N in time O(\sqrt{N}). This provides quantum speedups for any problem that can be expressed via Boolean formulas. This result can be also extended to span problems, a generalization of Boolean formulas. This provides an optimal quantum algorithm for any Boolean function in the black-box query model. The second new development is a quantum algorithm for solving systems of linear equations. In contrast with traditional algorithms that run in time O(N^{2.37...}) where N is the size of the system, the quantum algorithm runs in time O(\log^c N). It outputs a quantum state describing the solution of the system.Comment: 11 pages, 1 figure, to appear as an invited survey talk at MFCS'201

Unbounded-error quantum computation with small space bounds

We prove the following facts about the language recognition power of quantum Turing machines (QTMs) in the unbounded error setting: QTMs are strictly more powerful than probabilistic Turing machines for any common space bound $s$ satisfying $s(n)=o(\log \log n)$. For "one-way" Turing machines, where the input tape head is not allowed to move left, the above result holds for $s(n)=o(\log n)$. We also give a characterization for the class of languages recognized with unbounded error by real-time quantum finite automata (QFAs) with restricted measurements. It turns out that these automata are equal in power to their probabilistic counterparts, and this fact does not change when the QFA model is augmented to allow general measurements and mixed states. Unlike the case with classical finite automata, when the QFA tape head is allowed to remain stationary in some steps, more languages become recognizable. We define and use a QTM model that generalizes the other variants introduced earlier in the study of quantum space complexity.Comment: A preliminary version of this paper appeared in the Proceedings of the Fourth International Computer Science Symposium in Russia, pages 356--367, 200

Simulating quantum computation by contracting tensor networks

The treewidth of a graph is a useful combinatorial measure of how close the graph is to a tree. We prove that a quantum circuit with $T$ gates whose underlying graph has treewidth $d$ can be simulated deterministically in $T^{O(1)}\exp[O(d)]$ time, which, in particular, is polynomial in $T$ if $d=O(\log T)$. Among many implications, we show efficient simulations for log-depth circuits whose gates apply to nearby qubits only, a natural constraint satisfied by most physical implementations. We also show that one-way quantum computation of Raussendorf and Briegel (Physical Review Letters, 86:5188--5191, 2001), a universal quantum computation scheme with promising physical implementations, can be efficiently simulated by a randomized algorithm if its quantum resource is derived from a small-treewidth graph.Comment: 7 figure

Randomized and Quantum Algorithms Yield a Speed-Up for Initial-Value Problems

Quantum algorithms and complexity have recently been studied not only for discrete, but also for some numerical problems. Most attention has been paid so far to the integration problem, for which a speed-up is shown by quantum computers with respect to deterministic and randomized algorithms on a classical computer. In this paper we deal with the randomized and quantum complexity of initial-value problems. For this nonlinear problem, we show that both randomized and quantum algorithms yield a speed-up over deterministic algorithms. Upper bounds on the complexity in the randomized and quantum settings are shown by constructing algorithms with a suitable cost, where the construction is based on integral information. Lower bounds result from the respective bounds for the integration problem.Comment: LaTeX v. 2.09, 13 page