Ronald de Wolf

Quantum lower bounds by polynomials

We examine the number of queries to input variables that a quantum algorithm requires to compute Boolean functions on {0,1}N in the black-box model. We show that the exponential quantum speed-up obtained for partial functions (i.e., problems involving a promise on the input) by Deutsch and Jozsa, Simon, and Shor cannot be obtained for any total function: if a quantum algorithm computes some total Boolean function f with small error probability using T black-box queries, then there is a classical deterministic algorithm that computes f exactly with O(Ts6) queries. We also give asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings. Finally, we give new precise bounds for AND, OR, and PARITY. Our results are a quantum extension of the so-called polynomial method, which has been successfully applied in classical complexity theory, and also a quantum extension of results by Nisan about a polynomial relationship between randomized and deterministic decision tree complexity.

Complexity measures and decision tree complexity: a survey

We discuss several complexity measures for Boolean functions: certificate complexity, sensitivity, block sensitivity, and the degree of a representing or approximating polynomial. We survey the relations and biggest gaps known between these measures, and show how they give bounds for the decision tree complexity of Boolean functions on deterministic, randomized, and quantum computers.

Algorithmic Clustering of Music Based on String Compression

Cilibrasi, Vitanyi, and de Wolf Computer Music Journal, 28:4, pp. 49–67, Winter 2004 2004 Massachusetts Institute of Technology Rudi Cilibrasi,* Paul Vitanyi,*† and Ronald de Wolf* *Centrum voor Wiskunde en Informatica Kruislaan 413 1098 SJ Amsterdam, The Netherlands †Institute for Logic, Language, and Computation University of Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam, The Netherlands {Rudi.Cilibrasi, Paul.Vitanyi, Ronald.de.Wolf}@cwi.nl Algorithmic Clustering of Music Based on String Compression

Nonlocality and communication complexity

Quantum information processing is the emerging field that defines and realizes computing devices that make use of quantum mechanical principles, like the superposition principle, entanglement, and interference. Until recently the common notion of computing was based on classical mechanics, and did not take into account all the possibilities that physically-realizable computing devices offer in principle. The field gained momentum after Peter Shor developed an efficient algorithm for factoring numbers, demonstrating the potential computing powers that quantum computing devices can unleash. In this review we study the information counterpart of computing. It was realized early on by Holevo, that quantum bits, the quantum mechanical counterpart of classical bits, cannot be used for efficient transformation of information, in the sense that arbitrary k-bit messages can not be compressed into messages of k − 1 qubits. The abstract form of the distributed computing setting is called communication complexity. It studies the amount of information, in terms of bits or in our case qubits, that two spatially separated computing devices need to exchange in order to perform some computational task. Surprisingly, quantum mechanics can be used to obtain dramatic advantages for such tasks. We review the area of quantum communication complexity, and show how it connects the foundational physics questions regarding non-locality with those of communication complexity studied in theoretical computer science. The first examples exhibiting the advantage of the use of qubits in distributed information-processing tasks were based on non-locality tests. However, by now the field has produced strong and interesting quantum protocols and algorithms of its own that demonstrate that entanglement, although it cannot be used to replace communication, can be used to reduce the communication exponentially. In turn, these new advances yield a new outlook on the foundations of physics, and could even yield new proposals for experiments that test the foundations of physics.

Quantum search on bounded-error inputs

Suppose we have n algorithms, quantum or classical, each computing some bit-value with bounded error probability. We describe a quantum algorithm that uses O(√n) repetitions of the base algorithms and with high probability finds the index of a 1-bit among these n bits (if there is such an index). This shows that it is not necessary to first significantly reduce the error probability in the base algorithms to O(1/poly(n)) (which would require O(√n log n) repetitions in total). Our technique is a recursive interleaving of amplitude amplification and error-reduction, and may be of more general interest. Essentially, it shows that quantum amplitude amplification can be made to work also with a bounded-error verifier. As a corollary we obtain optimal quantum upper bounds of O(√N) queries for all constant-depth AND-OR trees on N variables, improving upon earlier upper bounds of O(√Npolylog(N)).

A Brief Introduction to Fourier Analysis on the Boolean Cube

We give a brief introduction to the basic notions of Fourier analysis on the Boolean cube, illustrated and motivated by a number of applications to theoretical computer science. ACM Classification: F.0, F.2 AMS Classification: 42-02, 68-02, 68Q17, 68Q25, 68Q32

Exponential lower bound for 2-query locally decodable codes via a quantum argument

A locally decodable code encodes <i>n</i>-bit strings <i>x</i> in <i>m</i>-bit codewords <i>C(x)</i>, in such a way that one can recover any bit <i>x_i</i> from a corrupted codeword by querying only a few bits of that word. We use a <i>quantum</i> argument to prove that LDCs with 2 classical queries need exponential length: <i>m=2^Ω(n)</i>. Previously this was known only for linear codes (Goldreich et al. 02). Our proof shows that a 2-query LDC can be decoded with only 1 quantum query, and then proves an exponential lower bound for such 1-query locally quantum-decodable codes. We also show that <i>q</i> quantum queries allow more succinct LDCs than the best known LDCs with <i>q</i> classical queries. Finally, we give new classical lower bounds and quantum upper bounds for the setting of private information retrieval. In particular, we exhibit a quantum 2 server PIR scheme with <i>O(n^3/10)</i> qubits of communication, improving upon the <i>O(n^1/3)</i> bits of communication of the best known classical 2-server PIR.

Exponential separations for one-way quantum communication complexity, with applications to cryptography

We give an exponential separation between one-way quantum and classical communication protocols for twopartial Boolean functions, both of which are variants of the Boolean Hidden Matching Problem of Bar-Yossef et al. Earlier such an exponential separation was known only for a relational version of the Hidden Matching Problem. Our proofs use the Fourier coefficients inequality of Kahn, Kalai, and Linial. We give a number of applications of this separation. In particular, in the bounded-storage model of cryptography we exhibita scheme that is secure against adversaries with a certain amount of classical storage, but insecure against adversaries with a similar (or even much smaller) amount of quantum storage; in the setting of privacy amplification, we show that there are strong extractors that yield a classically secure key, but are insecure against a quantum adversary.

Quantum and Classical Strong Direct Product Theorems and Optimal Time-Space Tradeoffs

A strong direct product theorem says that if we want to compute

Quantum communication and complexity

k