arXiv subjects

Ronald de Wolf

Publications and source records attributed to Ronald de Wolf.

At least 73 records · Page 4Linked to original sources

arXiv2008

Simultaneous Communication Protocols with Quantum and Classical Messages

We study the simultaneous message passing (SMP) model of communication complexity, for the case where one party is quantum and the other is classical. We show that in an SMP protocol that computes some function with the first party sending q qubits and the second sending c classical bits, the quantum message can be replaced by a randomized message of O(qc) classical bits, as well as by a deterministic message of O(q c log q) classical bits. Our proofs rely heavily on earlier results due to Scott Aaronson. In particular, our results imply that quantum-classical protocols need to send Omega(sqrt{n/log n}) bits/qubits to compute Equality on n-bit strings, and hence are not significantly better than classical-classical protocols (and are much worse than quantum-quantum protocols such as quantum fingerprinting). This essentially answers a recent question of Wim van Dam. Our results also imply, more generally, that there are no superpolynomial separations between quantum-classical and classical-classical SMP protocols for functional problems. This contrasts with the situation for relational problems, where exponential gaps between quantum-classical and classical-classical SMP protocols are known. We show that this surprising situation cannot arise in purely classical models: there, an exponential separation for a relational problem can be converted into an exponential separation for a functional problem.

quant-ph↗

arXiv2008

Error-Correcting Data Structures

We study data structures in the presence of adversarial noise. We want to encode a given object in a succinct data structure that enables us to efficiently answer specific queries about the object, even if the data structure has been corrupted by a constant fraction of errors. This new model is the common generalization of (static) data structures and locally decodable error-correcting codes. The main issue is the tradeoff between the space used by the data structure and the time (number of probes) needed to answer a query about the encoded object. We prove a number of upper and lower bounds on various natural error-correcting data structure problems. In particular, we show that the optimal length of error-correcting data structures for the Membership problem (where we want to store subsets of size s from a universe of size n) is closely related to the optimal length of locally decodable codes for s-bit strings.

cs.DS↗

arXiv2008

A Hypercontractive Inequality for Matrix-Valued Functions with Applications to Quantum Computing and LDCs

The Bonami-Beckner hypercontractive inequality is a powerful tool in Fourier analysis of real-valued functions on the Boolean cube. In this paper we present a version of this inequality for matrix-valued functions on the Boolean cube. Its proof is based on a powerful inequality by Ball, Carlen, and Lieb. We also present a number of applications. First, we analyze maps that encode $n$ classical bits into $m$ qubits, in such a way that each set of $k$ bits can be recovered with some probability by an appropriate measurement on the quantum encoding; we show that if $m<0.7 n$, then the success probability is exponentially small in $k$. This result may be viewed as a direct product version of Nayak's quantum random access code bound. It in turn implies strong direct product theorems for the one-way quantum communication complexity of Disjointness and other problems. Second, we prove that error-correcting codes that are locally decodable with 2 queries require length exponential in the length of the encoded string. This gives what is arguably the first ``non-quantum'' proof of a result originally derived by Kerenidis and de Wolf using quantum information theory, and answers a question by Trevisan.

quant-ph↗

arXiv2008

Locally Decodable Quantum Codes

We study a quantum analogue of locally decodable error-correcting codes. A q-query locally decodable quantum code encodes n classical bits in an m-qubit state, in such a way that each of the encoded bits can be recovered with high probability by a measurement on at most q qubits of the quantum code, even if a constant fraction of its qubits have been corrupted adversarially. We show that such a quantum code can be transformed into a classical q-query locally decodable code of the same length that can be decoded well on average (albeit with smaller success probability and noise-tolerance). This shows, roughly speaking, that q-query quantum codes are not significantly better than q-query classical codes, at least for constant or small q.

quant-ph↗

arXiv2008

A note on quantum algorithms and the minimal degree of epsilon-error polynomials for symmetric functions

The degrees of polynomials representing or approximating Boolean functions are a prominent tool in various branches of complexity theory. Sherstov recently characterized the minimal degree deg_{\eps}(f) among all polynomials (over the reals) that approximate a symmetric function f:{0,1}^n-->{0,1} up to worst-case error \eps: deg_{\eps}(f) = ~Θ(deg_{1/3}(f) + \sqrt{n\log(1/\eps)}). In this note we show how a tighter version (without the log-factors hidden in the ~Θ-notation), can be derived quite easily using the close connection between polynomials and quantum algorithms.

quant-ph↗

arXiv2008

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 a partial Boolean function (a variant of the Boolean Hidden Matching Problem of Bar-Yossef et al.) Earlier such an exponential separation was known only for a relational problem. The communication problem corresponds to a \emph{strong extractor} that fails against a small amount of \emph{quantum} information about its random source. Our proof uses the Fourier coefficients inequality of Kahn, Kalai, and Linial. We also give a number of applications of this separation. In particular, we show that there are privacy amplification schemes that are secure against classical adversaries but not against quantum adversaries; and we give the first example of a key-expansion scheme in the model of bounded-storage cryptography that is secure against classical memory-bounded adversaries but not against quantum ones.

quant-ph↗

arXiv2008

Upper Bounds on the Noise Threshold for Fault-tolerant Quantum Computing

We prove new upper bounds on the tolerable level of noise in a quantum circuit. We consider circuits consisting of unitary k-qubit gates each of whose input wires is subject to depolarizing noise of strength p, as well as arbitrary one-qubit gates that are essentially noise-free. We assume that the output of the circuit is the result of measuring some designated qubit in the final state. Our main result is that for p>1-Θ(1/\sqrt{k}), the output of any such circuit of large enough depth is essentially independent of its input, thereby making the circuit useless. For the important special case of k=2, our bound is p>35.7%. Moreover, if the only allowed gate on more than one qubit is the two-qubit CNOT gate, then our bound becomes 29.3%. These bounds on p are notably better than previous bounds, yet are incomparable because of the somewhat different circuit model that we are using. Our main technique is the use of a Pauli basis decomposition, which we believe should lead to further progress in deriving such bounds.

quant-ph↗

arXiv2006

Exponential Separation of Quantum and Classical One-Way Communication Complexity for a Boolean Function

We give an exponential separation between one-way quantum and classical communication complexity for a Boolean function. Earlier such a separation was known only for a relation. A very similar result was obtained earlier but independently by Kerenidis and Raz [KR06]. Our version of the result gives an example in the bounded storage model of cryptography, where the key is secure if the adversary has a certain amount of classical storage, but is completely insecure if he has a similar amount of quantum storage.

quant-ph↗

arXiv2006

A New Quantum Lower Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs

We give a new version of the adversary method for proving lower bounds on quantum query algorithms. The new method is based on analyzing the eigenspace structure of the problem at hand. We use it to prove a new and optimal strong direct product theorem for 2-sided error quantum algorithms computing k independent instances of a symmetric Boolean function: if the algorithm uses significantly less than k times the number of queries needed for one instance of the function, then its success probability is exponentially small in k. We also use the polynomial method to prove a direct product theorem for 1-sided error algorithms for k threshold functions with a stronger bound on the success probability. Finally, we present a quantum algorithm for evaluating solutions to systems of linear inequalities, and use our direct product theorems to show that the time-space tradeoff of this algorithm is close to optimal.

quant-ph↗

arXiv2006

Lower Bounds on Matrix Rigidity via a Quantum Argument

The rigidity of a matrix measures how many of its entries need to be changed in order to reduce its rank to some value. Good lower bounds on the rigidity of an explicit matrix would imply good lower bounds for arithmetic circuits as well as for communication complexity. Here we reprove the best known bounds on the rigidity of Hadamard matrices, due to Kashin and Razborov, using tools from quantum computing. Our proofs are somewhat simpler than earlier ones (at least for those familiar with quantum) and give slightly better constants. More importantly, they give a new approach to attack this longstanding open problem.

quant-ph↗

arXiv2006

Robust Polynomials and Quantum Algorithms

We define and study the complexity of robust polynomials for Boolean functions and the related fault-tolerant quantum decision trees, where input bits are perturbed by noise. We compare several different possible definitions. Our main results are * For every n-bit Boolean function f there is an n-variate polynomial p of degree O(n) that robustly approximates it, in the sense that p(x) remains close to f(x) if we slightly vary each of the n inputs of the polynomial. * There is an O(n)-query quantum algorithm that robustly recovers n noisy input bits. Hence every n-bit function can be quantum computed with O(n) queries in the presence of noise. This contrasts with the classical model of Feige et al., where functions such as parity need Theta(n*log n) queries. We give several extensions and applications of these results.

quant-ph↗

arXiv2006

Strengths and Weaknesses of Quantum Fingerprinting

We study the power of quantum fingerprints in the simultaneous message passing (SMP) setting of communication complexity. Yao recently showed how to simulate, with exponential overhead, classical shared-randomness SMP protocols by means of quantum SMP protocols without shared randomness ($Q^\parallel$-protocols). Our first result is to extend Yao's simulation to the strongest possible model: every many-round quantum protocol with unlimited shared entanglement can be simulated, with exponential overhead, by $Q^\parallel$-protocols. We apply our technique to obtain an efficient $Q^\parallel$-protocol for a function which cannot be efficiently solved through more restricted simulations. Second, we tightly characterize the power of the quantum fingerprinting technique by making a connection to arrangements of homogeneous halfspaces with maximal margin. These arrangements have been well studied in computational learning theory, and we use some strong results obtained in this area to exhibit weaknesses of quantum fingerprinting. In particular, this implies that for almost all functions, quantum fingerprinting protocols are exponentially worse than classical deterministic SMP protocols.

quant-ph↗

arXiv2005

Bounded-Error Quantum State Identification and Exponential Separations in Communication Complexity

We consider the problem of bounded-error quantum state identification: given either state α_0 or state α_1, we are required to output `0', `1' or `?' ("don't know"), such that conditioned on outputting `0' or `1', our guess is correct with high probability. The goal is to maximize the probability of not outputting `?'. We prove a direct product theorem: if we're given two such problems, with optimal probabilities a and b, respectively, and the states in the first problem are pure, then the optimal probability for the joint bounded-error state identification problem is O(ab). Our proof is based on semidefinite programming duality and may be of wider interest. Using this result, we present two exponential separations in the simultaneous message passing model of communication complexity. Both are shown in the strongest possible sense. First, we describe a relation that can be computed with O(log n) classical bits of communication in the presence of shared randomness, but needs Omega(n^{1/3}) communication if the parties don't share randomness, even if communication is quantum. This shows the optimality of Yao's recent exponential simulation of shared-randomness protocols by quantum protocols without shared randomness. Second, we describe a relation that can be computed with O(log n) classical bits of communication in the presence of shared entanglement, but needs Omega((n/log n)^{1/3}) communication if the parties share randomness but no entanglement, even if communication is quantum. This is the first example in communication complexity of a situation where entanglement buys you much more than quantum communication does.

quant-ph↗

arXiv2005

Improved Lower Bounds for Locally Decodable Codes and Private Information Retrieval

We prove new lower bounds for locally decodable codes and private information retrieval. We show that a 2-query LDC encoding n-bit strings over an l-bit alphabet, where the decoder only uses b bits of each queried position of the codeword, needs code length m = exp(Omega(n/(2^b Sum_{i=0}^b {l choose i}))) Similarly, a 2-server PIR scheme with an n-bit database and t-bit queries, where the user only needs b bits from each of the two l-bit answers, unknown to the servers, satisfies t = Omega(n/(2^b Sum_{i=0}^b {l choose i})). This implies that several known PIR schemes are close to optimal. Our results generalize those of Goldreich et al. who proved roughly the same bounds for linear LDCs and PIRs. Like earlier work by Kerenidis and de Wolf, our classical lower bounds are proved using quantum computational techniques. In particular, we give a tight analysis of how well a 2-input function can be computed from a quantum superposition of both inputs.

quant-ph↗

arXiv2004

Quantum Communication Cannot Simulate a Public Coin

We study the simultaneous message passing model of communication complexity. Building on the quantum fingerprinting protocol of Buhrman et al., Yao recently showed that a large class of efficient classical public-coin protocols can be turned into efficient quantum protocols without public coin. This raises the question whether this can be done always, i.e. whether quantum communication can always replace a public coin in the SMP model. We answer this question in the negative, exhibiting a communication problem where classical communication with public coin is exponentially more efficient than quantum communication. Together with a separation in the other direction due to Bar-Yossef et al., this shows that the quantum SMP model is incomparable with the classical public-coin SMP model. In addition we give a characterization of the power of quantum fingerprinting by means of a connection to geometrical tools from machine learning, a quadratic improvement of Yao's simulation, and a nearly tight analysis of the Hamming distance problem from Yao's paper.

quant-ph↗

arXiv2004

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

A strong direct product theorem says that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then our overall success probability will be exponentially small in k. We establish such theorems for the classical as well as quantum query complexity of the OR function. This implies slightly weaker direct product results for all total functions. We prove a similar result for quantum communication protocols computing k instances of the Disjointness function. Our direct product theorems imply a time-space tradeoff T^2*S=Omega(N^3) for sorting N items on a quantum computer, which is optimal up to polylog factors. They also give several tight time-space and communication-space tradeoffs for the problems of Boolean matrix-vector multiplication and matrix multiplication.

quant-ph↗

arXiv2004

Nondeterministic Quantum Query and Quantum Communication Complexities

We study nondeterministic quantum algorithms for Boolean functions f. Such algorithms have positive acceptance probability on input x iff f(x)=1. In the setting of query complexity, we show that the nondeterministic quantum complexity of a Boolean function is equal to its ``nondeterministic polynomial'' degree. We also prove a quantum-vs-classical gap of 1 vs n for nondeterministic query complexity for a total function. In the setting of communication complexity, we show that the nondeterministic quantum complexity of a two-party function is equal to the logarithm of the rank of a nondeterministic version of the communication matrix. This implies that the quantum communication complexities of the equality and disjointness functions are n+1 if we do not allow any error probability. We also exhibit a total function in which the nondeterministic quantum communication complexity is exponentially smaller than its classical counterpart.

cs.CC↗

arXiv2003

Quantum Symmetrically-Private Information Retrieval

Private information retrieval systems (PIRs) allow a user to extract an item from a database that is replicated over k>=1 servers, while satisfying various privacy constraints. We exhibit quantum k-server symmetrically-private information retrieval systems (QSPIRs) that use sublinear communication, do not use shared randomness among the servers, and preserve privacy against honest users and dishonest servers. Classically, SPIRs without shared randomness do not exist at all.

quant-ph↗