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Ashwin Nayak

Publications and source records attributed to Ashwin Nayak.

46 records · Page 3Linked to original sources
arXiv2006

Interaction in Quantum Communication

In some scenarios there are ways of conveying information with many fewer, even exponentially fewer, qubits than possible classically. Moreover, some of these methods have a very simple structure--they involve only few message exchanges between the communicating parties. It is therefore natural to ask whether every classical protocol may be transformed to a ``simpler'' quantum protocol--one that has similar efficiency, but uses fewer message exchanges. We show that for any constant k, there is a problem such that its k+1 message classical communication complexity is exponentially smaller than its k message quantum communication complexity. This, in particular, proves a round hierarchy theorem for quantum communication complexity, and implies, via a simple reduction, an Omega(N^{1/k}) lower bound for k message quantum protocols for Set Disjointness for constant k. Enroute, we prove information-theoretic lemmas, and define a related measure of correlation, the informational distance, that we believe may be of significance in other contexts as well.

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arXiv2002

Weak coin flipping with small bias

This note presents a quantum protocol that demonstrates that_weak_ coin flipping with bias approximately 0.239, less than 1/4, is possible. A bias of 1/4 was the smallest known, and followed from the strong coin flipping protocol of [Ambainis 2001]. Protocols with yet smaller bias, approximately 0.207, have independently been discovered [Ambainis 2001, Spekkens and Rudolph 2002]. We also present an alternative strong coin flipping protocol with bias 1/4 with analysis simpler than that of [Ambainis 2001].

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arXiv2002

On communication over an entanglement-assisted quantum channel

Shared entanglement is a resource available to parties communicating over a quantum channel, much akin to public coins in classical communication protocols. Whereas shared randomness does not help in the transmission of information, or significantly reduce the classical complexity of computing functions (as compared to private-coin protocols), shared entanglement leads to startling phenomena such as ``quantum teleportation'' and ``superdense coding.'' The problem of characterising the power of prior entanglement has puzzled many researchers. In this paper, we revisit the problem of transmitting classical bits over an entanglement-assisted quantum channel. We derive a new, optimal bound on the number of quantum bits required for this task, for any given probability of error. All known lower bounds in the setting of bounded error entanglement-assisted communication are based on sophisticated information theoretic arguments. In contrast, our result is derived from first principles, using a simple linear algebraic technique.

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arXiv2002

On bit-commitment based quantum coin flipping

In this paper, we focus on a special framework for quantum coin flipping protocols,_bit-commitment based protocols_, within which almost all known protocols fit. We show a lower bound of 1/16 for the bias in any such protocol. We also analyse a sequence of multi-round protocol that tries to overcome the drawbacks of the previously proposed protocols, in order to lower the bias. We show an intricate cheating strategy for this sequence, which leads to a bias of 1/4. This indicates that a bias of 1/4 might be optimal in such protocols, and also demonstrates that a cleverer proof technique may be required to show this optimality.

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arXiv2000

Quantum Walk on the Line

Motivated by the immense success of random walk and Markov chain methods in the design of classical algorithms, we consider_quantum_ walks on graphs. We analyse in detail the behaviour of unbiased quantum walk on the line, with the example of a typical walk, the ``Hadamard walk''. We show that after t time steps, the probability distribution on the line induced by the Hadamard walk is almost uniformly distributed over the interval [-t/sqrt(2),t/sqrt(2)]. This implies that the same walk defined on the circle mixes in_linear_ time. This is in direct contrast with the quadratic mixing time for the corresponding classical walk. We conclude by indicating how our techniques may be applied to more general graphs.

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arXiv2000

Interaction in Quantum Communication Complexity

One of the most intriguing facts about communication using quantum states is that these states cannot be used to transmit more classical bits than the number of qubits used, yet there are ways of conveying information with exponentially fewer qubits than possible classically. Moreover, these methods have a very simple structure---they involve little interaction between the communicating parties. We look more closely at the ways in which information encoded in quantum states may be manipulated, and consider the question as to whether every classical protocol may be transformed to a ``simpler'' quantum protocol of similar efficiency. By a simpler protocol, we mean a protocol that uses fewer message exchanges. We show that for any constant k, there is a problem such that its k+1 message classical communication complexity is exponentially smaller than its k message quantum communication complexity, thus answering the above question in the negative. Our result builds on two primitives, local transitions in bi-partite states (based on previous work) and average encoding which may be of significance in other applications as well.

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arXiv1999

Optimal lower bounds for quantum automata and random access codes

Consider the finite regular language L_n = {w0 : w \in {0,1}^*, |w| \le n}. It was shown by Ambainis, Nayak, Ta-Shma and Vazirani that while this language is accepted by a deterministic finite automaton of size O(n), any one-way quantum finite automaton (QFA) for it has size 2^{Omega(n/log n)}. This was based on the fact that the evolution of a QFA is required to be reversible. When arbitrary intermediate measurements are allowed, this intuition breaks down. Nonetheless, we show a 2^{Omega(n)} lower bound for such QFA for L_n, thus also improving the previous bound. The improved bound is obtained by simple entropy arguments based on Holevo's theorem. This method also allows us to obtain an asymptotically optimal (1-H(p))n bound for the dense quantum codes (random access codes) introduced by Ambainis et al. We then turn to Holevo's theorem, and show that in typical situations, it may be replaced by a tighter and more transparent in-probability bound.

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arXiv1998

Dense Quantum Coding and a Lower Bound for 1-way Quantum Automata

We consider the possibility of encoding m classical bits into much fewer n quantum bits so that an arbitrary bit from the original m bits can be recovered with a good probability, and we show that non-trivial quantum encodings exist that have no classical counterparts. On the other hand, we show that quantum encodings cannot be much more succint as compared to classical encodings, and we provide a lower bound on such quantum encodings. Finally, using this lower bound, we prove an exponential lower bound on the size of 1-way quantum finite automata for a family of languages accepted by linear sized deterministic finite automata.

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arXiv1998

The quantum query complexity of approximating the median and related statistics

Let X = (x_0,...,x_{n-1})$ be a sequence of n numbers. For ε> 0, we say that x_i is an ε-approximate median if the number of elements strictly less than x_i, and the number of elements strictly greater than x_i are each less than (1+ε)n/2. We consider the quantum query complexity of computing an ε-approximate median, given the sequence X as an oracle. We prove a lower bound of Ω(\min{{1/ε},n}) queries for any quantum algorithm that computes an ε-approximate median with any constant probability greater than 1/2. We also show how an ε-approximate median may be computed with O({1/ε}\log({1\/ε}) \log\log({1/ε})) oracle queries, which represents an improvement over an earlier algorithm due to Grover. Thus, the lower bound we obtain is essentially optimal. The upper and the lower bound both hold in the comparison tree model as well. Our lower bound result is an application of the polynomial paradigm recently introduced to quantum complexity theory by Beals et al. The main ingredient in the proof is a polynomial degree lower bound for real multilinear polynomials that ``approximate'' symmetric partial boolean functions. The degree bound extends a result of Paturi and also immediately yields lower bounds for the problems of approximating the kth-smallest element, approximating the mean of a sequence of numbers, and that of approximately counting the number of ones of a boolean function. All bounds obtained come within polylogarithmic factors of the optimal (as we show by presenting algorithms where no such optimal or near optimal algorithms were known), thus demonstrating the power of the polynomial method.

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