Ashwin Nayak

One-dimensional quantum walks

We define and analyze quantum computational variants of random walks on one-dimensional lattices. In particular, we analyze a quantum analog of the symmetric random walk, which we call the <italic>Hadamard walk</italic>. Several striking differences between the quantum and classical cases are observed. For example, when unrestricted in either direction, the Hadamard walk has position that is nearly uniformly distributed in the range <italic>[-t/\sqrt 2, t/\sqrt 2]</italic> after <italic>t</italic> steps, which is in sharp contrast to the classical random walk, which has distance <italic>O(\sqrt t)</italic> from the origin with high probability. With an absorbing boundary immediately to the left of the starting position, the probability that the walk exits to the left is <italic>2/&pgr</italic>, and with an additional absorbing boundary at location <italic>n</italic>, the probability that the walk exits to the left actually increases, approaching <italic>1/\sqrt 2</italic> in the limit. In the classical case both values are 1.

Optimal lower bounds for quantum automata and random access codes

Consider the finite regular language L/sub n/={w0|w/spl isin/{0,1}*,|w|/spl les/n}. A. Ambainis et al. (1999) showed 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/sup /spl Omega/(n/logn)/. 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/sup /spl Omega/(n)/ lower bound for such QFA for L/sub n/, thus also improving the previous bound. The improved bound is obtained from simple entropy arguments based on A.S. Holevos (1973) 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 A. Ambainis et al. We then turn to Holevos theorem, and show that in typical situations, it may be replaced by a tighter and more transparent in-probability bound.

Dense quantum coding and quantum finite automata

We consider the possibility of encoding m classical bits into many fewer n quantum bits (qubits) so that an arbitrary bit from the original m bits can be recovered with good probability. We show that nontrivial quantum codes exist that have no classical counterparts. On the other hand, we show that quantum encoding cannot save more than a logarithmic additive factor over the best classical encoding. The proof is based on an entropy coalescence principle that is obtained by viewing Holevos theorem from a new perspective.In the existing implementations of quantum computing, qubits are a very expensive resource. Moreover, it is difficult to reinitialize existing bits during the computation. In particular, reinitialization is impossible in NMR quantum computing, which is perhaps the most advanced implementation of quantum computing at the moment. This motivates the study of quantum computation with restricted memory and no reinitialization, that is, of quantum finite automata. It was known that there are languages that are recognized by quantum finite automata with sizes exponentially smaller than those of corresponding classical automata. Here, we apply our technique to show the surprising result that there are languages for which quantum finite automata take exponentially more states than those of corresponding classical automata.

Interaction in quantum communication and the complexity of set disjointness

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 in some scenarios there are ways of conveying information with exponentially fewer qubits than possible classically [3, 26]. Moreover, these methods have a very simple structure---they involve only few message exchanges between the communicating parties. We consider the question as to 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 <italic>k</italic>, there is a problem such that its <italic>k+1</italic> message classical communication complexity is exponentially smaller than its <italic>k</italic> message quantum communication complexity, thus answering the above question in the negative. This in particular proves a round hierarchy theorem for quantum communication complexity, and implies via a simple reduction, an <italic>\Omega(N^{1/k})</italic> lower bound for <italic>k</italic> message protocols for Set Disjointness for constant~<italic>k</italic>. Our result builds on two primitives, <italic>local transitions in bi-partite states</italic> (based on previous work) and <italic>average encoding</italic> which may be of significance in other contexts as well.

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 nontrivial 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.

Experimental implementation of heat-bath algorithmic cooling using solid-state nuclear magnetic resonance

The counter-intuitive properties of quantum mechanics have the potential to revolutionize information processing by enabling the development of efficient algorithms with no known classical counterparts. Harnessing this power requires the development of a set of building blocks, one of which is a method to initialize the set of quantum bits (qubits) to a known state. Additionally, fresh ancillary qubits must be available during the course of computation to achieve fault tolerance. In any physical system used to implement quantum computation, one must therefore be able to selectively and dynamically remove entropy from the part of the system that is to be mapped to qubits. One such method is an ‘open-system’ cooling protocol in which a subset of qubits can be brought into contact with an external system of large heat capacity. Theoretical efforts have led to an implementation-independent cooling procedure, namely heat-bath algorithmic cooling. These efforts have culminated with the proposal of an optimal algorithm, the partner-pairing algorithm, which was used to compute the physical limits of heat-bath algorithmic cooling. Here we report the experimental realization of multi-step cooling of a quantum system via heat-bath algorithmic cooling. The experiment was carried out using nuclear magnetic resonance of a solid-state ensemble three-qubit system. We demonstrate the repeated repolarization of a particular qubit to an effective spin-bath temperature, and alternating logical operations within the three-qubit subspace to ultimately cool a second qubit below this temperature. Demonstration of the control necessary for these operations represents an important step forward in the manipulation of solid-state nuclear magnetic resonance qubits.

Quantum Complexity of Testing Group Commutativity

AbstractWe consider the problem of testing the commutativity of a black-box group specified by its k generators. The complexity (in terms of k) of this problem was first considered by Pak, who gave a randomized algorithm involving O(k) group operations. We construct a quite optimal quantum algorithm for this problem whose complexity is in

On the hitting times of quantum versus random walks

\tilde{O}(k^{2/3})

Bit-commitment-based quantum coin flipping

. The algorithm uses and highlights the power of the quantization method of Szegedy. For the lower bound of

Quantum complexity of testing group commutativity

\Omega(k^{2/3})