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Lov Grover

Publications and source records attributed to Lov Grover.

6 recordsLinked to original sources
arXiv2006

A new algorithm for fixed point quantum search

The standard quantum search lacks a feature, enjoyed by many classical algorithms, of having a fixed point, i.e. monotonic convergence towards the solution. Recently a fixed point quantum search algorithm has been discovered, referred to as the Phase-$π/3$ search algorithm, which gets around this limitation. While searching a database for a target state, this algorithm reduces the error probability from $ε$ to $ε^{2q+1}$ using $q$ oracle queries, which has since been proved to be asymptotically optimal. A different algorithm is presented here, which has the same worst-case behavior as the Phase-$π/3$ search algorithm but much better average-case behavior. Furthermore the new algorithm gives $ε^{2q+1}$ convergence for all integral $q$, whereas the Phase-$π/3$ search algorithm requires $q$ to be $(3^{n}-1)/2$ with $n$ a positive integer. In the new algorithm, the operations are controlled by two ancilla qubits, and fixed point behavior is achieved by irreversible measurement operations applied to these ancillas. It is an example of how measurement can allow us to bypass some restrictions imposed by unitarity on quantum computing.

quant-ph↗
arXiv2003

How significant are the known collision and element distinctness quantum algorithms?

Quantum search is a technique for searching N possibilities in only O(sqrt(N)) steps. It has been applied in the design of quantum algorithms for several structured problems. Many of these algorithms require significant amount of quantum hardware. In this paper we observe that if an algorithm requires O(P) hardware, it should be considered significant if and only if it produces a speedup of at least O(sqrt(P)) over a simple quantum search algorithm. This is because a speedup of $O(sqrt(P)) $ can be trivially obtained by dividing the search space into $O(P)$ separate parts and handing the problem to independent processors that do a quantum search. We argue that the known algorithms for collision and element distinctness fail to be non-trivial in this sense.

quant-ph↗
arXiv2003

On the communication complexity of establishing a shared reference frame

We discuss the communication complexity of establishing a shared reference frame, in particular examining the case of aligning spatial axes via the exchange of spin-1/2 particles. Unlike previous work we allow for multiple rounds of communication, and we give several simple examples demonstrating that nontrivial tradeoffs between the number of rounds and the type of communication required exist. We then give an explicit protocol for aligning spatial axes via the exchange of spin-1/2 particles which makes no use of either exchanged entangled states or of joint measurements. Rather it works by performing a simple type of distributed quantum computation. To facilitate comparison with previous work, we show that this protocol achieves a worst case fidelity for the much studied problem of "direction finding" that is asymptotically equivalent (up to polylog factors) to the optimal average case fidelity achievable via a single forward communication of entangled states.

quant-ph↗
arXiv2002

A 2 rebit gate universal for quantum computing

We show, within the circuit model, how any quantum computation can be efficiently performed using states with only real amplitudes (a result known within the Quantum Turing Machine model). This allows us to identify a 2-qubit (in fact 2-rebit) gate which is universal for quantum computing, although it cannot be used to perform arbitrary unitary transformations.

quant-ph↗
arXiv2002

Quantum searching a classical database (or how we learned to stop worrying and love the bomb)

We show how to perform a quantum search for a classical object, specifically for a classical object which performs no coherent evolution on the quantum computer being used for the search. We do so by using interaction free measurement as a subroutine in a quantum search algorithm. In addition to providing a simple example of how non-unitary processes which approximate unitary ones can be useful in a quantum algorithm, our procedure requires only one photon regardless of the size of the database, thereby establishing an upper bound on the amount of energy required to search an arbitrarily large database. Alternatively, our result can be interpreted as showing how to perform an interaction free measurement with a single photon on an arbitrarily large number of possible bomb positions simultaneously. We also provide a simple example demonstrating that in terms of the number of database queries, the procedure outlined here can outperform the best classical one.

quant-ph↗