Comment on "Probabilistic Quantum Memories"

This is a comment on two wrong Phys. Rev. Letters papers by C.A. Trugenberger. Trugenberger claimed that quantum registers could be used as exponentially large "associative" memories. We show that his scheme is no better than one where the quantum register is replaced with a classical one of equal size. We also point out that the Holevo bound and more recent bounds on "quantum random access codes" pretty much rule out powerful memories (for classical information) based on quantum states.Comment: REVTeX4, 1 page, published versio

Quantum complexities of ordered searching, sorting, and element distinctness

We consider the quantum complexities of the following three problems: searching an ordered list, sorting an un-ordered list, and deciding whether the numbers in a list are all distinct. Letting N be the number of elements in the input list, we prove a lower bound of \frac{1}{\pi}(\ln(N)-1) accesses to the list elements for ordered searching, a lower bound of \Omega(N\log{N}) binary comparisons for sorting, and a lower bound of \Omega(\sqrt{N}\log{N}) binary comparisons for element distinctness. The previously best known lower bounds are {1/12}\log_2(N) - O(1) due to Ambainis, \Omega(N), and \Omega(\sqrt{N}), respectively. Our proofs are based on a weighted all-pairs inner product argument. In addition to our lower bound results, we give a quantum algorithm for ordered searching using roughly 0.631 \log_2(N) oracle accesses. Our algorithm uses a quantum routine for traversing through a binary search tree faster than classically, and it is of a nature very different from a faster algorithm due to Farhi, Goldstone, Gutmann, and Sipser.Comment: This new version contains new results. To appear at ICALP '01. Some of the results have previously been presented at QIP '01. This paper subsumes the papers quant-ph/0009091 and quant-ph/000903

Quantum Simulation of Tunneling in Small Systems

A number of quantum algorithms have been performed on small quantum computers; these include Shor's prime factorization algorithm, error correction, Grover's search algorithm and a number of analog and digital quantum simulations. Because of the number of gates and qubits necessary, however, digital quantum particle simulations remain untested. A contributing factor to the system size required is the number of ancillary qubits needed to implement matrix exponentials of the potential operator. Here, we show that a set of tunneling problems may be investigated with no ancillary qubits and a cost of one single-qubit operator per time step for the potential evolution. We show that physically interesting simulations of tunneling using 2 qubits (i.e. on 4 lattice point grids) may be performed with 40 single and two-qubit gates. Approximately 70 to 140 gates are needed to see interesting tunneling dynamics in three-qubit (8 lattice point) simulations.Comment: 4 pages, 2 figure

Experimental simulation of quantum tunneling in small systems

It is well known that quantum computers are superior to classical computers in efficiently simulating quantum systems. Here we report the first experimental simulation of quantum tunneling through potential barriers, a widespread phenomenon of a unique quantum nature, via NMR techniques. Our experiment is based on a digital particle simulation algorithm and requires very few spin-1/2 nuclei without the need of ancillary qubits. The occurrence of quantum tunneling through a barrier, together with the oscillation of the state in potential wells, are clearly observed through the experimental results. This experiment has clearly demonstrated the possibility to observe and study profound physical phenomena within even the reach of small quantum computers.Comment: 17 pages and 8 figure

Grover’s algorithm for unstructured search

Grover’s algorithm is used to search for an element on an unsorted list with quadratic speed-up when compared to the best possible classical algorithm. This quantum algorithm has an enormous historical importance, and is also a fundamental building block in quantum computing3555CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO - CNPQCOORDENAÇÃO DE APERFEIÇOAMENTO DE PESSOAL DE NÍVEL SUPERIOR - CAPESFUNDAÇÃO CARLOS CHAGAS FILHO DE AMPARO À PESQUISA DO ESTADO DO RIO DE JANEIRO - FAPERJFUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO - FAPESPNão temNão temNão temNão temWe are grateful to our colleagues and students from the Federal University of Rio de Janeiro (UFRJ, Brazil), the National Laboratory for Scientific Computing (LNCC, Brazil), and the University of Campinas (UNICAMP, Brazil) for several important discussions and interesting ideas. We acknowledge CAPES, CNPq, FAPERJ, and FAPESP—Brazilian funding agencies—for the financial support to our research projects. We also thank the Brazilian Society of Computational and Applied Mathematics (SBMAC) for the opportunity to give a course on this subject that resulted in the first version of this monograph in Portuguese (http://www.sbmac.org.br/arquivos/notas/livro_08.pdf), which in turn evolved from our earliest tutorials (in arXiv quant-ph/0301079 and quant-ph/0303175