Quantum Mechanics helps in searching for a needle in a haystack

Quantum mechanics can speed up a range of search applications over unsorted data. For example imagine a phone directory containing N names arranged in completely random order. To find someone's phone number with a probability of 50%, any classical algorithm (whether deterministic or probabilistic) will need to access the database a minimum of O(N) times. Quantum mechanical systems can be in a superposition of states and simultaneously examine multiple names. By properly adjusting the phases of various operations, successful computations reinforce each other while others interfere randomly. As a result, the desired phone number can be obtained in only O(sqrt(N)) accesses to the database.Comment: Postscript, 4 pages. This is a modified version of the STOC paper (quant-ph/9605043) and is modified to make it more comprehensible to physicists. It appeared in Phys. Rev. Letters on July 14, 1997. (This paper was originally put out on quant-ph on June 13, 1997, the present version has some minor typographical changes

Heisenberg chains cannot mirror a state

Faithful exchange of quantum information can in future become a key part of many computational algorithms. Some Authors suggest to use chains of mutually coupled spins as channels for quantum communication. One can divide these proposals into the groups of assisted protocols, which require some additional action from the users, and natural ones, based on the concept of state mirroring. We show that mirror is fundamentally not the feature chains of spins-1/2 coupled by the Heisenberg interaction, but without local magnetic fields. This fact has certain consequences in terms of the natural state transfer

Quantum computers can search arbitrarily large databases by a single query

This paper shows that a quantum mechanical algorithm that can query information relating to multiple items of the database, can search a database in a single query (a query is defined as any question to the database to which the database has to return a (YES/NO) answer). A classical algorithm will be limited to the information theoretic bound of at least O(log N) queries (which it would achieve by using a binary search).Comment: Several enhancements to the original pape

Quantum computers can search rapidly by using almost any transformation

A quantum computer has a clear advantage over a classical computer for exhaustive search. The quantum mechanical algorithm for exhaustive search was originally derived by using subtle properties of a particular quantum mechanical operation called the Walsh-Hadamard (W-H) transform. This paper shows that this algorithm can be implemented by replacing the W-H transform by almost any quantum mechanical operation. This leads to several new applications where it improves the number of steps by a square-root. It also broadens the scope for implementation since it demonstrates quantum mechanical algorithms that can readily adapt to available technology.Comment: This paper is an adapted version of quant-ph/9711043. It has been modified to make it more readable for physicists. 9 pages, postscrip

Entangled Quantum States Generated by Shor's Factoring Algorithm

The intermediate quantum states of multiple qubits, generated during the operation of Shor's factoring algorithm are analyzed. Their entanglement is evaluated using the Groverian measure. It is found that the entanglement is generated during the pre-processing stage of the algorithm and remains nearly constant during the quantum Fourier transform stage. The entanglement is found to be correlated with the speedup achieved by the quantum algorithm compared to classical algorithms.Comment: 7 pages, 4 figures submitted to Phys. Rev.

High-pressure x-ray diffraction study of bulk and nanocrystalline PbMoO4

We studied the effects of high-pressure on the crystalline structure of bulk and nanocrystalline scheelite-type PbMoO4. We found that in both cases the compressibility of the materials is highly non-isotropic, being the c-axis the most compressible one. We also observed that the volume compressibility of nanocrystals becomes higher that the bulk one at 5 GPa. In addition, at 10.7(8) GPa we observed the onset of an structural phase transition in bulk PbMoO4. The high-pressure phase has a monoclinic structure similar to M-fergusonite. The transition is reversible and not volume change is detected between the low- and high-pressure phases. No additional structural changes or evidence of decomposition are found up to 21.1 GPa. In contrast nanocrystalline PbMoO4 remains in the scheelite structure at least up to 16.1 GPa. Finally, the equation of state for bulk and nanocrystalline PbMoO4 are also determined.Comment: 18 pages, 4 figure

Energy and Efficiency of Adiabatic Quantum Search Algorithms

We present the results of a detailed analysis of a general, unstructured adiabatic quantum search of a data base of $N$ items. In particular we examine the effects on the computation time of adding energy to the system. We find that by increasing the lowest eigenvalue of the time dependent Hamiltonian {\it temporarily} to a maximum of $\propto \sqrt{N}$, it is possible to do the calculation in constant time. This leads us to derive the general theorem which provides the adiabatic analogue of the $\sqrt{N}$ bound of conventional quantum searches. The result suggests that the action associated with the oracle term in the time dependent Hamiltonian is a direct measure of the resources required by the adiabatic quantum search.Comment: 6 pages, Revtex, 1 figure. Theorem modified, references and comments added, sections introduced, typos corrected. Version to appear in J. Phys.

Nested quantum search and NP-complete problems

A quantum algorithm is known that solves an unstructured search problem in a number of iterations of order $\sqrt{d}$, where $d$ is the dimension of the search space, whereas any classical algorithm necessarily scales as $O(d)$. It is shown here that an improved quantum search algorithm can be devised that exploits the structure of a tree search problem by nesting this standard search algorithm. The number of iterations required to find the solution of an average instance of a constraint satisfaction problem scales as $\sqrt{d^\alpha}$, with a constant $\alpha<1$ depending on the nesting depth and the problem considered. When applying a single nesting level to a problem with constraints of size 2 such as the graph coloring problem, this constant $\alpha$ is estimated to be around 0.62 for average instances of maximum difficulty. This corresponds to a square-root speedup over a classical nested search algorithm, of which our presented algorithm is the quantum counterpart.Comment: 18 pages RevTeX, 3 Postscript figure

Quantum Analogue Computing

We briefly review what a quantum computer is, what it promises to do for us, and why it is so hard to build one. Among the first applications anticipated to bear fruit is quantum simulation of quantum systems. While most quantum computation is an extension of classical digital computation, quantum simulation differs fundamentally in how the data is encoded in the quantum computer. To perform a quantum simulation, the Hilbert space of the system to be simulated is mapped directly onto the Hilbert space of the (logical) qubits in the quantum computer. This type of direct correspondence is how data is encoded in a classical analogue computer. There is no binary encoding, and increasing precision becomes exponentially costly: an extra bit of precision doubles the size of the computer. This has important consequences for both the precision and error correction requirements of quantum simulation, and significant open questions remain about its practicality. It also means that the quantum version of analogue computers, continuous variable quantum computers (CVQC) becomes an equally efficient architecture for quantum simulation. Lessons from past use of classical analogue computers can help us to build better quantum simulators in future.Comment: 10 pages, to appear in the Visions 2010 issue of Phil. Trans. Roy. Soc.

The Precise Formula in a Sine Function Form of the norm of the Amplitude and the Necessary and Sufficient Phase Condition for Any Quantum Algorithm with Arbitrary Phase Rotations

In this paper we derived the precise formula in a sine function form of the norm of the amplitude in the desired state, and by means of he precise formula we presented the necessary and sufficient phase condition for any quantum algorithm with arbitrary phase rotations. We also showed that the phase condition: identical rotation angles, is a sufficient but not a necessary phase condition.Comment: 16 pages. Modified some English sentences and some proofs. Removed a table. Corrected the formula for kol on page 10. No figure