2.4 The density operator 98 2.4.1 Ensembles of quantum states 99 2.4.2 General properties of the density operator 101 2.4.3 The reduced density operator 105 2.5 The Schmidt decomposition and purifications 109 2.6 EPR and the Bell inequality 111

An unsorted database contains N records, of which just one satisfies a particular property. The problem is to identify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) steps since on the average it will have to examine a large fraction of the N records. Quantum mechanical systems can do several operations simultaneously due to their wave like properties. This paper gives an O ( JN) step quantum mechanical algorithm for identifying that record. It is within a constant factor of the fastest possible quantum mechanical algorithm.

This paper extends the quantum search class of algorithms to the multiple solution case. It is shown that, like the basic search algorithm, these too can be represented as a rotation in an appropriately defined two dimensional vector space. This yields new applicationsan algorithm is presented that can create an arbitrarily specified quantum superposition on a space of size in steps. By making a measurement on this superposition, it is possible to obtain a sample according to an arbitrarily specified classical probability distribution in steps. A classical algorithm would need steps.

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 adapt to available technology. [S0031-9007(98)06052-9] PACS numbers: 03.67.Lx, 89.70. + c Quantum mechanical systems can be in a superposition of computational states and hence simultaneously carry out multiple computations in the same computer. In the last few years there has been extensive research on how to use this quantum parallelism to carry out meaningful computations. In any quantum mechanical computation the system is initialized to a state that is easy to prepare and caused to evolve unitarily, the answer to the computational problem is deduced by a final measurement that projects the system onto a unique state. The amplitude (and hence probability) of reaching a specified final state depends on the interference of all paths that take it from the initial to the final state-the system is thus very sensitive to any magnitude of phase disturbances that affect any of the paths leading to the desired final state. As a result, quantum mechanical algorithms are very delicate, and it is generally believed that an actual implementation would need elaborate procedures for correcting errors [1]. This paper shows that the quantum search algorithm is surprisingly robust to certain kinds of perturbations. It was originally derived by using the Walsh-Hadamard (W-H) transform and appeared to be a consequence of the special properties of this transform; this paper shows that similar results are obtained by substituting almost any unitary transformation in place of the W-H transform. Since all quantum mechanical operations are unitary, this means that almost any quantum mechanical system can be used-all that is needed is a valid quantum mechanical operation and a way of selectively inverting the phase of states. Meaningful computation can hence be carried out on the basis of universal properties of quantum mechanical operations; this is somewhat similar in spirit to [2], where circuit behavior of a certain class of neural networks was independent of the precise nature of the nonlinearity in each neuron.1. Quantum operations.-In a quantum computer, the logic circuitry and time steps are essentially classical, only the memory bits that hold the variables are in quantum superpositions -these are called qubits. There is a set of distinguished computational states in which all the bits are definite 0s or 1s. In a quantum mechanical algorithm, the quantum computer consisting of a number of qubits is prepared in some simple...

SummaryThis paper shows that a quantum mechanical algorithm that can query information relating to multiple items of the database, can search a database for a unique item satisfying a given condition, 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 queries, which it would achieve by using a binary search. Background