In order to demonstrate non-trivial quantum computations experimentally, such as the synthesis of arbitrary entangled states, it will be useful to understand how to decompose a desired quantum computation into the shortest possible sequence of one-qubit and two-qubit gates. We contribute to this effort by providing a method to construct an optimal quantum circuit for a general two-qubit gate that requires at most 3 CNOT gates and 15 elementary one-qubit gates. Moreover, if the desired two-qubit gate corresponds to a purely real unitary transformation, we provide a construction that requires at most 2 CNOTs and 12 one-qubit gates. We then prove that these constructions are optimal with respect to the family of CNOT, y-rotation, z-rotation, and phase gates.

We present here algorithmic cooling (via polarization heat bath)-a powerful method for obtaining a large number of highly polarized spins in liquid nuclear-spin systems at finite temperature. Given that spin-half states represent (quantum) bits, algorithmic cooling cleans dirty bits beyond the Shannon's bound on data compression, by using a set of rapidly thermal-relaxing bits. Such auxiliary bits could be implemented by using spins that rapidly get into thermal equilibrium with the environment, e.g., electron spins. Interestingly, the interaction with the environment, usually a most undesired interaction, is used here to our benefit, allowing a cooling mechanism. Cooling spins to a very low temperature without cooling the environment could lead to a breakthrough in NMR experiments, and our ''spin-refrigerating'' method suggests that this is possible. The scaling of NMR ensemble computers is currently one of the main obstacles to building larger-scale quantum computing devices, and our spin-refrigerating method suggests that this problem can be resolved.

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Any deterministic bipartite entanglement transformation involving finite copies of pure states and carried out using local operations and classical communication (LOCC) results in a net loss of entanglement. We show that for almost all such transformations, partial recovery of lost entanglement is achievable by using 2 × 2 auxiliary entangled states, no matter how large the dimensions of the parent states are. For the rest of the special cases of deterministic LOCC transformations, we show that the dimension of the auxiliary entangled state depends on the presence of equalities in the majorization relations of the parent states. We show that genuine recovery is still possible using auxiliary states in dimensions less than that of the parent states for all patterns of majorization relations except only one special case. Entanglement, shared among spatially separated parties, is a critical resource that enables efficient implementations of several quantum information processing [2] and distributed computation [3] tasks. To better exploit the power of entanglement, considerable effort has been put into understanding its transformation properties [4]- [7] and characterizing transformations allowed under local operations and classical communication (LOCC) . A central question is: what happens to the overall entanglement during transformations? In the asymptotic limit involving infinite number of copies of pure states, entanglement can be *

A novel universal and fault-tolerant basis (set of gates) for quantum computation is described. Such a set is necessary to perform quantum computation in a realistic noisy environment. The new basis consists of two single-qubit gates (Hadamard and σ z 1 4 ), and one double-qubit gate (Controlled-NOT). Since the set consisting of Controlled-NOT and Hadamard gates is not universal, the new basis achieves universality by including only one additional elementary (in the sense that it does not include angles that are irrational multiples of π) single-qubit gate, and hence, is potentially the simplest universal basis that one can construct. We also provide an alternative proof of universality for the only other known class of universal and fault-tolerant basis proposed in [24,16].