QIC Abstracts

 Vol.6 No.1 January 1, 2006
Research Articles:
Quantum computer with dipole-dipole interacting two-level systems (pp001-015)
         D. Petrosyan and G. Kurizki
A scalable, high-performance quantum processor can be implemented using near-resonant dipole-dipole interacting dopants in a transparent solid state host. In this scheme, the qubits are represented by ground and subradiant states of effective dimers formed by pairs of closely spaced two-level systems, while the two-qubit entanglement either relies on the coherent excitation exchange between the dimers or is mediated by external laser fields.

Quantum measurements and entropic bounds (pp016-045)
         A. Barchielli and G. Lupieri
While a positive operator valued measure gives the probabilities in a quantum measurement, an instrument gives both the probabilities and the a posteriori states. By interpreting the instrument as a quantum channel and by using the monotonicity theorem for relative entropies many bounds on the classical information extracted in a quantum measurement are obtained in a unified manner. In particular, it is shown that such bounds can all be stated as inequalities between mutual entropies. This approach based on channels gives rise to a unified picture of known and new bounds on the classical information (the bounds by Holevo, by Shumacher, Westmoreland and Wootters, by Hall, by Scutaru, a new upper bound and a new lower one). Some examples clarify the mutual relationships among the various bounds.

Quantum lower bounds for fanout (pp046-057)
         M. Fang, S. Fenner, F. Green, S. Homer, and Y. Zhang 
We consider the resource bounded quantum circuit model with circuits restricted by the number of qubits they act upon and by their depth. Focusing on natural universal sets of gates which are familiar from classical circuit theory, several new lower bounds for constant depth quantum circuits are proved. The main result is that parity (and hence fanout) requires log depth quantum circuits, when the circuits are composed of single qubit and arbitrary size Toffoli gates, and when they use only constantly many ancill\ae. Under this constraint, this bound is close to optimal. In the case of a non-constant number $a$ of ancill\ae\ and $n$ input qubits, we give a tradeoff between $a$ and the required depth, that results in a non-constant lower bound for fanout when $a = n^{1-o(1)}$. We also show that, regardless of the number of ancill\ae\, arbitrary arity Toffoli gates cannot be simulated exactly by a constant depth circuit family with gates of bounded arity.

A discrete local invariant for quantum gates (pp058-066)
         L. Koponen, V. Bergholm, and M.M. Salomaa
In this paper we study the properties of two-qubit gates. We review the most common parameterizations for the local equivalence classes of two-qubit gates and the connections between them. We then introduce a new discrete local invariant, namely the number of local degrees of freedom that a gate can bind. The value of this invariant is calculated analytically for all the local equivalence classes of two-qubit gates. We find that almost all two-qubit gates can bind the full six local degrees of freedom and are in this sense more effective than the controlled-NOT gate which only can bind four local degrees of freedom.

A new algorithm for producing quantum circuits using KAK decompositions (pp067-080)
         M.Y. Nakajima, Y. Kawano, and H. Sekigawa 
We provide a new algorithm that translates a unitary matrix into a quantum circuit according to the G=KAK theorem in Lie group theory. With our algorithm, any matrix decomposition corresponding to type-AIII KAK decompositions can be derived according to the given Cartan involution. Our algorithm contains, as its special cases, Cosine-Sine decomposition (CSD) and Khaneja-Glaser decomposition (KGD) in the sense that it derives the same quantum circuits as the ones obtained by them if we select suitable Cartan involutions and square root matrices. The selections of Cartan involutions for computing CSD and KGD will be shown explicitly. As an example, we show explicitly that our method can automatically reproduce the well-known efficient quantum circuit for the $n$-qubit quantum Fourier transform.

Mini Tutorial:
The Solovay-Kitaev algorithm (pp081-095) 
         C.M. Dawson and M.A. Nielsen     
This pedagogical review presents the proof of the Solovay-Kitaev theorem in the form of an efficient classical algorithm for compiling an arbitrary single-qubit gate into a sequence of gates from a fixed and finite set. The algorithm can be used, for example, to compile Shor's algorithm, which uses rotations of $\pi / 2^k$, into an efficient fault-tolerant form using only Hadamard, controlled-{\sc not}, and $\pi / 8$ gates. The algorithm runs in $O(\log^{2.71}(1/\epsilon))$ time, and produces as output a sequence of $O(\log^{3.97}(1/\epsilon))$ quantum gates which is guaranteed to approximate the desired quantum gate to an accuracy within $\epsilon > 0$. We also explain how the algorithm can be generalized to apply to multi-qubit gates and to gates from SU(d). 

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