International Journal of Quantum Information

Quantum-circuit optimization is essential for any practical realization of quantum computation, in order to beat decoherence. We present a scheme for implementing the final stage in the compilation of quantum circuits, i.e. for finding the actual physical realizations of the individual modules in the quantum-gate library. We find that numerical optimization can be efficiently utilized in order to generate the appropriate control-parameter sequences which produce the desired three-qubit modules within the Josephson charge-qubit model. Our work suggests ways in which one can in fact considerably reduce the number of gates required to implement a given quantum circuit, hence diminishing idle time and significantly accelerating the execution of quantum algorithms.

Let A={ρ₁,…,ρ_n} be a given set of quantum states. We consider the problem of finding necessary and sufficient conditions on another set B={σ₁,…,σ_n} that guarantee the existence of a physical transformation taking ρ_i to σ_i for all i. Uhlmann has given an elegant such condition when both sets comprise pure states. We give a simple proof of this condition and develop some consequences. Then we consider multi-probabilistic transformations between sets of pure states which leads to conditions for the problem of transformability between A and B when one set is pure and the other is arbitrary.

We review the status of Bell's inequalities in quantum information, stressing mainly the links with quantum key distribution and distillation of entanglement. We also prove that for all the eavesdropping attacks using one qubit, and for a family of attacks of two qubits, acting on half of a maximally entangled state of two qubits, the violation of a Bell inequality implies the possibility of an efficient secret-key extraction.

We first consider various methods for the indirect implementation of unitary gates. We apply these methods to rederive the universality of 4-qubit measurements based on a scheme much simpler than Nielsen's original construction [quant-ph/0108020]. Then, we prove the universality of simple discrete sets of 2-qubit measurements, again using a scheme simplifying the initial construction [quant-ph/0111122]. Finally, we show how to use a single 4-qubit measurement to achieve universal quantum computation, and outline a proof for the universality of almost all maximally entangling 4-qubit measurements.

We present a way to apply quantum logic to the study of quantum programs. This is made possible by using an extension of the usual propositional language in order to make transformations performed on the system appear explicitly. This way, the evolution of the system becomes part of the logical study. We show how both unitary operations and two-valued measurements can be included in this formalism and can thus be handled logically.

We present families of quantum error-correcting codes which are optimal in the sense that the minimum distance is maximal. These maximum distance separable (MDS) codes are defined over q-dimensional quantum systems, where q is an arbitrary prime power. It is shown that codes with parameters 〚n, n - 2d + 2, d〛_q exist for all 3≤n≤q and 1≤d≤n/2+1. We also present quantum MDS codes with parameters 〚q², q²-2d+2, d〛_q for 1≤d≤q which additionally give rise to shortened codes 〚q²-s, q²-2d+2-s, d〛_q for some s.

Stochastic local operations and classical communication (SLOCC), also called local filtering operations, are a convenient, useful set of quantum operations in grasping essential properties of entanglement. We give a quick overview of the characteristics of multipartite entanglement in terms of SLOCC, illustrating the 2-qubit and the rest (2×2×n) quantum system. This not only includes celebrated results of 3-qubit pure states, but also has implications to 2-qubit mixed states.

Effects of quantum and classical correlations on game theory are studied to clarify the new aspects brought into game theory by the quantum mechanical toolbox. In this study, we compare quantum correlation represented by a maximally entangled state and classical correlation that is generated through phase damping processes on the maximally entangled state. Thus, this also sheds light on the behavior of games under the influence of noisy sources. It is observed that the quantum correlation can always resolve the dilemmas in non-zero sum games and attain the maximum sum of both players' payoffs, while the classical correlation cannot necessarily resolve the dilemmas.

We show how the Quantum Fast Fourier Transform (QFFT) can be made exact for arbitrary orders (first showing it for large primes). Most quantum algorithms only need a good approximation of the quantum Fourier transform of order 2ⁿ to succeed with high probability, and this QFFT can in fact be done exactly. Kitaev¹ showed how to approximate the Fourier transform for any order. Here we show how his construction can be made exact by using the technique known as "amplitude amplification". Although unlikely to be of any practical use, this construction allows one to make Shor's discrete logarithm quantum algorithm exact. Thus we have the first example of an exact non black box fast quantum algorithm, thereby giving more evidence that "quantum" need not be probabilistic.

We also show that in a certain sense the family of circuits for the exact QFFT is uniform. Namely, the parameters of the gates can be approximated efficiently.

When the electromagnetic field is detected by stimulated emission, rather than by absorption, antinormally ordered photodetection can be realized. One of the distinct features of this photodetection scheme is its sensitivity to zero-point fluctuation due to the existence of the spontaneous emission. We have recently succeeded in experimentally demonstrating the antinormally ordered photodetection by exploiting nondegenerate stimulated parametric down-conversion process. To properly account for the experiment, the detection process needs to be treated with time-dependent and continuous-mode operators because of the broadband nature of the parametric down-conversion process and the wide spectrum of the pump that we used. Here, we theoretically analyze the antinormally ordered intensity correlation of the continuous-mode fields by pursuing the detection process in the Heisenberg picture. It is shown that the excess positive correlation due to zero-point fluctuation reduces because of the frequency-distinguishability of the two emitted photon pairs.

Quantum Fourier transform (QFT) is a key function to realize quantum computers. A QFT followed by measurement was demonstrated on a simple circuit based on fiber-optics. The QFT was shown to be robust against imperfections in the rotation gate. Error probability was estimated to be 0.01 per qubit, which corresponded to error-free operation on 100 qubits. The error probability can be further reduced by taking the majority of the accumulated results. The reduction of error probability resulted in a successful QFT demonstration on 1024 qubits.

Multipartite entangled states are key elements for quantum information processing. Here we experimentally investigate the particular properties of a polarization-entangled four-photon state, which can be generated directly by second order parametric down-conversion. The perfect correlations and the invariance under local transformations enable one to encode one qubit of quantum information in a decoherence-free subspace and thus to communicate it safely over noisy quantum channels. Furthermore, we present an experimental method to detect genuine fourpartite entanglement using entanglement witness operators. The implementation of such operators requires only of few local polarization measurements but uniquely proves the genuine multipartite entanglement.