On the logic of pairs of quantum systems

C. Piron

Farid Mansurovich Ablayev, S. N. Andrianov, Sergey A. Moiseev +1 more

We propose an effective realization of the universal set of elementary quantum gates in solid state quantum computer based on macroscopic (or mesoscopic) resonance systems — multiatomic coherent ensembles, squids or quantum dots in quantum electrodynamic cavity. We exploit an encoding of logical qubits by the pairs of the macroscopic two- or three-level atoms that is working in a Hilbert subspace of all states inherent to these atomic systems. In this subspace, logical single qubit gates are realized by the controlled reversible transfer of single atomic excitation in the pair via the exchange of virtual photons and by the frequency shift of one of the atomic ensembles in a pair. In the case of two-level systems, the logical two-qubit gates are performed by the controlling of Lamb shift magnitude in one atomic ensemble, allowing/blocking the excitation transfer in a pair, respectively, that is controlled by the third atomic system of another pair. When using three-level systems, we describe the NOT-gate in the atomic pair controlled by the transfer of working atomic excitation to the additional third level caused by direct impact of the control pair excitation. Finally, we discuss advantages of the proposed physical system for accelerated computation of some useful quantum gates.

Phil Catabrese

The algebraic construct of 'forming ordered pairs of the members of a set' has been used and reused with amazing success. Among these developments are Integer Fractions to better measure distances with pairs of whole numbers; Dedekind Cuts - an early way to define real numbers with ordered pairs of subsets of fractions; Vectors; Quatemians; and others like complex numbers - ordered pairs of real numbers. In the realm of Boolean Logic, ordered pairs of Boolean propositions (alternately ordered pairs of subsets of some universe set) transcend the limitations of pure Boolean Iogic to represent partially true conditional statements such as those used by conditiona1 probability. Pure Boolean logic is unable to adequately represent uncertain conditional logic, but ordered pairs of BooIean propositions can and do represent this knowledge in the logic of Boolean Fractions, also called Conditional Events. A given B can not be represented by A or not B unless P(AJB) and P(B) are both 1. Otherwise, P(A1B) < P(A or not B). But A given B can be represented by an ordered pair of propositions (AJB). These Boolean Fractions apply to the natural language and logic of A given B, they apply to classical conditional probability calculations, to expert system if-then rules, and now to the logic of quantum measurements, a logic that Boolean algebra could not represent, but which Boolean Fractions can represent without Hilbert space.

Peter D. Finch

Apostol Vourdas

Sheh‐Yi Sheu, Hua-Yi Hsu, Dah‐Yen Yang

Quantum computers (QCs) adopt an n‐state quantum mechanical system to manipulate the superposition state. However, molecular transistors are not used to build up the quantum logic gate. Herein, it is demonstrated that DNA, RNA, and protein are promising media for QCs, and one uses residue pairs, including nucleotide base pairs and amino acid pairs, via proton‐coupled electron transfer to fabricate a quantum logic gate. In the residue pair, the proton transfer between donor and acceptor states fulfills a qubit. Both the DNA‐CG (3‐qubit) nucleobase pair and nucleotide base pair obey the Toffoli gate. AT (2‐qubit) nucleotide base pair behaves as a SWAP gate and a CNOT gate. Furthermore, the AU and RNA‐CG nucleotide base pairs follow the CNOT gate and Toffoli gate, respectively. In addition, a pair of amino acids achieves 1‐qubit and satisfies the Pauli‐X, ‐Y, and ‐Z gates. The generators of universal quantum logic gates are obtained.