The mathematics of a quantum Hamiltonian computing half adder Boolean logic gate

Abstract

The mathematics behind the quantum Hamiltonian computing (QHC) approach of designing Boolean logic gates with a quantum system are given. Using the quantum eigenvalue repulsion effect, the QHC AND, NAND, OR, NOR, XOR, and NXOR Hamiltonian Boolean matrices are constructed. This is applied to the construction of a QHC half adder Hamiltonian matrix requiring only six quantum states to fullfil a half Boolean logical truth table. The QHC design rules open a nano-architectronic way of constructing Boolean logic gates inside a single molecule or atom by atom at the surface of a passivated semi-conductor.

Quantum Hamiltonian Computing protocols for molecular electronics Boolean logic gates

“…Note that for the µ(α, β) expression for XOR, this identification gives e = k and e 2 = k 2 , so that no solution exists with P αβ (0) ∝ 1 − µ(α, β) issued from the calculating block (3). But as demonstrated in [22], other 3 × 3 calculating blocks can be found for the XOR and one of the corresponding polynomials is given in Table I last column. Indeed, one can find families of polynomials in α and β not related to µ(α, β) which take the value 0 if and only if the output is 1, and then determine structural parameters for matrices such that the determinant coincides with these polynomials.…”

Quantum half-adder Boolean logic gate with a nano-graphene molecule and graphene nano-electrodes

“…In electronics, the output of a given gate along the circuit is generally required to distribute its output signal at many locations of the circuit at the same time. Our present molecule QHC design can be considered as a practical implementation of the formal QHC 1/2-adder [16] where it was shown how to position the logical inputs along the QHC quantum graph to avoid this cascading problem. Two fluorine atoms were also added to the NGM to optimize, as presented in Fig.…”

“…To solve those gain and exponential decay problems still with the objective to embark a calculating unit on a single molecule, a new quantum computing approach was proposed [13].When prepared in a non-stationary state, the quantum Hamiltonian computing (QHC) logic gate beneficiates from the spontaneous Heisenberg-Rabi quantum oscillations of the quantum system to run a quantum computation with classical inputs [14]. By measuring the effecting Heisenberg-Rabi oscillations frequency using metallic nano-electrodes [15], different Boolean logic operations can be performed in parallel and within the same quantum system [16]. The functioning of a Boolean QHC gate is based on the quantum level repulsion effect together with the control of constructive and destructive quantum interferences affecting tunneling transport.…”

“…t(E) is the top left block of the multi-channel elastic scattering matrix describing the scattering process through the central NGM. The complete multichannel scattering matrix was calculated exactly using the elastic scattering quantum chemistry (ESQC) method[16,34,35]. A full valence semi-empirical Hamiltonian was used in ESQC to describe the central molecule along with its 4 short GNR molecular wires and the graphene electrodes.…”

Quantum arithmetics via computation with minimized external control: The half-adder

“…Because of its importance, different schemes have been proposed to implement half-adders in different experimental systems, such as linear optics [21], nanographene molecules [22], 1D cellular automaton [23], and superconducting [24] or atomic [25] systems. Successful implementations of a quantum half-adder have also been demonstrated experimentally, for example, using nuclear magnetic resonance 7/2-spin systems [26], although with a 10% error rate, mainly due to radio frequency inhomogeneity and pulse imperfections-since multiple pulses are required to perform the half-adder, these imperfections accumulate.…”