Controlled full adder or subtractor by vibrational quantum computing

A controlled full addition or subtraction can be realized by a unitary transformation on a register of four qubits. The fourth qubit is then used as a control qubit to enforce the addition or the subtraction of two binary digits and a carry or a borrow. The transformation can be decomposed into six elementary gates. The network differs from the adder network of four elementary gates by including two new controlled-NOT gates. The scheme is general and its implementation using vibrational computing has the advantage that the single global transformation that connects the inputs to the outputs can be driven in one step by a single laser shot. This decreases the time of operation and allows for a better use of the optical resources and for an improvement of the fidelity. The laser pulses are optimized by optimal control theory.

Figure 1

(Color online) Network for a factorized controlled adder-subtractor. The state of the modified qubit after each gate is written above the line for the adder and below the line for the subtractor. The control qubit $Q_4$ initially contains 0 for the adder and 1 for the subtractor. The upper arrow indicates the original adder sequence (in blue), the two added $\text{CNOT}$ gates are ${\text{CNOT}}_{\text{a}}$ and ${\text{CNOT}}_{\text{b}}$, the first two $\text{CNOT}$ on the left (in red).

Figure 2

(Color online) Direct controlled adder-subtractor implemented in the ${\text{SCCl}}_2$ molecule by OCT with phase constraint. The control qubit is the fourth one. (a) An example of addition $\left(C_{\text{in}}=0,A=1,B=1,0\right)\to \left(C_{\text{in}}=0,A=1,\text{the}\text{sum}\text{is}0,C_{\text{out}}=1\right)$ (upper bold line in Table ). (b) Corresponding subtraction $\left(B_{\text{in}}=0,A=1,B=1,1\right)\to \left({\overline{B}}_{\text{in}}=1,A=1,\text{the}\text{difference}\text{is}0,B_{\text{out}}=1\right)$ (lower bold line in Table ).

Figure 3

(Color online) (a) Primary ordinate axis: spectrum (in arbitrary units) of the unfiltered optimal field driving the direct controlled adder-subtractor implemented in the ${\text{SCCl}}_2$ by OCT with phase constraint. Inset with secondary axis: the filtering function in yellow dashed line. (b) Spectrum of the optimal field after filtering.

Figure 4

(Color online) Optimal field of the direct controlled adder-subtractor implemented in the ${\text{SCCl}}_2$ molecule by OCT with phase constraint.

Figure 5

(Color online) Controlled factorized adder-subtractor implemented in the ${\text{SCCl}}_2$ molecule by OCT without phase constraint. (a) Addition $\left(C_{\text{in}}=0,A=1,B=1,0\right)\to \left(C_{\text{in}}=0,A=1,S=0,C_{\text{out}}=1\right)$ (line 7 of Table ); (b) subtraction $\left(B_{\text{in}}=0,A=1,B=1,1\right)\to \left({\overline{B}}_{\text{in}}=0,A=1,D=0,B_{\text{out}}=0\right)$ (line 15 of Table ).