Construction of two-qubit gates using B Gate

Abstract In the circuit model of quantum computation, an entangling two-qubit gate and a set of single-qubit gates are used as universal gate set or basis gates for doing quantum computation. CNOT and <?CDATA $\sqrt{\mathrm{iSWAP}}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msqrt> <mml:mrow> <mml:mi>iSWAP</mml:mi> </mml:mrow> </mml:msqrt> </mml:math> gates, the perfect entanglers that can create maximally entangled two-qubit states in one application, are broadly used entangling two-qubit basis gates in quantum computers. In this paper, we analyze the potentiality of <?CDATA $\sqrt{{\rm{B}}}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msqrt> <mml:mrow> <mml:mi mathvariant="normal">B</mml:mi> </mml:mrow> </mml:msqrt> </mml:math> gate, an unexplored non-perfect entangler of the form, <?CDATA $\exp \left(\tfrac{i\pi ({\hat{\sigma }}_{x}\otimes {\hat{\sigma }}_{x})}{8}+\tfrac{i\pi ({\hat{\sigma }}_{y}\otimes {\hat{\sigma }}_{y})}{16}\right)$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>exp</mml:mi> <mml:mfenced close=")" open="("> <mml:mrow> <mml:mstyle displaystyle="false"> <mml:mfrac> <mml:mrow> <mml:mi>i</mml:mi> <mml:mi>π</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mi>σ</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>ˆ</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> <mml:mo>⨂</mml:mo> <mml:msub> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mi>σ</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>ˆ</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>8</mml:mn> </mml:mrow> </mml:mfrac> </mml:mstyle> <mml:mo>+</mml:mo> <mml:mstyle displaystyle="false"> <mml:mfrac> <mml:mrow> <mml:mi>i</mml:mi> <mml:mi>π</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mi>σ</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>ˆ</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>y</mml:mi> </mml:mrow> </mml:msub> <mml:mo>⨂</mml:mo> <mml:msub> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mi>σ</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>ˆ</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>y</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>16</mml:mn> </mml:mrow> </mml:mfrac> </mml:mstyle> </mml:mrow> </mml:mfenced> </mml:math> , as an entangling two-qubit basis gate in quantum computers by studying its ability to generate other two-qubit gates. We derive a necessary condition for a two-qubit gate to be generated by n applications of <?CDATA $\sqrt{{\rm{B}}}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msqrt> <mml:mrow> <mml:mi mathvariant="normal">B</mml:mi> </mml:mrow> </mml:msqrt> </mml:math> gate. Using this condition, we show that the gates that can be generated by two and three applications of <?CDATA $\sqrt{{\rm{B}}}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msqrt> <mml:mrow> <mml:mi mathvariant="normal">B</mml:mi> </mml:mrow> </mml:msqrt> </mml:math> gate are contained in the 50% and 92.97% of the volume of Weyl chamber of two-qubit gates, respectively. We prove that two applications of <?CDATA $\sqrt{{\rm{B}}}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msqrt> <mml:mrow> <mml:mi mathvariant="normal">B</mml:mi> </mml:mrow> </mml:msqrt> </mml:math> gate can generate both <?CDATA $\sqrt{\mathrm{SWAP}}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msqrt> <mml:mrow> <mml:mi>SWAP</mml:mi> </mml:mrow> </mml:msqrt> </mml:math> and <?CDATA ${\sqrt{\mathrm{SWAP}}}^{\dagger }$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mrow> <mml:msqrt> <mml:mrow> <mml:mi>SWAP</mml:mi> </mml:mrow> </mml:msqrt> </mml:mrow> <mml:mrow> <mml:mo>†</mml:mo> </mml:mrow> </mml:msup> </mml:math> which is not possible for CNOT and <?CDATA $\sqrt{\mathrm{iSWAP}}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msqrt> <mml:mrow> <mml:mi>iSWAP</mml:mi> </mml:mrow> </mml:msqrt> </mml:math> gates; further, we conjecture that three applications of <?CDATA $\sqrt{{\rm{B}}}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msqrt> <mml:mrow> <mml:mi mathvariant="normal">B</mml:mi> </mml:mrow> </mml:msqrt> </mml:math> gate can generate all perfect entanglers. Finally, we discuss about the construction of a three independent parameter universal two-qubit quantum circuit using four <?CDATA $\sqrt{{\rm{B}}}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msqrt> <mml:mrow> <mml:mi mathvariant="normal">B</mml:mi> </mml:mrow> </mml:msqrt> </mml:math> gates that can generate all two-qubit gates. In the end, we mention about the schemes to implement <?CDATA $\sqrt{{\rm{B}}}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msqrt> <mml:mrow> <mml:mi mathvariant="normal">B</mml:mi> </mml:mrow> </mml:msqrt> </mml:math> gate in ion-trap quantum computers.