Solving independent set problems with photonic quantum circuits

An independent set (IS) is a set of vertices in a graph such that no edge connects any two vertices. In adiabatic quantum computation [E. Farhi, et al ., Science 292, 472–475 (2001); A. Das, B. K. Chakrabarti, Rev. Mod. Phys. 80, 1061–1081 (2008)], a given graph G ( V , E ) can be naturally mapped onto a many-body Hamiltonian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msubsup> <mml:mi mathvariant="script">H</mml:mi> <mml:mrow> <mml:mi mathvariant="bold">IS</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="bold">G</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="bold">V</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="bold">E</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> , with edges <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="bold">E</mml:mi> </mml:math> being the two-body interactions between adjacent vertices <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="bold">V</mml:mi> </mml:math> . Thus, solving the IS problem is equivalent to finding all the computational basis ground states of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msubsup> <mml:mi mathvariant="script">H</mml:mi> <mml:mrow> <mml:mi mathvariant="bold">IS</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="bold">G</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="bold">V</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="bold">E</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> . Very recently, non-Abelian adiabatic mixing (NAAM) has been proposed to address this task, exploiting an emergent non-Abelian gauge symmetry of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msubsup> <mml:mi mathvariant="script">H</mml:mi> <mml:mrow> <mml:mi mathvariant="bold">IS</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="bold">G</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="bold">V</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="bold">E</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> [B. Wu, H. Yu, F. Wilczek, Phys. Rev. A 101, 012318 (2020)]. Here, we solve a representative IS problem <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="bold">G</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn mathvariant="bold">8</mml:mn> <mml:mo>,</mml:mo> <mml:mn mathvariant="bold">7</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> by simulating the NAAM digitally using a linear optical quantum network, consisting of three C-Phase gates, four deterministic two-qubit gate arrays (DGA), and ten single rotation gates. The maximum IS has been successfully identified with sufficient Trotterization steps and a carefully chosen evolution path. Remarkably, we find IS with a total probability of 0.875(16), among which the nontrivial ones have a considerable weight of about 31.4%. Our experiment demonstrates the potential advantage of NAAM for solving IS-equivalent problems.