In　this　paper,　we　propose　a　bio-molecular　algorithm　with　O(　n2　+　m　)　biological　operations,　O(　2n　)　DNA　strands,　O(　n　)　tubes　and　the　longest　DNA　strand,　O(　n　),　for　solving　the　independent-set　problem　for　any　graph　G　with　m　edges　and　n　vertices.　Next,　we　show　that　a　new　kind　of　the　straightforward　Boolean　circuit　yielded　from　the　bio-molecular　solutions　with　m　NAND　gates,　(　m　+n　×　(　n}　+　1　))　AND　gates　and　((　n　×　(　n　+　1　))/2)　NOT　gates　can　find　the　maximal　independent-set(s)　to　the　independent-set　problem　for　any　graph　G　with　m　edges　and　n　vertices.　We　show　that　a　new　kind　of　the　proposed　quantum-molecular　algorithm　can　find　the　maximal　independent　set(s)　with　the　lower　bound　Ω　(　2n/2　)　queries　and　the　upper　bound　O(　2n/2　)　queries.　This　work　offers　an　obvious　evidence　for　that　to　solve　the　independent-set　problem　in　any　graph　G　with　m　edges　and　n　vertices,　bio-molecular　computers　are　able　to　generate　a　new　kind　of　the　straightforward　Boolean　circuit　such　that　by　means　of　implementing　it　quantum　computers　can　give　a　quadratic　speed-up.　This　work　also　offers　one　obvious　evidence　that　quantum　computers　can　significantly　accelerate　the　speed　and　enhance　the　scalability　of　bio-molecular　computers.　Next,　the　element　distinctness　problem　with　input　of　n}　bits　is　to　determine　whether　the　given　2n　real　numbers　are　distinct　or　not.　The　quantum　lower　bound　of　solving　the　element　distinctness　problem　is　Omega　(　2n×　(2/3))　queries　in　the　case　of　a　quantum　walk　algorithm.　We　further　show　that　the　proposed　quantum-molecular　algorithm　reduces　the　quantum　lower　bound　to　Ω　((　2n/2)/(　21/2))　queries.　Furthermore,　to　justify　the　feasibility　of　the　proposed　quantum-molecular　algorithm,　we　successfully　solve　a　typical　independent　set　problem　for　a　graph　G　with　two　vertices　and　one　edge　by　carrying　out　experiments　on　the　backend　ibmqx4　with　five　quantum　bits　and　the　backend　simulator　with　32　quantum　bits　on　IBM＇s　quantum　computer.　©　2002-2011　IEEE.