{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2025-12-14T01:33:56.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2025-12-14T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":61286,"title":"The Heisenberg Entanglement Protocol","description":"Abstract:\r\nA quantum signal must be routed through  nodes. The cost of transmission is not purely geometric but depends on the phase coherence between entanglement pairs and the fundamental limits of quantum mechanics.\r\nKey Mechanics:\r\nEntanglement Phase Cost: The cost to jump from node  to  is propotional to the absolute sine of their phase different: .\r\nQuantum Tunneling: If the Euclidean distance between two nodes is a prime number (rounded), a tunnel opens. This allows a jump with a fixed cost of 0.5, but at the expense of a 20% reduction in remaining ATP.\r\nHeisenberg Uncertainly Barrier: A jump is forbidden if the product of \"Position Uncertainly\" () and \"Phase Momentum\" ( ) is less than the effective Planck constant .\r\nTrivalent Constraints: You must reach the destination before ATP reaching 0 and within a maximum of steps.\r\nGoal: Find the minimum total cost path while satisfying the uncertainty principle.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 298.1px; display: block; min-width: 0px; padding-block-start: 0px; padding-inline-start: 2px; padding-left: 2px; padding-top: 0px; perspective-origin: 468.5px 149.05px; transform-origin: 468.5px 149.05px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 444.5px 10.5px; text-align: left; transform-origin: 444.5px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eAbstract\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 444.5px 21px; text-align: left; transform-origin: 444.5px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA quantum signal must be routed through \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(33, 33, 33);\"\u003eN\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e nodes. The cost of transmission is not purely geometric but depends on the phase coherence between entanglement pairs and the fundamental limits of quantum mechanics.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 444.5px 10.5px; text-align: left; transform-origin: 444.5px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eKey Mechanics\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003col style=\"block-size: 145.1px; font-family: Helvetica, Arial, sans-serif; list-style-type: decimal; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 451.5px 72.55px; transform-origin: 451.5px 72.55px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 41.9px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 423.5px 20.95px; text-align: left; transform-origin: 423.5px 20.95px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eEntanglement Phase Cost\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e: The cost to jump from node \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(33, 33, 33);\"\u003eu\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(33, 33, 33);\"\u003ev\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is propotional to the absolute sine of their phase different: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg 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\" width=\"224\" height=\"20\" style=\"width: 224px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 40.8667px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 423.5px 20.4333px; text-align: left; transform-origin: 423.5px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eQuantum Tunneling\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e: If the Euclidean distance between two nodes is a prime number (rounded), a tunnel opens. This allows a jump with a fixed cost of 0.5, but at the expense of a 20% reduction in remaining ATP.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 41.9px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 423.5px 20.95px; text-align: left; transform-origin: 423.5px 20.95px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eHeisenberg Uncertainly Barrier\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e: A jump is forbidden if the product of \"Position Uncertainly\" (\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"92.5\" height=\"18\" style=\"width: 92.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e) and \"Phase Momentum\" (\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg 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width=\"91\" height=\"20\" style=\"width: 91px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e ) is less than the effective Planck constant \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg 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width=\"24\" height=\"20\" style=\"width: 24px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 423.5px 10.2167px; text-align: left; transform-origin: 423.5px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eTrivalent Constraints\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e: You must reach the destination before \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eATP\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e reaching 0 and within a maximum of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(33, 33, 33);\"\u003eN\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esteps.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 444.5px 10.5px; text-align: left; transform-origin: 444.5px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eGoal: Find the minimum total cost path while satisfying the uncertainty principle.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [cost, path] = solve_heisenberg_entanglement(nodes,params)\r\n  %nodes - struct array(.x, .y, .z, .phase)\r\n  %params - struct(.start_id, .end_id, .ATP_start, .h_eff)\r\n  \r\n  %cost - minimum total cost(-1 if unreachable)\r\n  %path - vector of node indices\r\n  \r\n  cost = -1;\r\n  path = [];\r\nend","test_suite":"%% \r\nn1 = struct('x',{0, 11}, 'y',{0, 0}, 'z',{0, 0}, 'phase',{0, 0.5});\r\np1.start_id = 1; p1.end_id = 2; p1.ATP_start = 100; p1.h_eff = 1.0;\r\n[c1, path1] = solve_heisenberg_entanglement(n1, p1);\r\n\r\nassert(abs(c1 - 0.5) \u003c 1e-4, 'Phải nhận diện được Quantum Tunneling khi khoảng cách là số nguyên tố!');\r\nassert(isequal(path1, [1 2]), 'Lộ trình đơn giản 1-2 bị sai');\r\n\r\n%% \r\nn2 = struct('x',{0, 1}, 'y',{0, 0}, 'z',{0, 0}, 'phase',{0, 0.1});\r\np2.start_id = 1; p2.end_id = 2; p2.ATP_start = 100; p2.h_eff = 2.0; % 1 * 0.1 \u003c 2.0\r\n[c2, path2] = solve_heisenberg_entanglement(n2, p2);\r\n\r\nassert(c2 == -1, 'Vi phạm nguyên lý bất định nhưng vẫn cho phép đi là sai!');\r\nassert(isempty(path2), 'Path phải trống khi không tìm thấy đường');\r\n\r\n%% \r\nn3 = struct('x',{0, 10, 20}, 'y',{0, 0, 0}, 'z',{0, 0, 0}, 'phase',{0, 1, 2});\r\np3.start_id = 1; p3.end_id = 3; p3.ATP_start = 1.0; p3.h_eff = 0.1; \r\n[c3, path3] = solve_heisenberg_entanglement(n3, p3);\r\n\r\nassert(c3 == -1, 'Phải trả về -1 khi cạn kiệt ATP');\r\n\r\n%% \r\nn4 = struct('x',{0, 10, 10, 20}, 'y',{0, 5, -5, 0}, 'z',{0, 0, 0, 0}, 'phase',{0, 0.1, 1.5, 0.2});\r\np4.start_id = 1; p4.end_id = 4; p4.ATP_start = 100; p4.h_eff = 0.1;\r\n[c4, path4] = solve_heisenberg_entanglement(n4, p4);\r\n\r\nassert(c4 \u003c 5, 'Chưa tìm được đường tối ưu về chi phí năng lượng (Phase Cost)');\r\n\r\n%% \r\nrng(77); n = 15;\r\nfor i=1:n, n5(i).x=rand*50; n5(i).y=rand*50; n5(i).z=rand*50; n5(i).phase=rand*2*pi; end\r\np5.start_id=1; p5.end_id=n; p5.ATP_start=100; p5.h_eff=1.5;\r\n[c5, path5] = solve_heisenberg_entanglement(n5, p5);\r\n\r\nassert(c5 \u003e 0 || c5 == -1, 'Lỗi thực thi trên bản đồ phức tạp');","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":4945722,"edited_by":4945722,"edited_at":"2026-03-20T15:25:56.000Z","deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":"2026-03-20T15:25:56.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2026-03-20T08:28:01.000Z","updated_at":"2026-03-22T17:56:51.000Z","published_at":"2026-03-20T08:28:01.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml 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w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eN\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e nodes. The cost of transmission is not purely geometric but depends on the phase coherence between entanglement pairs and the fundamental limits of quantum mechanics.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eKey Mechanics\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eEntanglement Phase Cost\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e: The cost to jump from node \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ev\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is propotional to the absolute sine of their phase different: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eCost = \\\\left|\\\\sin(\\\\theta_{u} - \\\\theta_{v}\\\\right| \\\\times \\\\text{Distance}(u,v)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eQuantum Tunneling\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e: If the Euclidean distance between two nodes is a prime number (rounded), a tunnel opens. This allows a jump with a fixed cost of 0.5, but at the expense of a 20% reduction in remaining ATP.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHeisenberg Uncertainly Barrier\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e: A jump is forbidden if the product of \\\"Position Uncertainly\\\" (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta x = \\\\text{Distance}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e) and \\\"Phase Momentum\\\" (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta p = \\\\left|\\\\theta_{u} - \\\\theta_{v}\\\\right|\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e ) is less than the effective Planck constant \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\hbar_{eff}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eTrivalent Constraints\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e: You must reach the destination before \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eATP\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e reaching 0 and within a maximum of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eN\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003esteps.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGoal: Find the minimum total cost path while satisfying the uncertainty principle.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":61286,"title":"The Heisenberg Entanglement Protocol","description":"Abstract:\r\nA quantum signal must be routed through  nodes. The cost of transmission is not purely geometric but depends on the phase coherence between entanglement pairs and the fundamental limits of quantum mechanics.\r\nKey Mechanics:\r\nEntanglement Phase Cost: The cost to jump from node  to  is propotional to the absolute sine of their phase different: .\r\nQuantum Tunneling: If the Euclidean distance between two nodes is a prime number (rounded), a tunnel opens. This allows a jump with a fixed cost of 0.5, but at the expense of a 20% reduction in remaining ATP.\r\nHeisenberg Uncertainly Barrier: A jump is forbidden if the product of \"Position Uncertainly\" () and \"Phase Momentum\" ( ) is less than the effective Planck constant .\r\nTrivalent Constraints: You must reach the destination before ATP reaching 0 and within a maximum of steps.\r\nGoal: Find the minimum total cost path while satisfying the uncertainty principle.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 298.1px; display: block; min-width: 0px; padding-block-start: 0px; padding-inline-start: 2px; padding-left: 2px; padding-top: 0px; perspective-origin: 468.5px 149.05px; transform-origin: 468.5px 149.05px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 444.5px 10.5px; text-align: left; transform-origin: 444.5px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eAbstract\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 444.5px 21px; text-align: left; transform-origin: 444.5px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA quantum signal must be routed through \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(33, 33, 33);\"\u003eN\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e nodes. The cost of transmission is not purely geometric but depends on the phase coherence between entanglement pairs and the fundamental limits of quantum mechanics.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 444.5px 10.5px; text-align: left; transform-origin: 444.5px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eKey Mechanics\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003col style=\"block-size: 145.1px; font-family: Helvetica, Arial, sans-serif; list-style-type: decimal; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 451.5px 72.55px; transform-origin: 451.5px 72.55px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 41.9px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 423.5px 20.95px; text-align: left; transform-origin: 423.5px 20.95px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eEntanglement Phase Cost\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e: The cost to jump from node \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(33, 33, 33);\"\u003eu\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(33, 33, 33);\"\u003ev\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is propotional to the absolute sine of their phase different: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg 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\" width=\"224\" height=\"20\" style=\"width: 224px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 40.8667px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 423.5px 20.4333px; text-align: left; transform-origin: 423.5px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eQuantum Tunneling\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e: If the Euclidean distance between two nodes is a prime number (rounded), a tunnel opens. This allows a jump with a fixed cost of 0.5, but at the expense of a 20% reduction in remaining ATP.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 41.9px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 423.5px 20.95px; text-align: left; transform-origin: 423.5px 20.95px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eHeisenberg Uncertainly Barrier\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e: A jump is forbidden if the product of \"Position Uncertainly\" (\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"92.5\" height=\"18\" style=\"width: 92.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e) and \"Phase Momentum\" (\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg 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width=\"91\" height=\"20\" style=\"width: 91px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e ) is less than the effective Planck constant \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg 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width=\"24\" height=\"20\" style=\"width: 24px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 423.5px 10.2167px; text-align: left; transform-origin: 423.5px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eTrivalent Constraints\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e: You must reach the destination before \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eATP\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e reaching 0 and within a maximum of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(33, 33, 33);\"\u003eN\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esteps.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 444.5px 10.5px; text-align: left; transform-origin: 444.5px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eGoal: Find the minimum total cost path while satisfying the uncertainty principle.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [cost, path] = solve_heisenberg_entanglement(nodes,params)\r\n  %nodes - struct array(.x, .y, .z, .phase)\r\n  %params - struct(.start_id, .end_id, .ATP_start, .h_eff)\r\n  \r\n  %cost - minimum total cost(-1 if unreachable)\r\n  %path - vector of node indices\r\n  \r\n  cost = -1;\r\n  path = [];\r\nend","test_suite":"%% \r\nn1 = struct('x',{0, 11}, 'y',{0, 0}, 'z',{0, 0}, 'phase',{0, 0.5});\r\np1.start_id = 1; p1.end_id = 2; p1.ATP_start = 100; p1.h_eff = 1.0;\r\n[c1, path1] = solve_heisenberg_entanglement(n1, p1);\r\n\r\nassert(abs(c1 - 0.5) \u003c 1e-4, 'Phải nhận diện được Quantum Tunneling khi khoảng cách là số nguyên tố!');\r\nassert(isequal(path1, [1 2]), 'Lộ trình đơn giản 1-2 bị sai');\r\n\r\n%% \r\nn2 = struct('x',{0, 1}, 'y',{0, 0}, 'z',{0, 0}, 'phase',{0, 0.1});\r\np2.start_id = 1; p2.end_id = 2; p2.ATP_start = 100; p2.h_eff = 2.0; % 1 * 0.1 \u003c 2.0\r\n[c2, path2] = solve_heisenberg_entanglement(n2, p2);\r\n\r\nassert(c2 == -1, 'Vi phạm nguyên lý bất định nhưng vẫn cho phép đi là sai!');\r\nassert(isempty(path2), 'Path phải trống khi không tìm thấy đường');\r\n\r\n%% \r\nn3 = struct('x',{0, 10, 20}, 'y',{0, 0, 0}, 'z',{0, 0, 0}, 'phase',{0, 1, 2});\r\np3.start_id = 1; p3.end_id = 3; p3.ATP_start = 1.0; p3.h_eff = 0.1; \r\n[c3, path3] = solve_heisenberg_entanglement(n3, p3);\r\n\r\nassert(c3 == -1, 'Phải trả về -1 khi cạn kiệt ATP');\r\n\r\n%% \r\nn4 = struct('x',{0, 10, 10, 20}, 'y',{0, 5, -5, 0}, 'z',{0, 0, 0, 0}, 'phase',{0, 0.1, 1.5, 0.2});\r\np4.start_id = 1; p4.end_id = 4; p4.ATP_start = 100; p4.h_eff = 0.1;\r\n[c4, path4] = solve_heisenberg_entanglement(n4, p4);\r\n\r\nassert(c4 \u003c 5, 'Chưa tìm được đường tối ưu về chi phí năng lượng (Phase Cost)');\r\n\r\n%% \r\nrng(77); n = 15;\r\nfor i=1:n, n5(i).x=rand*50; n5(i).y=rand*50; n5(i).z=rand*50; n5(i).phase=rand*2*pi; end\r\np5.start_id=1; p5.end_id=n; p5.ATP_start=100; p5.h_eff=1.5;\r\n[c5, path5] = solve_heisenberg_entanglement(n5, p5);\r\n\r\nassert(c5 \u003e 0 || c5 == -1, 'Lỗi thực thi trên bản đồ phức tạp');","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":4945722,"edited_by":4945722,"edited_at":"2026-03-20T15:25:56.000Z","deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":"2026-03-20T15:25:56.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2026-03-20T08:28:01.000Z","updated_at":"2026-03-22T17:56:51.000Z","published_at":"2026-03-20T08:28:01.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eAbstract\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA quantum signal must be routed through \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eN\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e nodes. The cost of transmission is not purely geometric but depends on the phase coherence between entanglement pairs and the fundamental limits of quantum mechanics.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eKey Mechanics\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eEntanglement Phase Cost\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e: The cost to jump from node \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ev\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is propotional to the absolute sine of their phase different: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eCost = \\\\left|\\\\sin(\\\\theta_{u} - \\\\theta_{v}\\\\right| \\\\times \\\\text{Distance}(u,v)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eQuantum Tunneling\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e: If the Euclidean distance between two nodes is a prime number (rounded), a tunnel opens. This allows a jump with a fixed cost of 0.5, but at the expense of a 20% reduction in remaining ATP.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHeisenberg Uncertainly Barrier\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e: A jump is forbidden if the product of \\\"Position Uncertainly\\\" (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta x = \\\\text{Distance}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e) and \\\"Phase Momentum\\\" (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta p = \\\\left|\\\\theta_{u} - \\\\theta_{v}\\\\right|\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e ) is less than the effective Planck constant \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\hbar_{eff}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eTrivalent Constraints\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e: You must reach the destination before \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eATP\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e reaching 0 and within a maximum of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eN\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003esteps.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGoal: Find 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