{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"critpt","formal_name":"CritPt","introduction":"CritPt evaluates scientific understanding and multi-step reasoning and computation on research-level physics problems. Its public dataset contains 70 challenges with problem descriptions and code templates.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://critpt.com/","indexing_mode":"noindex","profile":{"resources":[],"task_format":"","scoring":"","metric":"","size":"","answer_access":"","license":"","citation":"","maintainer":"","released":"","why_hard":"","related":[]}},"task_id":"e7c2a6f9-4626-5efc-956d-d363145896bd","task_key":"train--Challenge~5f30~5fmain","task_revision_id":"2","upstream_id":"Challenge_30_main","short_description":"Let $H_b$, $H_B$, $H_f$, $H_P$ each be a finite-dimensional Hilbert space, with…","config":"","split":"train","body":"{\"code_template\":\"import sympy as sp\\nfrom sympy.physics.quantum import Ket, Bra, Dagger\\n\\nphi = Ket('phi')\\npsi = Ket('psi')\\nphi_star = Ket('phi*')\\npsi_star = Ket('psi*')\\nd_P, d_B = sp.symbols('d_P d_B')\\nd = sp.symbols('d')\\n\\ndef answer(phi, psi, phi_star, psi_star, d_P, d_B, d):\\n    r\\\"\\\"\\\"\\n    Return the expression of \\\\overline{\\\\lvert \\\\langle\\\\phi|V^\\\\dagger V|\\\\psi\\\\rangle \\\\rvert^2} in Sympy format.\\n\\n    Inputs\\n    ----------\\n    phi: sympy.Symbol, state vector |\\\\phi\\\\rangle_b\\n    psi: sympy.Symbol, state vector |\\\\psi\\\\rangle_b\\n    phi_star: sympy.Symbol, state vector |\\\\phi^*\\\\rangle_b\\n    psi_star: sympy.Symbol, state vector |\\\\psi^*\\\\rangle_b\\n    d_P: sympy.Symbol, $d_P := \\\\dim H_P$\\n    d_B: sympy.Symbol, $d_B := \\\\dim H_B$\\n    d: sympy.Symbol, $d = \\\\dim(H_b \\\\otimes H_f)$\\n\\n    Outputs\\n    ----------\\n    expr: sympy.Expr, \\\\overline{\\\\lvert \\\\langle\\\\phi|V^\\\\dagger V|\\\\psi\\\\rangle \\\\rvert^2},\\n        where the overline indicates the average over $O$.\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    expr = ...  # a SymPy expression of inputs\\n    # ---------------------------------------------------------------\\n\\n    return expr\",\"problem_description\":\"# Problem setup:\\nLet $H_b$, $H_B$, $H_f$, $H_P$ each be a finite-dimensional Hilbert space, with $\\\\dim H_b > \\\\dim H_B$ and $\\\\dim(H_b \\\\otimes H_f) = \\\\dim(H_B \\\\otimes H_P)$.\\nDefine the linear map $V: H_b \\\\to H_B$ as follows,\\n\\\\begin{equation}\\nV = \\\\sqrt{\\\\dim(H_P)} \\\\langle 0|_P O |0\\\\rangle_f,\\n\\\\end{equation}\\nwhere $|0\\\\rangle_f$ and $|0\\\\rangle_P$ are fiducial states on $H_f$ and $H_P$, respectively, and $O$ is an operator acting on $H_b \\\\otimes H_f$ drawn at random from the orthogonal group $O(d)$ with $d = \\\\dim(H_b \\\\otimes H_f)$.\\n\\nLet $d_P := \\\\dim H_P$ and $d_B := \\\\dim H_B$. And for any state $|\\\\phi\\\\rangle$,  denote its complex conjugate by $|\\\\phi^*\\\\rangle$.\\n\\n# Main problem:\\n\\nGiven two states $|\\\\psi\\\\rangle_b \\\\in H_b$ and $|\\\\phi\\\\rangle_b \\\\in H_b$, calculate the quantity\\n\\\\begin{equation}\\n\\t\\\\overline{\\\\lvert \\\\langle\\\\phi|V^\\\\dagger V|\\\\psi\\\\rangle \\\\rvert^2},\\n\\\\end{equation}\\nwhere the overline indicates the average over $O$.\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}