{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"critpt","formal_name":"CritPt","introduction":"研究水準の物理問題で、科学的理解と多段階の推論・計算を評価するベンチマークです。公開データには70の課題があり、問題文とコード雛形を組み合わせて解答を構成します。\n\nCritPt 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"},"task_id":"47d9e4aa-1c42-5976-a55c-556b55602b63","task_key":"train--Challenge~5f15~5fmain","task_revision_id":"1","upstream_id":"Challenge_15_main","short_description":"The Hamiltonian is","config":"","split":"train","body":"{\"code_template\":\"import sympy as sp\\n\\nN, l = sp.symbols('N l', positive=True, integer=True)\\np = sp.symbols('p', real=True)\\n\\ndef answer(N, p, l):\\n    r\\\"\\\"\\\"\\n    Return the expression of $\\\\mathcal{S}_{0}$ in Sympy format.\\n\\n    Inputs\\n    ----------\\n    N: sympy.Symbol, site number, $N$\\n    p: sympy.Symbol, noise rate, $p\\\\in[0,1]$\\n    l: sympy.Symbol, string length, $l$\\n\\n    Outputs\\n    ----------\\n    S0: sympy.Expr, string order parameter, $\\\\mathcal{S}_{0}$\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    S0 = ...  # a SymPy expression of inputs\\n    # ---------------------------------------------------------------\\n\\n    return S0\",\"problem_description\":\"# Problem setup:\\nThe Hamiltonian is\\n\\\\begin{equation}\\nH=\\\\sum_{i=1}^N\\\\left[\\\\boldsymbol{S}_{i}\\\\cdot\\\\boldsymbol{S}_{i+1}+\\\\frac{1}{3}\\\\left(\\\\boldsymbol{S}_{i}\\\\cdot\\\\boldsymbol{S}_{i+1}\\\\right)^{2}\\\\right],\\\\\\n\\\\end{equation}\\nwhere $\\\\boldsymbol{S}_i=(S_{x, i},\\\\ S_{y, i},\\\\ S_{z, i})$ is the $3\\\\times 3$ identity matrix, with $S_{\\\\{x, y, z\\\\}}$ the spin operators along the $x,\\\\ y$, and $z$ directions. The subscript $i$ denotes the site index, and $N$ is the number of sites. In particular, the subscript $i$ can be neglected when unambiguous. We then take the ground state of this model and apply the following noise:\\n\\\\begin{equation}\\n\\\\rho = \\\\mathcal{E}_1\\\\circ\\\\mathcal{E}_2\\\\circ...\\\\mathcal{E}_N[\\\\rho_{0}],\\\\ \\\\text{and}\\\\  \\\\mathcal{E}_i[\\\\cdot]=\\\\sum_{\\\\alpha} K_{\\\\alpha,i}(\\\\cdot) K_{\\\\alpha,i}^{\\\\dagger},\\n\\\\end{equation}\\nwhere $\\\\rho_0$ is the ground state of $H$, and the Kraus operators $\\\\{K_{\\\\alpha,i}\\\\}$ of the quantum channel $\\\\mathcal{E}_i$ act locally on the spin-$1$ Hilbert space at site $i$ as $\\\\{\\\\sqrt{1-p}\\\\mathbb{I}_3,\\\\sqrt{p}S_x S_y, \\\\sqrt{p}S_yS_z,\\\\sqrt{p} S_z S_x\\\\}$. Here, $\\\\mathbb{I}_3$ is a $3\\\\times 3$ identity matrix for the spin-$1$ operator, and $p\\\\in [0,1]$ represents the noise rate.\\n\\n# Main problem:\\nExactly calculate the quantity\\n\\\\begin{equation}\\n\\\\mathcal{S}_{0}=\\\\text{Tr}\\\\left[ \\\\rho \\\\mathbb{I}_{3}\\\\otimes\\\\left(\\\\otimes_{i=j}^{j+l-1}R_{z}\\\\right)\\\\otimes \\\\mathbb{I}_{3}\\\\right],\\n\\\\end{equation}\\nwhere $R_{z}=e^{i\\\\pi S_{z}}$, and $l$ is the string length. This represents the string order parameter with length $l$.\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}