{"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":"94ee7dfb-6a27-5e60-abe2-e039051c77eb","task_key":"train--Challenge~5f34~5fmain","task_revision_id":"1","upstream_id":"Challenge_34_main","short_description":"We will explore a simple example of a quantum tensor network, a tensor network…","config":"","split":"train","body":"{\"code_template\":\"import sympy as sp\\n\\na, b = sp.symbols('a b')\\n\\ndef answer(a, b):\\n    r\\\"\\\"\\\"\\n    Return the expression of the expectation value in SymPy format.\\n\\n    Inputs\\n    ----------\\n    a: sympy.Symbol, single-qubit rotation angle in $U_{jk}$, $a$\\n    b: sympy.Symbol, two-qubit entangling angle in $U_{jk}$, $b$\\n\\n    Output\\n    ----------\\n    expectation: sympy.Expr, the expectation value of the two-point correlation function of the qMPS in the thermodynamic limit, $\\\\lim_{N\\\\rightarrow \\\\infty} \\\\langle Z_{N-2} Z_{N} \\\\rangle$\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULT BELOW ------------------\\n    expectation = ...   # a SymPy expression of inputs\\n    # ---------------------------------------------------------------\\n\\n    return expectation\\n\",\"problem_description\":\"# Problem setup:\\nWe will explore a simple example of a quantum tensor network, a tensor network made of unitary tensors that can be executed as a quantum circuit on a quantum computer. In particular, we will perform a numerical computation of a quantum matrix product state (qMPS) with bond dimension $\\\\chi=2$.\\n\\nConsider a qMPS circuit defined on $N+1$ qubits labelled $0,\\\\ldots,N$, with all qubits starting in the $|0\\\\rangle$ state. In the qMPS circuit, the two-qubit gate\\n$$\\n\\\\begin{align*}\\nU_{jk} &=e^{-i b (X_j X_k + Z_j Z_k)/2}e^{-i a X_k/2},\\n\\\\end{align*}\\n$$\\nwhich acts on qubits $(j,k)$, is used, where $X_j,Z_j$ are Pauli matrices acting on qubit $j$. (Note that the above equation describes matrix multiplication, where operations occur in the opposite order as gates in a quantum circuit.) In the circuit that generates the qMPS state, the $U_{jk}$ gate is applied to qubits $(0,1),(0,2),(0,3),\\\\ldots,(0,N)$ in that order. Suppose that $0 < a,b < \\\\pi/2.$\\n\\n# Main problem:\\nWhat is the expectation value of the two-point correlation function $\\\\lim_{N\\\\rightarrow \\\\infty} \\\\langle Z_{N-2} Z_{N} \\\\rangle$  of the qMPS in the thermodynamic limit as a function of $a$ and $b$?\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}