z i#SrSSKJr SSKrSSKJr SSKJr SSKr SSK J r SSK J r SSKJr SSKrS S KJrJr \R((a SS K Jr SS KJr "S S\5rSSjrg)z#The Suzuki-Trotter product formula.) annotationsN)Callable)chain)ParameterExpression)QuantumCircuit) SparsePauliOp)ProductFormulareorder_paulis)ParameterValueType)PauliEvolutionGatec^\rSrSrSrS SS.S U4SjjjjrS Sjr\S5rSr U=r $) SuzukiTrotter"aThe (higher order) Suzuki-Trotter product formula. The Suzuki-Trotter formulas improve the error of the Lie-Trotter approximation. For example, the second order decomposition is .. math:: e^{A + B} \approx e^{B/2} e^{A} e^{B/2}. Higher order decompositions are based on recursions, see Ref. [1] for more details. In this implementation, the operators are provided as sum terms of a Pauli operator. For example, in the second order Suzuki-Trotter decomposition we approximate .. math:: e^{-it(XI + ZZ)} = e^{-it/2 XI}e^{-it ZZ}e^{-it/2 XI} + \mathcal{O}(t^3). References: [1]: D. Berry, G. Ahokas, R. Cleve and B. Sanders, "Efficient quantum algorithms for simulating sparse Hamiltonians" (2006). `arXiv:quant-ph/0508139 `_ [2]: N. Hatano and M. Suzuki, "Finding Exponential Product Formulas of Higher Orders" (2005). `arXiv:math-ph/0506007 `_ F)"atomic_evolution_sparse_observablec j>US:aUS-S:Xa[SUS35e[T U] UUUUUUUUS9 g)a Args: order: The order of the product formula. reps: The number of time steps. insert_barriers: Whether to insert barriers between the atomic evolutions. cx_structure: How to arrange the CX gates for the Pauli evolutions, can be ``"chain"``, where next neighbor connections are used, or ``"fountain"``, where all qubits are connected to one. This only takes effect when ``atomic_evolution is None``. atomic_evolution: A function to apply the evolution of a single :class:`~.quantum_info.Pauli`, or :class:`.SparsePauliOp` of only commuting terms, to a circuit. The function takes in three arguments: the circuit to append the evolution to, the Pauli operator to evolve, and the evolution time. By default, a single Pauli evolution is decomposed into a chain of ``CX`` gates and a single ``RZ`` gate. wrap: Whether to wrap the atomic evolutions into custom gate objects. This only takes effect when ``atomic_evolution is None``. preserve_order: If ``False``, allows reordering the terms of the operator to potentially yield a shallower evolution circuit. Not relevant when synthesizing operator with a single term. atomic_evolution_sparse_observable: If a custom ``atomic_evolution`` is passed, which does not yet support :class:`.SparseObservable`\ s as input, set this argument to ``False`` to automatically apply a conversion to :class:`.SparsePauliOp`. This argument is supported until Qiskit 2.2, at which point all atomic evolutions are required to support :class:`.SparseObservable`\ s as input. Raises: ValueError: If order is not even r zfSuzuki product formulae are symmetric and therefore only defined for when the order is 1 or even, not .)preserve_orderrN) ValueErrorsuper__init__) selforderrepsinsert_barriers cx_structureatomic_evolutionwraprr __class__s c/home/james-whalen/.local/lib/python3.13/site-packages/qiskit/synthesis/evolution/suzuki_trotter.pyrSuzukiTrotter.__init__>sbV 19a88=waA        )/Q  cv^^ URnURm UU 4Sjn[U[5(aUVs/sH oC"U5PM nnOU"U5Vs/sHof/PM nnTR TR U5nTR [[R"U55-nU$s snfs snf)aExpand the Hamiltonian into a Suzuki-Trotter sequence of sparse gates. For example, the Hamiltonian ``H = IX + ZZ`` for an evolution time ``t`` and 1 repetition for an order 2 formula would get decomposed into a list of 3-tuples containing ``(pauli, indices, rz_rotation_angle)``, that is: .. code-block:: text ("X", [0], t), ("ZZ", [0, 1], 2t), ("X", [0], t) Note that the rotation angle contains a factor of 2, such that that evolution of a Pauli :math:`P` over time :math:`t`, which is :math:`e^{itP}`, is represented by ``(P, indices, 2 * t)``. For ``N`` repetitions, this sequence would be repeated ``N`` times and the coefficients divided by ``N``. Args: evolution: The evolution gate to expand. Returns: The Pauli network implementing the Trotter expansion. c &>[U[5(aUR5OUR5nUVVVs/sH&up#nX#[U5T-S-TR- 4PM( nnnnTR (d [ U5$U$s snnnf)Nr) isinstancerto_sparse_list real_or_failrrr )operator sparse_listpauliindicescoeffpaulisrtimes r!r',SuzukiTrotter.expand..to_sparse_listsh 66''),,. .9-8)EEe!4t!;a!?$))!KL-8 &&%f--Ms-B ) r)r/r&list_recurserrr from_iterable) r evolution operatorsr'r) non_commutingopproduct_formula flattenedr/s ` @r!expandSuzukiTrotter.expandzs4&& ~~  i & &FOPi(^H5iMPM-;9,EF,EbT,EMF-- MBIIU%8%8%I JJ QGs B1 B6cUS:XaU$US:XaVUSSVVVVs/sH!nUVVVs/sH up4oSXES- 4PM snnnPM# nnnnnUS/nXg-[[U55-$SSSSUS- - -- - nS[RUS- UVVVVs/sH!nUVVVs/sH up4oSXEU-4PM snnnPM# snnnn5-n [RUS- UVVVVs/sH(nUVVVs/sHup4nX4USSU-- -4PM snnnPM* snnnn5n X-U -$s snnnfs snnnnfs snnnfs snnnnfs snnnfs snnnnf)Nr r)r1reversedrr2) rgrouped_paulisr.labelqubitsr-halvesfull reductionouterinners r!r2SuzukiTrotter._recurses A:! ! aZ-Sb11FIOO0Du+O1 #2&'D=4(8#99 9QqEAI!778I .. #1"0U[[TZV#wwu~ 1%8LM NNr#)d)rS __future__rtypingcollections.abcr itertoolsrnumpyrX"qiskit.circuit.parameterexpressionrqiskit.circuit.quantumcircuitrqiskit.quantum_inforqiskitr8r r TYPE_CHECKINGr &qiskit.circuit.library.pauli_evolutionr rr(rIr#r!rnsL*" $B8-; @Ir)Nr)j Or#