ó Óz iæmãóî•SrSSKJr SSKJrJrJr SSKrSSKJ r J r J r J r SSK Jr SSKJrJrJr SSKJr S S KJr S S KJr SSS jjr"S S\ 5rSSSjjrg)zThe Grover operator.é)Ú annotations)ÚListÚOptionalÚUnionN)ÚQuantumCircuitÚQuantumRegisterÚAncillaRegisterÚ AncillaQubit)Ú QiskitError)Ú StatevectorÚOperatorÚ DensityMatrix)Údeprecate_funcé)ÚMCXGate)Ú DiagonalGatecó•[U[5(aURUSS9nO[URUS9n[U[5(a4[ SUR -5nURXvR5 OURUSS9 U(aUR5 Uc@[UR5VV s/sHup‰[U [5(aMUPM nnn UcURU5 OURUR5SS9 U(aUR5 Uc[U5n UR!U5 U S:XaUR#US5 OG[%U S- 5n URUS5 URX³5 URUS5 UR!U5 Oj[U[&[(45(a?[ UR R+55nURXvR5 OURUSS9 U(aUR5 UcURU5 OURUSS9 [,R.UlU$s sn nf) u‘Construct the Grover operator. Grover's search algorithm [1, 2] consists of repeated applications of the so-called Grover operator used to amplify the amplitudes of the desired output states. This operator, :math:`\mathcal{Q}`, consists of the phase oracle, :math:`\mathcal{S}_f`, zero phase-shift or zero reflection, :math:`\mathcal{S}_0`, and an input state preparation :math:`\mathcal{A}`: .. math:: \mathcal{Q} = \mathcal{A} \mathcal{S}_0 \mathcal{A}^\dagger \mathcal{S}_f In the standard Grover search we have :math:`\mathcal{A} = H^{\otimes n}`: .. math:: \mathcal{Q} = H^{\otimes n} \mathcal{S}_0 H^{\otimes n} \mathcal{S}_f = D \mathcal{S_f} The operation :math:`D = H^{\otimes n} \mathcal{S}_0 H^{\otimes n}` is also referred to as diffusion operator. In this formulation we can see that Grover's operator consists of two steps: first, the phase oracle multiplies the good states by -1 (with :math:`\mathcal{S}_f`) and then the whole state is reflected around the mean (with :math:`D`). This class allows setting a different state preparation, as in quantum amplitude amplification (a generalization of Grover's algorithm), :math:`\mathcal{A}` might not be a layer of Hardamard gates [3]. The action of the phase oracle :math:`\mathcal{S}_f` is defined as .. math:: \mathcal{S}_f: |x\rangle \mapsto (-1)^{f(x)}|x\rangle where :math:`f(x) = 1` if :math:`x` is a good state and 0 otherwise. To highlight the fact that this oracle flips the phase of the good states and does not flip the state of a result qubit, we call :math:`\mathcal{S}_f` a phase oracle. Note that you can easily construct a phase oracle from a bitflip oracle by sandwiching the controlled X gate on the result qubit by a X and H gate. For instance .. parsed-literal:: Bitflip oracle Phaseflip oracle q_0: ──■── q_0: ────────────■──────────── ┌─┴─┠┌───â”┌───â”┌─┴─â”┌───â”┌───┠out: ┤ X ├ out: ┤ X ├┤ H ├┤ X ├┤ H ├┤ X ├ └───┘ └───┘└───┘└───┘└───┘└───┘ There is some flexibility in defining the oracle and :math:`\mathcal{A}` operator. Before the Grover operator is applied in Grover's algorithm, the qubits are first prepared with one application of the :math:`\mathcal{A}` operator (or Hadamard gates in the standard formulation). Thus, we always have operation of the form :math:`\mathcal{A} \mathcal{S}_f \mathcal{A}^\dagger`. Therefore it is possible to move bitflip logic into :math:`\mathcal{A}` and leaving the oracle only to do phaseflips via Z gates based on the bitflips. One possible use-case for this are oracles that do not uncompute the state qubits. The zero reflection :math:`\mathcal{S}_0` is usually defined as .. math:: \mathcal{S}_0 = 2 |0\rangle^{\otimes n} \langle 0|^{\otimes n} - \mathbb{I}_n where :math:`\mathbb{I}_n` is the identity on :math:`n` qubits. By default, this class implements the negative version :math:`2 |0\rangle^{\otimes n} \langle 0|^{\otimes n} - \mathbb{I}_n`, since this can simply be implemented with a multi-controlled Z sandwiched by X gates on the target qubit and the introduced global phase does not matter for Grover's algorithm. Examples: We can construct a Grover operator just from the phase oracle: .. plot:: :alt: Circuit diagram output by the previous code. :include-source: :context: from qiskit.circuit import QuantumCircuit from qiskit.circuit.library import grover_operator oracle = QuantumCircuit(2) oracle.z(0) # good state = first qubit is |1> grover_op = grover_operator(oracle, insert_barriers=True) grover_op.draw("mpl") We can also modify the state preparation: .. plot:: :alt: Circuit diagram output by the previous code. :include-source: :context: close-figs oracle = QuantumCircuit(1) oracle.z(0) # the qubit state |1> is the good state state_preparation = QuantumCircuit(1) state_preparation.ry(0.2, 0) # non-uniform state preparation grover_op = grover_operator(oracle, state_preparation) grover_op.draw("mpl") In addition, we can also mark which qubits the zero reflection should act on. This is useful in case that some qubits are just used as scratch space but should not affect the oracle: .. plot:: :alt: Circuit diagram output by the previous code. :include-source: :context: close-figs oracle = QuantumCircuit(4) oracle.z(3) reflection_qubits = [0, 3] state_preparation = QuantumCircuit(4) state_preparation.cry(0.1, 0, 3) state_preparation.ry(0.5, 3) grover_op = grover_operator(oracle, state_preparation, reflection_qubits=reflection_qubits) grover_op.draw("mpl") The oracle and the zero reflection can also be passed as :mod:`qiskit.quantum_info` objects: .. plot:: :alt: Circuit diagram output by the previous code. :include-source: :context: close-figs from qiskit.quantum_info import Statevector, DensityMatrix, Operator mark_state = Statevector.from_label("011") reflection = 2 * DensityMatrix.from_label("000") - Operator.from_label("III") grover_op = grover_operator(oracle=mark_state, zero_reflection=reflection) grover_op.draw("mpl") For a large number of qubits, the multi-controlled X gate used for the zero-reflection can be synthesized in different fashions. Depending on the number of available qubits, the compiler will choose a different implementation: .. code-block:: python from qiskit import transpile, Qubit from qiskit.circuit import QuantumCircuit from qiskit.circuit.library import grover_operator oracle = QuantumCircuit(10) oracle.z(oracle.qubits) grover_op = grover_operator(oracle) # without extra qubit space, the MCX synthesis is expensive basis_gates = ["u", "cx"] tqc = transpile(grover_op, basis_gates=basis_gates) is_2q = lambda inst: len(inst.qubits) == 2 print("2q depth w/o scratch qubits:", tqc.depth(filter_function=is_2q)) # > 350 # add extra bits that can be used as scratch space grover_op.add_bits([Qubit() for _ in range(num_qubits)]) tqc = transpile(grover_op, basis_gates=basis_gates) print("2q depth w/ scratch qubits:", tqc.depth(filter_function=is_2q)) # < 100 Args: oracle: The phase oracle implementing a reflection about the bad state. Note that this is not a bitflip oracle, see the docstring for more information. state_preparation: The operator preparing the good and bad state. For Grover's algorithm, this is a n-qubit Hadamard gate and for amplitude amplification or estimation the operator :math:`\mathcal{A}`. zero_reflection: The reflection about the zero state, :math:`\mathcal{S}_0`. reflection_qubits: Qubits on which the zero reflection acts on. insert_barriers: Whether barriers should be inserted between the reflections and A. name: The name of the circuit. References: [1] L. K. Grover (1996), A fast quantum mechanical algorithm for database search, `arXiv:quant-ph/9605043 `_. [2] I. Chuang & M. Nielsen, Quantum Computation and Quantum Information, Cambridge: Cambridge University Press, 2000. Chapter 6.1.2. [3] Brassard, G., Hoyer, P., Mosca, M., & Tapp, A. (2000). Quantum Amplitude Amplification and Estimation. `arXiv:quant-ph/0005055 `_. Údrop)ÚnameÚ vars_mode©réÿÿÿÿT©Úinplacerr)Ú isinstancerÚcopy_empty_likeÚ num_qubitsr rÚdataÚappendÚqubitsÚcomposeÚbarrierÚ enumerater ÚhÚinverseÚlenÚxÚzrr rÚdiagonalÚnumpyÚpiÚ global_phase) ÚoracleÚstate_preparationÚzero_reflectionÚreflection_qubitsÚinsert_barriersrÚcircuitr)ÚiÚqubitÚnum_reflectionÚmcxs Ú`/home/james-whalen/.local/lib/python3.13/site-packages/qiskit/circuit/library/grover_operator.pyÚgrover_operatorr8s!€ôx&œ.×)Ñ)Ø×(Ñ(¨d¸fÐ(ÐE‰ä  ×!2Ñ!2¸Ñ>ˆô &œ+×&Ñ&Ü ¨¯ © Ñ 3Ó4ˆØ‰x§¡Õ0à‰˜¨ˆÑ-æ؉Ôð Ñ ä'¨¯©Ô7ô Ú7‘(!¼zÈ%ÔQ]×?^AÑ7ð ñ ðÑ Ø ‰ Ð#Õ$à‰Ð)×1Ñ1Ó3¸TˆÑBæ؉ÔðÑÜÐ.Ó/ˆà ‰ Ð#Ô$Ø ˜QÓ Ø I‰IØ! !Ñ$õ ô˜.¨1Ñ,Ó-ˆCà I‰IÐ'¨Ñ+Ô ,Ø N‰N˜3Ô 2Ø I‰IÐ'¨Ñ+Ô ,Ø ‰ Ð#Õ$ä O¤h´ Ð%>× ?Ñ ?Ü × 4Ñ 4× =Ñ =Ó ?Ó@ˆØ‰x§¡Õ0𠉘°ˆÑ6æ؉ÔðÑ Ø ‰ Ð#Õ$à‰Ð)°4ˆÑ8ô!Ÿ8™8€GÔà €Nùóa s ÃJÃ%JcóÂ^•\rSrSrSr\"SSSS9SSU4Sjjj5r\S5r\SS j5r \SS j5r \S 5r S r S r U=r$)ÚGroverOperatori u.&The Grover operator. Grover's search algorithm [1, 2] consists of repeated applications of the so-called Grover operator used to amplify the amplitudes of the desired output states. This operator, :math:`\mathcal{Q}`, consists of the phase oracle, :math:`\mathcal{S}_f`, zero phase-shift or zero reflection, :math:`\mathcal{S}_0`, and an input state preparation :math:`\mathcal{A}`: .. math:: \mathcal{Q} = \mathcal{A} \mathcal{S}_0 \mathcal{A}^\dagger \mathcal{S}_f In the standard Grover search we have :math:`\mathcal{A} = H^{\otimes n}`: .. math:: \mathcal{Q} = H^{\otimes n} \mathcal{S}_0 H^{\otimes n} \mathcal{S}_f = D \mathcal{S_f} The operation :math:`D = H^{\otimes n} \mathcal{S}_0 H^{\otimes n}` is also referred to as diffusion operator. In this formulation we can see that Grover's operator consists of two steps: first, the phase oracle multiplies the good states by -1 (with :math:`\mathcal{S}_f`) and then the whole state is reflected around the mean (with :math:`D`). This class allows setting a different state preparation, as in quantum amplitude amplification (a generalization of Grover's algorithm), :math:`\mathcal{A}` might not be a layer of Hardamard gates [3]. The action of the phase oracle :math:`\mathcal{S}_f` is defined as .. math:: \mathcal{S}_f: |x\rangle \mapsto (-1)^{f(x)}|x\rangle where :math:`f(x) = 1` if :math:`x` is a good state and 0 otherwise. To highlight the fact that this oracle flips the phase of the good states and does not flip the state of a result qubit, we call :math:`\mathcal{S}_f` a phase oracle. Note that you can easily construct a phase oracle from a bitflip oracle by sandwiching the controlled X gate on the result qubit by a X and H gate. For instance .. code-block:: text Bitflip oracle Phaseflip oracle q_0: ──■── q_0: ────────────■──────────── ┌─┴─┠┌───â”┌───â”┌─┴─â”┌───â”┌───┠out: ┤ X ├ out: ┤ X ├┤ H ├┤ X ├┤ H ├┤ X ├ └───┘ └───┘└───┘└───┘└───┘└───┘ There is some flexibility in defining the oracle and :math:`\mathcal{A}` operator. Before the Grover operator is applied in Grover's algorithm, the qubits are first prepared with one application of the :math:`\mathcal{A}` operator (or Hadamard gates in the standard formulation). Thus, we always have operation of the form :math:`\mathcal{A} \mathcal{S}_f \mathcal{A}^\dagger`. Therefore it is possible to move bitflip logic into :math:`\mathcal{A}` and leaving the oracle only to do phaseflips via Z gates based on the bitflips. One possible use-case for this are oracles that do not uncompute the state qubits. The zero reflection :math:`\mathcal{S}_0` is usually defined as .. math:: \mathcal{S}_0 = 2 |0\rangle^{\otimes n} \langle 0|^{\otimes n} - \mathbb{I}_n where :math:`\mathbb{I}_n` is the identity on :math:`n` qubits. By default, this class implements the negative version :math:`2 |0\rangle^{\otimes n} \langle 0|^{\otimes n} - \mathbb{I}_n`, since this can simply be implemented with a multi-controlled Z sandwiched by X gates on the target qubit and the introduced global phase does not matter for Grover's algorithm. Examples: >>> from qiskit.circuit import QuantumCircuit >>> from qiskit.circuit.library import GroverOperator >>> oracle = QuantumCircuit(2) >>> oracle.z(0) # good state = first qubit is |1> >>> grover_op = GroverOperator(oracle, insert_barriers=True) >>> grover_op.decompose().draw() ┌───┠░ ┌───┠░ ┌───┠┌───┠░ ┌───┠state_0: ┤ Z ├─░─┤ H ├─░─┤ X ├───────■──┤ X ├──────░─┤ H ├ └───┘ â–‘ ├───┤ â–‘ ├───┤┌───â”┌─┴─â”├───┤┌───┠░ ├───┤ state_1: ──────░─┤ H ├─░─┤ X ├┤ H ├┤ X ├┤ H ├┤ X ├─░─┤ H ├ â–‘ └───┘ â–‘ └───┘└───┘└───┘└───┘└───┘ â–‘ └───┘ >>> oracle = QuantumCircuit(1) >>> oracle.z(0) # the qubit state |1> is the good state >>> state_preparation = QuantumCircuit(1) >>> state_preparation.ry(0.2, 0) # non-uniform state preparation >>> grover_op = GroverOperator(oracle, state_preparation) >>> grover_op.decompose().draw() ┌───â”┌──────────â”┌───â”┌───â”┌───â”┌─────────┠state_0: ┤ Z ├┤ RY(-0.2) ├┤ X ├┤ Z ├┤ X ├┤ RY(0.2) ├ └───┘└──────────┘└───┘└───┘└───┘└─────────┘ >>> oracle = QuantumCircuit(4) >>> oracle.z(3) >>> reflection_qubits = [0, 3] >>> state_preparation = QuantumCircuit(4) >>> state_preparation.cry(0.1, 0, 3) >>> state_preparation.ry(0.5, 3) >>> grover_op = GroverOperator(oracle, state_preparation, ... reflection_qubits=reflection_qubits) >>> grover_op.decompose().draw() ┌───┠┌───┠state_0: ──────────────────────■──────┤ X ├───────■──┤ X ├──────────■──────────────── │ └───┘ │ └───┘ │ state_1: ──────────────────────┼──────────────────┼─────────────────┼──────────────── │ │ │ state_2: ──────────────────────┼──────────────────┼─────────────────┼──────────────── ┌───â”┌──────────â”┌────┴─────â”┌───â”┌───â”┌─┴─â”┌───â”┌───â”┌────┴────â”┌─────────┠state_3: ┤ Z ├┤ RY(-0.5) ├┤ RY(-0.1) ├┤ X ├┤ H ├┤ X ├┤ H ├┤ X ├┤ RY(0.1) ├┤ RY(0.5) ├ └───┘└──────────┘└──────────┘└───┘└───┘└───┘└───┘└───┘└─────────┘└─────────┘ >>> mark_state = Statevector.from_label('011') >>> diffuse_operator = 2 * DensityMatrix.from_label('000') - Operator.from_label('III') >>> grover_op = GroverOperator(oracle=mark_state, zero_reflection=diffuse_operator) >>> grover_op.decompose().draw(fold=70) ┌─────────────────┠┌───┠» state_0: ┤0 ├──────┤ H ├──────────────────────────» │ │┌─────┴───┴─────┠┌───┠» state_1: ┤1 UCRZ(0,pi,0,0) ├┤0 ├─────┤ H ├──────────» │ ││ UCRZ(pi/2,0) │┌────┴───┴────â”┌───â”» state_2: ┤2 ├┤1 ├┤ UCRZ(-pi/4) ├┤ H ï └─────────────────┘└───────────────┘└─────────────┘└───┘» « ┌─────────────────┠┌───┠«state_0: ┤0 ├──────┤ H ├───────────────────────── « │ │┌─────┴───┴─────┠┌───┠«state_1: ┤1 UCRZ(pi,0,0,0) ├┤0 ├────┤ H ├────────── « │ ││ UCRZ(pi/2,0) │┌───┴───┴────â”┌───┠«state_2: ┤2 ├┤1 ├┤ UCRZ(pi/4) ├┤ H ├ « └─────────────────┘└───────────────┘└────────────┘└───┘ .. seealso:: The :func:`.grover_operator` implements the same functionality but keeping the :class:`.MCXGate` abstract, such that the compiler may choose the optimal decomposition. We recommend using :func:`.grover_operator` for performance reasons, which does not wrap the circuit into an opaque gate. References: [1] L. K. Grover (1996), A fast quantum mechanical algorithm for database search, `arXiv:quant-ph/9605043 `_. [2] I. Chuang & M. Nielsen, Quantum Computation and Quantum Information, Cambridge: Cambridge University Press, 2000. Chapter 6.1.2. [3] Brassard, G., Hoyer, P., Mosca, M., & Tapp, A. (2000). Quantum Amplitude Amplification and Estimation. `arXiv:quant-ph/0005055 `_. z2.1z3Use qiskit.circuit.library.grover_operator instead.z in Qiskit 3.0)ÚsinceÚadditional_msgÚremoval_timelinecój>•[T U]US9 [U[5(aSSKJn U"SUR -5nXl[U[[45(a&SSKJn U"UR R55nX0l X@l X l XPlX`lUR!5 g)aá Args: oracle: The phase oracle implementing a reflection about the bad state. Note that this is not a bitflip oracle, see the docstring for more information. state_preparation: The operator preparing the good and bad state. For Grover's algorithm, this is a n-qubit Hadamard gate and for amplitude amplification or estimation the operator :math:`\mathcal{A}`. zero_reflection: The reflection about the zero state, :math:`\mathcal{S}_0`. reflection_qubits: Qubits on which the zero reflection acts on. insert_barriers: Whether barriers should be inserted between the reflections and A. mcx_mode: The mode to use for building the default zero reflection. name: The name of the circuit. rr)ÚDiagonalrN)ÚsuperÚ__init__rr Úqiskit.circuit.libraryr?rÚ_oracler rr)Ú_zero_reflectionÚ_reflection_qubitsÚ_state_preparationÚ_insert_barriersÚ _mcx_modeÚ_build) Úselfr-r.r/r0r1Úmcx_moderr?Ú __class__s €r7rAÚGroverOperator.__init__´s•ø€ô8 ‰Ñ˜dÐÑ#ô fœk× *Ñ *Ý 7á˜r f§k¡kÑ1Ó2ˆFØŒ ä o¬´-Ð'@× AÑ AÝ 7á& ×';Ñ';×'DÑ'DÓ'FÓGˆOØ /Ôà"3ÔØ"3ÔØ /ÔØ!Œð ‰ ócó¶•URb UR$URRURR- n[ [ U55$)zHReflection qubits, on which S0 is applied (if S0 is not user-specified).)rEr-rÚ num_ancillasÚlistÚrange©rJÚnum_state_qubitss r7r0Ú GroverOperator.reflection_qubitsçsL€ð × "Ñ "Ñ .Ø×*Ñ*Ð *àŸ;™;×1Ñ1°D·K±K×4LÑ4LÑLÐÜ”EÐ*Ó+Ó,Ð,rNcóΕURb UR$URRURR- n[XRUR 5$)z3The subcircuit implementing the reflection about 0.)rDr-rrPr0rHrSs r7r/ÚGroverOperator.zero_reflectionðsS€ð × Ñ Ñ ,Ø×(Ñ(Ð (àŸ;™;×1Ñ1°D·K±K×4LÑ4LÑLÐÜÐ 0×2HÑ2HÈ$Ï.É.ÓYÐYrNcóÜ•URb UR$URRURR- n[ USS9nUR UR 5 U$)z8The subcircuit implementing the A operator or Hadamards.ÚHr)rFr-rrPrr$r0)rJrTÚ hadamardss r7r.Ú GroverOperator.state_preparationùs`€ð × "Ñ "Ñ .Ø×*Ñ*Ð *àŸ;™;×1Ñ1°D·K±K×4LÑ4LÑLÐÜ"Ð#3¸#Ñ>ˆ à ‰ D×*Ñ*Ô+ØÐrNcó•UR$)z9The oracle implementing a reflection about the bad state.)rC)rJs r7r-ÚGroverOperator.oracles€ð|‰|ÐrNcóF•URRURR- n[[ USS9SS9n[ R "URRURRURR/5nUS:”aUR[USS95 URUR[[URR55SS9 UR(aUR5 URURR!5[[URR55SS9 UR(aUR5 URUR[[URR55SS9 UR(aUR5 URUR[[URR55SS9 [ R"UlUR"UR&6 UR)5nURX@R.SS9 g![*a UR-5nN7f=f) NÚstaterÚQrÚancillaTr)r r)r-rrPrrr*Úmaxr/r.Ú add_registerr r!rQrRrGr"r%r+r,ÚqregsÚto_gater Úto_instructionr )rJrTr2rPÚcircuit_wrappeds r7rIÚGroverOperator._build s €ØŸ;™;×1Ñ1°D·K±K×4LÑ4LÑLÐÜ ¤Ð1AÈÑ!PÐWZÑ[ˆÜ—y’yà— ‘ ×(Ñ(Ø×$Ñ$×1Ñ1Ø×&Ñ&×3Ñ3ð ó ˆ ð ˜!Ó Ø × Ñ ¤°ÀIÑ!NÔ Oà‰˜Ÿ ™ ¤T¬%°· ± ×0FÑ0FÓ*GÓ%HÐRVˆÑWØ × × Ø O‰OÔ Ø‰Ø × "Ñ "× *Ñ *Ó ,Ü ”t×-Ñ-×8Ñ8Ó9Ó :Øð ñ ð × × Ø O‰OÔ Ø‰Ø × Ñ ¤$¤u¨T×-AÑ-A×-LÑ-LÓ'MÓ"NÐX\ð ñ ð × × Ø O‰OÔ Ø‰Ø × "Ñ "¤D¬¨t×/EÑ/E×/PÑ/PÓ)QÓ$RÐ\`ð ñ ô %Ÿx™xˆÔà ×Ò˜7Ÿ=™=Ñ)ð 7Ø%Ÿo™oÓ/ˆOð ‰ _¯[©[À$ˆ ÒGøôó 7Ø%×4Ñ4Ó6ŠOð 7úsÉJÊJ ÊJ )rGrHrCrErFrD)NNNFÚ noancillar`)r-z"Union[QuantumCircuit, Statevector]r.zOptional[QuantumCircuit]r/z8Optional[Union[QuantumCircuit, DensityMatrix, Operator]]r0zOptional[List[int]]r1ÚboolrKÚstrrrkÚreturnÚNone)rlr)Ú__name__Ú __module__Ú __qualname__Ú__firstlineno__Ú__doc__rrAÚpropertyr0r/r.r-rIÚ__static_attributes__Ú __classcell__)rLs@r7r:r: sîø†ñQñfØØLØ(ñð7;ØTXØ15Ø %Ø#Øð,à2ð,ð4ð,ðRð ,ð /ð ,ð ð ,ðð,ðð,ð ÷,ó ð ,ð\ñ-óð-ðóZóðZðó óð ðñóð÷)Hð)HrNr:cóâ•[US5n[USS9n[R"[ U5S- U5nUS:”a[ US5nUR U5 O [ S5nURU5 [ U5S:XaURS5 OCURUS5 URUSSUSUSSUS9 URUS5 URU5 U$) Nr_ÚS_0rrrrar)Úmode) rrrÚget_num_ancilla_qubitsr&r rcr'r(r$r6)rTr rKÚqr_stateÚ reflectionrPÚ qr_ancillas r7rDrD7sÚ€ôÐ/°Ó9€HÜ ¨uÑ5€Jä×1Ò1´#°f³+À±/À8ÓL€LØaÓÜ$ \°9Ó=ˆ Ø×Ñ  Õ+ä$ QÓ'ˆ à‡LLÔÜ ˆ6ƒ{aÓØ ‰ Qà ‰ V˜B‘ZÔ Ø‰v˜c˜r{ F¨2¡J° ¹1° ÀHˆÑMØ ‰ V˜B‘ZÔ Ø‡LLÔà ÐrN)NNNFr`) r-zQuantumCircuit | Statevectorr.zQuantumCircuit | Noner/z0QuantumCircuit | DensityMatrix | Operator | Noner0zlist[int] | Noner1rjrrk)N)rTÚintr z List[int]rKz Optional[str]rlr)rrÚ __future__rÚtypingrrrr*Úqiskit.circuitrrr r Úqiskit.exceptionsr Úqiskit.quantum_infor r rÚqiskit.utils.deprecationrÚstandard_gatesrÚgeneralized_gatesrr8r:rD©rNr7Úr‡sÂðñå"ß(Ñ(Û çYÓYÝ)ßDÑDÝ3Ý#Ý+ð 04ØHLØ*.Ø!Øð BØ (ðBà,ðBðFðBð(ð Bð ð Bð õ BôJSH^ôSHðpIMðØðØ#,ðØ8EðàörN