§ ¹³Ùg6oãóÊ—dZddlmZddlmZmZmZddlZddlm Z m Z m Z m Z ddl mZddlmZmZmZddlmZd d lmZd d lmZ d&d'd„ZGd„de ¦«Z d(d)d%„ZdS)*zThe Grover operator.é)Ú annotations)ÚListÚOptionalÚUnionN)ÚQuantumCircuitÚQuantumRegisterÚAncillaRegisterÚ AncillaQubit)Ú QiskitError)Ú StatevectorÚOperatorÚ DensityMatrix)Údeprecate_funcé)ÚMCXGate)Ú DiagonalGateFÚQÚoracleúQuantumCircuit | StatevectorÚstate_preparationúQuantumCircuit | NoneÚzero_reflectionú0QuantumCircuit | DensityMatrix | Operator | NoneÚreflection_qubitsúlist[int] | NoneÚinsert_barriersÚboolÚnameÚstrcór—t|t¦«r| |d¬¦«}nt|j|¬¦«}t|t¦«r3t d|jz¦«}| ||j¦«n|  |d¬¦«|r|  ¦«|€d„t|j¦«D¦«}|€|  |¦«n)|  |  ¦«d¬¦«|r|  ¦«|€ºt|¦«}| |¦«|d kr| |d ¦«n^t#|d z ¦«} |  |d¦«| | |¦«|  |d¦«| |¦«nut|t$t&f¦«rBt |j ¦«¦«}| ||j¦«n|  |d¬¦«|r|  ¦«|€|  |¦«n|  |d¬¦«t*j|_|S) uå 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 `_. 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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 `_. z1.3z3Use qiskit.circuit.library.grover_operator instead.T)ÚsinceÚadditional_msgÚpendingNFÚ noancillarrú"Union[QuantumCircuit, Statevector]rúOptional[QuantumCircuit]rú8Optional[Union[QuantumCircuit, DensityMatrix, Operator]]rúOptional[List[int]]rrÚmcx_moderrÚreturnÚNonec󬕗t¦« |¬¦«t|t¦«rddlm}|d|jz¦«}||_t|ttf¦«r(ddlm}||j  ¦«¦«}||_ ||_ ||_ ||_||_| ¦«dS)aA 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. r#r)ÚDiagonalr$N)ÚsuperÚ__init__r)r Úqiskit.circuit.libraryrSr2Ú_oracler rr=Ú_zero_reflectionÚ_reflection_qubitsÚ_state_preparationÚ_insert_barriersÚ _mcx_modeÚ_build) ÚselfrrrrrrOrrSÚ __class__s €r-rUzGroverOperator.__init__®såø€õ8 ‰Œ×Ò˜dÐÑ#Ô#Ð#õ fkÑ *Ô *ð 3Ø 7Ð 7Ð 7Ð 7Ð 7Ð 7àX˜r f¤kÑ1Ñ2Ô2ˆF؈Œ å o­µ-Ð'@Ñ AÔ Að HØ 7Ð 7Ð 7Ð 7Ð 7Ð 7à&˜h Ô';×'DÒ'DÑ'FÔ'FÑGÔGˆOØ /ˆÔà"3ˆÔØ"3ˆÔØ /ˆÔØ!ˆŒð Š ‰ Œ ˆ ˆ ˆ r/cóˆ—|j|jS|jj|jjz }t t |¦«¦«S)zHReflection qubits, on which S0 is applied (if S0 is not user-specified).)rYrr1Ú num_ancillasÚlistÚrange©r^Únum_state_qubitss r-rz GroverOperator.reflection_qubitsásB€ð Ô "Ð .ØÔ*Ð *àœ;Ô1°D´KÔ4LÑLÐÝ•EÐ*Ñ+Ô+Ñ,Ô,Ð,r/rcó†—|j|jS|jj|jjz }t||j|j¦«S)z3The subcircuit implementing the reflection about 0.)rXrr1rarr\rds r-rzGroverOperator.zero_reflectionêsC€ð Ô Ð ,ØÔ(Ð (àœ;Ô1°D´KÔ4LÑLÐÝÐ 0°$Ô2HÈ$Ì.ÑYÔYÐYr/cóª—|j|jS|jj|jjz }t |d¬¦«}| |j¦«|S)z8The subcircuit implementing the A operator or Hadamards.NÚHr#)rZrr1rarr8r)r^reÚ hadamardss r-rz GroverOperator.state_preparationósY€ð Ô "Ð .ØÔ*Ð *àœ;Ô1°D´KÔ4LÑLÐÝ"Ð#3¸#Ð>Ñ>Ô>ˆ à Š DÔ*Ñ+Ô+Ð+ØÐr/có—|jS)z9The oracle implementing a reflection about the bad state.)rW)r^s r-rzGroverOperator.oracleÿs €ðŒ|Ðr/cóò—|jj|jjz }tt |d¬¦«d¬¦«}t j|jj|jj|jjg¦«}|dkr$|  t|d¬¦«¦«|  |jtt|jj¦«¦«d¬¦«|jr| ¦«|  |j ¦«tt|jj¦«¦«d¬¦«|jr| ¦«|  |jtt|jj¦«¦«d¬¦«|jr| ¦«|  |jtt|jj¦«¦«d¬¦«t j|_|j |jŽ | ¦«}n$#t*$r| ¦«}YnwxYw|  ||jd¬¦«dS) NÚstater#rrÚancillaTr%)r4r&)rr1rarrr>ÚmaxrrÚ add_registerr r5rbrcr[r6r9r?r@ÚqregsÚto_gater Úto_instructionr4)r^rerAraÚcircuit_wrappeds r-r]zGroverOperator._buildsb€Øœ;Ô1°D´KÔ4LÑLÐÝ ¥Ð1AÈÐ!PÑ!PÔ!PÐWZÐ[Ñ[Ô[ˆÝ”yà” Ô(ØÔ$Ô1ØÔ&Ô3ð ñ ô ˆ ð ˜!Ò Ð Ø × Ò ¥°ÀIÐ!NÑ!NÔ!NÑ OÔ OÐ OàŠ˜œ ¥T­%°´ Ô0FÑ*GÔ*GÑ%HÔ%HÐRVˆÑWÔWÐWØ Ô ð Ø OŠOÑ Ô Ð ØŠØ Ô "× *Ò *Ñ ,Ô ,Ý •tÔ-Ô8Ñ9Ô9Ñ :Ô :Øð ñ ô ð ð Ô ð Ø OŠOÑ Ô Ð ØŠØ Ô ¥$¥u¨TÔ-AÔ-LÑ'MÔ'MÑ"NÔ"NÐX\ð ñ ô ð ð Ô ð Ø OŠOÑ Ô Ð ØŠØ Ô "¥D­¨tÔ/EÔ/PÑ)QÔ)QÑ$RÔ$RÐ\`ð ñ ô ð õ %œxˆÔàˆÔ˜7œ=Ð)Ð)ð 7Ø%ŸošoÑ/Ô/ˆOˆOøÝð 7ð 7ð 7Ø%×4Ò4Ñ6Ô6ˆOˆOˆOð 7øøøð Š _¨T¬[À$ˆ ÑGÔGÐGÐGÐGsÈ!H6È6IÉI)NNNFrJr)rrKrrLrrMrrNrrrOrrrrPrQ)rPr) Ú__name__Ú __module__Ú __qualname__Ú__doc__rrUÚpropertyrrrrr]Ú __classcell__)r_s@r-rFrFsø€€€€€ðNðNð`€^ØØLØðñôð7;ØTXØ15Ø %Ø#Øð,ð,ð,ð,ð,ð,ñ ôð ,ð\ð-ð-ñ„Xð-ððZðZðZñ„XðZðð ð ð ñ„Xð ðððñ„Xðð)Hð)Hð)Hð)Hð)Hð)Hð)Hr/rFreÚintr4ú List[int]rOú Optional[str]rPrcóx—t|d¦«}t|d¬¦«}tjt |¦«dz |¦«}|dkr&t |d¦«}| |¦«nt d¦«}| |¦«t |¦«dkr| d¦«ne|  |d¦«|  |dd…|d|dd…|¬¦«|  |d¦«| |¦«|S) NrlÚS_0r#rrrmr$)Úmode) rrrÚget_num_ancilla_qubitsr:r ror;r<r8rC)rer4rOÚqr_stateÚ reflectionraÚ qr_ancillas r-rXrX1s&€õÐ/°Ñ9Ô9€HÝ ¨uÐ5Ñ5Ô5€JåÔ1µ#°f±+´+À±/À8ÑLÔL€LØaÒÐÝ$ \°9Ñ=Ô=ˆ Ø×Ò  Ñ+Ô+Ð+Ð+å$ QÑ'Ô'ˆ à‡L‚LÑÔÐÝ ˆ6{„{aÒÐØ Š Q‰Œˆˆà Š V˜B”ZÑ Ô Ð ØŠv˜c˜r˜c”{ F¨2¤J° ¸1¸1¸1´ ÀHˆÑMÔMÐMØ Š V˜B”ZÑ Ô Ð Ø‡L‚LÑÔÐà Ðr/)NNNFr) rrrrrrrrrrrr)N)rerzr4r{rOr|rPr)rwÚ __future__rÚtypingrrrr>Úqiskit.circuitrrr r Úqiskit.exceptionsr Úqiskit.quantum_infor r rÚqiskit.utils.deprecationrÚstandard_gatesrÚgeneralized_gatesrrDrFrXr(r/r-úrŒseððÐà"Ð"Ð"Ð"Ð"Ð"Ø(Ð(Ð(Ð(Ð(Ð(Ð(Ð(Ð(Ð(Ø € € € àYÐYÐYÐYÐYÐYÐYÐYÐYÐYÐYÐYØ)Ð)Ð)Ð)Ð)Ð)ØDÐDÐDÐDÐDÐDÐDÐDÐDÐDØ3Ð3Ð3Ð3Ð3Ð3Ø#Ð#Ð#Ð#Ð#Ð#Ø+Ð+Ð+Ð+Ð+Ð+ð 04ØHLØ*.Ø!Øð ðððððDPHðPHðPHðPHðPH^ñPHôPHðPHðjIMðððððððr/