Add grover_operator function

qiskit/circuit/library/__init__.py

@@ -327,6 +327,8 @@
    HamiltonianGate
    UnitaryOverlap
 
+.. autofunction:: grover_operator
+
 
 N-local circuits
 ================
@@ -573,6 +575,6 @@
 from .hidden_linear_function import HiddenLinearFunction
 from .iqp import IQP
 from .phase_estimation import PhaseEstimation
-from .grover_operator import GroverOperator
+from .grover_operator import GroverOperator, grover_operator
 from .phase_oracle import PhaseOracle
 from .overlap import UnitaryOverlap

qiskit/circuit/library/grover_operator.py

@@ -16,10 +16,265 @@
 from typing import List, Optional, Union
 import numpy
 
-from qiskit.circuit import QuantumCircuit, QuantumRegister, AncillaRegister
+from qiskit.circuit import QuantumCircuit, QuantumRegister, AncillaRegister, AncillaQubit
 from qiskit.exceptions import QiskitError
 from qiskit.quantum_info import Statevector, Operator, DensityMatrix
+from qiskit.utils.deprecation import deprecate_func
 from .standard_gates import MCXGate
+from .generalized_gates import DiagonalGate
+
+
+def grover_operator(
+    oracle: QuantumCircuit | Statevector,
+    state_preparation: QuantumCircuit | None = None,
+    zero_reflection: QuantumCircuit | DensityMatrix | Operator | None = None,
+    reflection_qubits: list[int] | None = None,
+    insert_barriers: bool = False,
+    name: str = "Q",
+):
+    r"""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: ────────────■────────────
+             ┌─┴─┐              ┌───┐┌───┐┌─┴─┐┌───┐┌───┐
+        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::
+            :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::
+            :include-source:
+            :context:
+
+            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::
+            :include-source:
+            :context:
+
+            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::
+            :include-source:
+            :context:
+
+            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 fashion. 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)])
+            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 <https://arxiv.org/abs/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 <http://arxiv.org/abs/quant-ph/0005055>`_.
+    """
+    # We inherit the ancillas/qubits structure from the oracle, if it is given as circuit.
+    if isinstance(oracle, QuantumCircuit):
+        circuit = oracle.copy_empty_like(name=name, vars_mode="drop")
+    else:
+        circuit = QuantumCircuit(oracle.num_qubits, name=name)
+
+    # (1) Add the oracle.
+    # If the oracle is given as statevector, turn it into a circuit that implements the
+    # reflection about the state.
+    if isinstance(oracle, Statevector):
+        diagonal = DiagonalGate((-1) ** oracle.data)
+        circuit.append(diagonal, circuit.qubits)
+    else:
+        circuit.compose(oracle, inplace=True)
+
+    if insert_barriers:
+        circuit.barrier()
+
+    # (2) Add the inverse state preparation.
+    # For this we need to know the target qubits that we apply the zero reflection to.
+    # If the reflection qubits are not given, we assume they are the qubits that are not
+    # of type ``AncillaQubit`` in the oracle.
+    if reflection_qubits is None:
+        reflection_qubits = [
+            i for i, qubit in enumerate(circuit.qubits) if not isinstance(qubit, AncillaQubit)
+        ]
+
+    if state_preparation is None:
+        circuit.h(reflection_qubits)  # H is self-inverse
+    else:
+        circuit.compose(state_preparation.inverse(), inplace=True)
+
+    if insert_barriers:
+        circuit.barrier()
+
+    # (3) Add the zero reflection.
+    if zero_reflection is None:
+        num_reflection = len(reflection_qubits)
+
+        circuit.x(reflection_qubits)
+        if num_reflection == 1:
+            circuit.z(
+                reflection_qubits[0]
+            )  # MCX does not support 0 controls, hence this is separate
+        else:
+            mcx = MCXGate(num_reflection - 1)
+
+            circuit.h(reflection_qubits[-1])
+            circuit.append(mcx, reflection_qubits)
+            circuit.h(reflection_qubits[-1])
+        circuit.x(reflection_qubits)
+
+    elif isinstance(zero_reflection, (Operator, DensityMatrix)):
+        diagonal = DiagonalGate(zero_reflection.data.diagonal())
+        circuit.append(diagonal, circuit.qubits)
+
+    else:
+        circuit.compose(zero_reflection, inplace=True)
+
+    if insert_barriers:
+        circuit.barrier()
+
+    # (4) Add the state preparation.
+    if state_preparation is None:
+        circuit.h(reflection_qubits)
+    else:
+        circuit.compose(state_preparation, inplace=True)
+
+    # minus sign
+    circuit.global_phase = numpy.pi
+
+    return circuit
 
 
 class GroverOperator(QuantumCircuit):
@@ -150,6 +405,11 @@ class GroverOperator(QuantumCircuit):
         «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.
+
     References:
         [1]: L. K. Grover (1996), A fast quantum mechanical algorithm for database search,
             `arXiv:quant-ph/9605043 <https://arxiv.org/abs/quant-ph/9605043>`_.
@@ -160,6 +420,11 @@ class GroverOperator(QuantumCircuit):
             `arXiv:quant-ph/0005055 <http://arxiv.org/abs/quant-ph/0005055>`_.
     """
 
+    @deprecate_func(
+        since="1.3",
+        additional_msg="Use qiskit.circuit.library.grover_operator instead.",
+        pending=True,
+    )
     def __init__(
         self,
         oracle: Union[QuantumCircuit, Statevector],

releasenotes/notes/clib-grover-op-cb032144e899ed0d.yaml

@@ -0,0 +1,8 @@
+---
+features_circuits:
+  - |
+    Added :func:`.grover_operator` to construct a Grover operator circuit, used in e.g.
+    Grover's algorithm and amplitude estimation/amplification. This function is similar to
+    :class:`.GroverOperator`, but does not require choosing the implementation of the
+    multi-controlled X gate a-priori and let's the compiler choose the optimal decomposition
+    instead.