GroverOperator

class GroverOperator(oracle, state_preparation=None, zero_reflection=None, reflection_qubits=None, insert_barriers=False, mcx_mode='noancilla', name='Q')

Bases: qiskit.circuit.quantumcircuit.QuantumCircuit

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, $\mathcal{Q}$, consists of the phase oracle, $\mathcal{S}_f$, zero phase-shift or zero reflection, $\mathcal{S}_0$, and an input state preparation $\mathcal{A}$:

In the standard Grover search we have $\mathcal{A} = H^{\otimes n}$:

The operation $D = H^{\otimes n} \mathcal{S}_0 H^{\otimes n}$ 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 $\mathcal{S}_f$) and then the whole state is reflected around the mean (with $D$).

This class allows setting a different state preparation, as in quantum amplitude amplification (a generalization of Grover’s algorithm), $\mathcal{A}$ might not be a layer of Hardamard gates [3].

The action of the phase oracle $\mathcal{S}_f$ is defined as

where $f(x) = 1$ if $x$ 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 $\mathcal{S}_f$ 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

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 $\mathcal{A}$ operator. Before the Grover operator is applied in Grover’s algorithm, the qubits are first prepared with one application of the $\mathcal{A}$ operator (or Hadamard gates in the standard formulation). Thus, we always have operation of the form $\mathcal{A} \mathcal{S}_f \mathcal{A}^\dagger$. Therefore it is possible to move bitflip logic into $\mathcal{A}$ 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 $\mathcal{S}_0$ is usually defined as

where $\mathbb{I}_n$ is the identity on $n$ qubits. By default, this class implements the negative version $2 |0\rangle^{\otimes n} \langle 0|^{\otimes n} - \mathbb{I}_n$, 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 ├
         └───┘ ░ ├───┤ ░ ├───┤┌───┐┌─┴─┐├───┤┌───┐ ░ ├───┤
state_1: ──────░─┤ H ├─░─┤ X ├┤ H ├┤ X ├┤ H ├┤ X ├─░─┤ H ├
               ░ └───┘ ░ └───┘└───┘└───┘└───┘└───┘ ░ └───┘

>>> 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()
         ┌───┐┌──────────┐┌───┐┌───┐┌───┐┌─────────┐
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: ──────────────────────┼──────────────────┼─────────────────┼────────────────
         ┌───┐┌──────────┐┌────┴─────┐┌───┐┌───┐┌─┴─┐┌───┐┌───┐┌────┴────┐┌─────────┐
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) │┌────┴───┴────┐┌───┐»
state_2: ┤2                ├┤1              ├┤ UCRZ(-pi/4) ├┤ H ├»
         └─────────────────┘└───────────────┘└─────────────┘└───┘»
«         ┌─────────────────┐      ┌───┐
«state_0: ┤0                ├──────┤ H ├─────────────────────────
«         │                 │┌─────┴───┴─────┐    ┌───┐
«state_1: ┤1 UCRZ(pi,0,0,0) ├┤0              ├────┤ H ├──────────
«         │                 ││  UCRZ(pi/2,0) │┌───┴───┴────┐┌───┐
«state_2: ┤2                ├┤1              ├┤ UCRZ(pi/4) ├┤ H ├
«         └─────────────────┘└───────────────┘└────────────┘└───┘

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.

Parameters

• oracle (Union[QuantumCircuit, Statevector]) – 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 (Optional[QuantumCircuit]) – 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 $\mathcal{A}$.
• zero_reflection (Union[QuantumCircuit, DensityMatrix, Operator, None]) – The reflection about the zero state, $\mathcal{S}_0$.
• reflection_qubits (Optional[List[int]]) – Qubits on which the zero reflection acts on.
• insert_barriers (bool) – Whether barriers should be inserted between the reflections and A.
• mcx_mode (str) – The mode to use for building the default zero reflection.
• name (str) – The name of the circuit.

Attributes

ancillas

calibrations

clbits

data

extension_lib

global_phase

header

instances

metadata

num_ancillas

num_clbits

num_parameters

num_qubits

op_start_times

oracle

parameters

>>> from qiskit.circuit import QuantumCircuit, Parameter
>>> a, b, elephant = Parameter("a"), Parameter("b"), Parameter("elephant")
>>> circuit = QuantumCircuit(1)
>>> circuit.rx(b, 0)
>>> circuit.rz(elephant, 0)
>>> circuit.ry(a, 0)
>>> circuit.parameters  # sorted alphabetically!
ParameterView([Parameter(a), Parameter(b), Parameter(elephant)])

>>> from qiskit.circuit import QuantumCircuit, Parameter
>>> angles = [Parameter("angle_1"), Parameter("angle_2"), Parameter("angle_10")]
>>> circuit = QuantumCircuit(1)
>>> circuit.u(*angles, 0)
>>> circuit.draw()
   ┌─────────────────────────────┐
q: ┤ U(angle_1,angle_2,angle_10) ├
   └─────────────────────────────┘
>>> circuit.parameters
ParameterView([Parameter(angle_1), Parameter(angle_10), Parameter(angle_2)])

>>> from qiskit.circuit import QuantumCircuit, Parameter, ParameterVector
>>> x = ParameterVector("x", 12)
>>> circuit = QuantumCircuit(1)
>>> for x_i in x:
...     circuit.rx(x_i, 0)
>>> circuit.parameters
ParameterView([
    ParameterVectorElement(x[0]), ParameterVectorElement(x[1]),
    ParameterVectorElement(x[2]), ParameterVectorElement(x[3]),
    ..., ParameterVectorElement(x[11])
])

prefix

qubits

reflection_qubits

state_preparation

zero_reflection