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GroverOperator

GroverOperator(oracle, state_preparation=None, zero_reflection=None, reflection_qubits=None, insert_barriers=False, mcx_mode='noancilla', name='Q') GitHub(opens in a new tab)

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, Q\mathcal{Q}, consists of the phase oracle, Sf\mathcal{S}_f, zero phase-shift or zero reflection, S0\mathcal{S}_0, and an input state preparation A\mathcal{A}:

Q=AS0ASf\mathcal{Q} = \mathcal{A} \mathcal{S}_0 \mathcal{A}^\dagger \mathcal{S}_f

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

Q=HnS0HnSf=DSf\mathcal{Q} = H^{\otimes n} \mathcal{S}_0 H^{\otimes n} \mathcal{S}_f = D \mathcal{S_f}

The operation D=HnS0HnD = 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 Sf\mathcal{S}_f) and then the whole state is reflected around the mean (with DD).

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

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

Sf:x(1)f(x)x\mathcal{S}_f: |x\rangle \mapsto (-1)^{f(x)}|x\rangle

where f(x)=1f(x) = 1 if xx 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 Sf\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 A\mathcal{A} operator. Before the Grover operator is applied in Grover’s algorithm, the qubits are first prepared with one application of the A\mathcal{A} operator (or Hadamard gates in the standard formulation). Thus, we always have operation of the form ASfA\mathcal{A} \mathcal{S}_f \mathcal{A}^\dagger. Therefore it is possible to move bitflip logic into A\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 S0\mathcal{S}_0 is usually defined as

S0=20n0nIn\mathcal{S}_0 = 2 |0\rangle^{\otimes n} \langle 0|^{\otimes n} - \mathbb{I}_n

where In\mathbb{I}_n is the identity on nn qubits. By default, this class implements the negative version 20n0nIn2 |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                ô1              ô UCRZ(-pi/4) ô H ï
         └─────────────────┘└───────────────┘└─────────────┘└───┘»
«         ┌─────────────────┐      ┌───┐
«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(opens in a new tab).

[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(opens in a new tab).

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 A\mathcal{A}.
  • zero_reflection (Union[QuantumCircuit, DensityMatrix, Operator, None]) – The reflection about the zero state, S0\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

Returns a list of ancilla bits in the order that the registers were added.

Return type

List[AncillaQubit]

calibrations

Return calibration dictionary.

The custom pulse definition of a given gate is of the form

{‘gate_name’: {(qubits, params): schedule}}

Return type

dict

clbits

Returns a list of classical bits in the order that the registers were added.

Return type

List[Clbit]

data

Return the circuit data (instructions and context).

Returns

a list-like object containing the tuples for the circuit’s data.

Each tuple is in the format (instruction, qargs, cargs), where instruction is an Instruction (or subclass) object, qargs is a list of Qubit objects, and cargs is a list of Clbit objects.

Return type

QuantumCircuitData

extension_lib

= 'include "qelib1.inc";'

global_phase

Return the global phase of the circuit in radians.

Return type

Union[ParameterExpression, float]

= 'OPENQASM 2.0;'

instances

= 9

metadata

The user provided metadata associated with the circuit

The metadata for the circuit is a user provided dict of metadata for the circuit. It will not be used to influence the execution or operation of the circuit, but it is expected to be passed between all transforms of the circuit (ie transpilation) and that providers will associate any circuit metadata with the results it returns from execution of that circuit.

Return type

dict

num_ancillas

Return the number of ancilla qubits.

Return type

int

num_clbits

Return number of classical bits.

Return type

int

num_parameters

Convenience function to get the number of parameter objects in the circuit.

Return type

int

num_qubits

Return number of qubits.

Return type

int

oracle

The oracle implementing a reflection about the bad state.

parameters

Convenience function to get the parameters defined in the parameter table.

Return type

ParameterView

prefix

= 'circuit'

qubits

Returns a list of quantum bits in the order that the registers were added.

Return type

List[Qubit]

reflection_qubits

Reflection qubits, on which S0 is applied (if S0 is not user-specified).

state_preparation

The subcircuit implementing the A operator or Hadamards.

Return type

QuantumCircuit

zero_reflection

The subcircuit implementing the reflection about 0.

Return type

QuantumCircuit

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