grover_operator

class qiskit.circuit.library.grover_operator(oracle, state_preparation=None, zero_reflection=None, reflection_qubits=None, insert_barriers=False, name='Q')

Bases:

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, $\mathcal{Q}$, 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

We can construct a Grover operator just from the phase oracle:

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:

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:

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 qiskit.quantum_info objects:

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:

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

Parameters

• oracle (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 (QuantumCircuit | None) – 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 (QuantumCircuit |DensityMatrix |Operator | None) – The reflection about the zero state, $\mathcal{S}_0$.
• reflection_qubits (list[int] | None) – Qubits on which the zero reflection acts on.
• insert_barriers (bool) – Whether barriers should be inserted between the reflections and A.
• name (str) – 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.