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# Grover

class Grover(iterations=None, growth_rate=None, sample_from_iterations=False, quantum_instance=None, sampler=None)

GitHub(opens in a new tab)

Grover’s Search algorithm.

Note

If you want to learn more about the theory behind Grover’s Search algorithm, check out the Qiskit Textbook(opens in a new tab). or the Qiskit Tutorials(opens in a new tab) for more concrete how-to examples.

Grover’s Search [1, 2] is a well known quantum algorithm that can be used for searching through unstructured collections of records for particular targets with quadratic speedup compared to classical algorithms.

Given a set $X$ of $N$ elements $X=\{x_1,x_2,\ldots,x_N\}$ and a boolean function $f : X \rightarrow \{0,1\}$, the goal of an unstructured-search problem is to find an element $x^* \in X$ such that $f(x^*)=1$.

The search is called unstructured because there are no guarantees as to how the database is ordered. On a sorted database, for instance, one could perform binary search to find an element in $\mathbb{O}(\log N)$ worst-case time. Instead, in an unstructured-search problem, there is no prior knowledge about the contents of the database. With classical circuits, there is no alternative but to perform a linear number of queries to find the target element. Conversely, Grover’s Search algorithm allows to solve the unstructured-search problem on a quantum computer in $\mathcal{O}(\sqrt{N})$ queries.

To carry out this search a so-called oracle is required, that flags a good element/state. The action of the oracle $\mathcal{S}_f$ is

$\mathcal{S}_f |x\rangle = (-1)^{f(x)} |x\rangle,$

i.e. it flips the phase of the state $|x\rangle$ if $x$ is a hit. The details of how $S_f$ works are unimportant to the algorithm; Grover’s search algorithm treats the oracle as a black box.

This class supports oracles in form of a QuantumCircuit.

With the given oracle, Grover’s Search constructs the Grover operator to amplify the amplitudes of the good states:

$\mathcal{Q} = H^{\otimes n} \mathcal{S}_0 H^{\otimes n} \mathcal{S}_f = D \mathcal{S}_f,$

where $\mathcal{S}_0$ flips the phase of the all-zero state and acts as identity on all other states. Sometimes the first three operands are summarized as diffusion operator, which implements a reflection over the equal superposition state.

If the number of solutions is known, we can calculate how often $\mathcal{Q}$ should be applied to find a solution with very high probability, see the method optimal_num_iterations. If the number of solutions is unknown, the algorithm tries different powers of Grover’s operator, see the iterations argument, and after each iteration checks if a good state has been measured using good_state.

The generalization of Grover’s Search, Quantum Amplitude Amplification [3], uses a modified version of $\mathcal{Q}$ where the diffusion operator does not reflect about the equal superposition state, but another state specified via an operator $\mathcal{A}$:

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

For more information, see the GroverOperator in the circuit library.

References

[1]: L. K. Grover (1996), A fast quantum mechanical algorithm for database search,

[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

• iterations (Union[List[int], Iterator[int], int, None]) – Specify the number of iterations/power of Grover’s operator to be checked. * If an int, only one circuit is run with that power of the Grover operator. If the number of solutions is known, this option should be used with the optimal power. The optimal power can be computed with Grover.optimal_num_iterations. * If a list, all the powers in the list are run in the specified order. * If an iterator, the powers yielded by the iterator are checked, until a maximum number of iterations or maximum power is reached. * If None, the AmplificationProblem provided must have an is_good_state, and circuits are run until that good state is reached.
• growth_rate (Optional[float]) – If specified, the iterator is set to increasing powers of growth_rate, i.e. to int(growth_rate ** 1), int(growth_rate ** 2), ... until a maximum number of iterations is reached.
• sample_from_iterations (bool) – If True, instead of taking the values in iterations as powers of the Grover operator, a random integer sample between 0 and smaller value than the iteration is used as a power, see [1], Section 4.
• quantum_instance (Union[Backend, QuantumInstance, None]) – Pending deprecation: A Quantum Instance or Backend to run the circuits.
• sampler (Optional[BaseSampler]) – A Sampler to use for sampling the results of the circuits.

Raises

• ValueError – If growth_rate is a float but not larger than 1.
• ValueError – If both iterations and growth_rate is set.

References

[1]: Boyer et al., Tight bounds on quantum searching

https://arxiv.org/abs/quant-ph/9605034(opens in a new tab)

## Methods

### amplify

Grover.amplify(amplification_problem)

Run the Grover algorithm.

Parameters

amplification_problem (AmplificationProblem) – The amplification problem.

Return type

GroverResult

Returns

The result as a GroverResult, where e.g. the most likely state can be queried as result.top_measurement.

Raises

• ValueError – If a quantum instance or sampler is not set.
• AlgorithmError – If a sampler job fails.
• TypeError – If is_good_state is not provided and is required (i.e. when iterations
• is None** or a list)** –

### construct_circuit

Grover.construct_circuit(problem, power=None, measurement=False)

Construct the circuit for Grover’s algorithm with power Grover operators.

Parameters

• problem (AmplificationProblem) – The amplification problem for the algorithm.
• power (Optional[int]) – The number of times the Grover operator is repeated. If None, this argument is set to the first item in iterations.
• measurement (bool) – Boolean flag to indicate if measurement should be included in the circuit.

Returns

the QuantumCircuit object for the constructed circuit

Return type

QuantumCircuit

Raises

ValueError – If no power is passed and the iterations are not an integer.

### optimal_num_iterations

static Grover.optimal_num_iterations(num_solutions, num_qubits)

Return the optimal number of iterations, if the number of solutions is known.

Parameters

• num_solutions (int) – The number of solutions.
• num_qubits (int) – The number of qubits used to encode the states.

Return type

int

Returns

The optimal number of iterations for Grover’s algorithm to succeed.

## Attributes

### quantum_instance

Pending deprecation; Get the quantum instance.

Return type

Optional[QuantumInstance]

Returns

The quantum instance used to run this algorithm.

### sampler

Get the sampler.

Return type

Optional[BaseSampler]

Returns

The sampler used to run this algorithm.