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Quantum Computing

8.3 Grover's algorithm: implementation and geometric interpretation

3 min readLast Updated on July 23, 2024

Grover's algorithm is a quantum search technique that finds marked items in an unsorted database quadratically faster than classical methods. It uses quantum superposition and interference to amplify the probability of measuring the desired state.

The algorithm's implementation involves initializing a superposition, applying an oracle to mark the target state, and using a diffusion operator to amplify its amplitude. This process is repeated for an optimal number of iterations before measurement, maximizing the chance of finding the solution.

Grover's Algorithm Implementation

Steps of Grover's algorithm implementation

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  • Initialize the system
    • Prepare a uniform superposition of all possible states using the Hadamard gate on each qubit (equal amplitudes for all states)
    • Apply the oracle to mark the desired state by flipping its phase (changes the sign of the amplitude)
  • Perform the Grover iteration
    • Apply the diffusion operator to amplify the amplitude of the marked state (increases the probability of measuring the desired state)
    • Repeat the oracle and diffusion operator application for the optimal number of iterations (determined by the number of marked and total states)
  • Measure the final state
    • Perform a measurement on the qubits to obtain the result with high probability (close to 1 for the optimal number of iterations)

Oracle and diffusion operator roles

  • Oracle
    • A black box function that identifies the solution to the search problem (encodes the search criteria)
    • Marks the desired state by flipping its phase, leaving all other states unchanged (distinguishes the target state)
    • Implemented using a unitary operator specific to the problem (requires knowledge of the solution)
  • Diffusion operator
    • Amplifies the amplitude of the marked state while reducing the amplitudes of the non-marked states (increases the likelihood of measuring the desired state)
    • Consists of the following steps:
      1. Apply the Hadamard gate to each qubit
      2. Apply a phase flip to all states except the 0|0\rangle state
      3. Apply the Hadamard gate to each qubit again
    • Increases the probability of measuring the marked state (constructive interference for the target state, destructive interference for others)

Geometric Interpretation and Optimization

Geometric interpretation of vector rotation

  • Grover's algorithm can be visualized as a rotation of the state vector in a two-dimensional plane (geometric representation of the quantum state)
  • The plane is spanned by two vectors:
    • The uniform superposition state s|s\rangle (initial state with equal amplitudes)
    • The marked state w|w\rangle (desired state to be found)
  • Each iteration of the algorithm rotates the state vector by an angle θ\theta towards the marked state (gradually aligns the state with the target)
    • The angle θ\theta is determined by the number of marked states and the total number of states (depends on the problem size)
    • θ=2arcsin(M/N)\theta = 2 \arcsin(\sqrt{M/N}), where MM is the number of marked states and NN is the total number of states
  • After the optimal number of iterations, the state vector is closely aligned with the marked state, maximizing the probability of measuring the correct result (near-certain measurement of the desired state)

Optimal iterations for success probability

  • The optimal number of iterations, rr, is approximately π/4θ\pi/4\theta (balances the rotation to align with the marked state)
    • rπ4N/Mr \approx \frac{\pi}{4}\sqrt{N/M}, where NN is the total number of states and MM is the number of marked states
  • Performing the Grover iteration more or fewer times than the optimal number reduces the probability of success (overshoots or undershoots the target state)
  • The probability of measuring the marked state after the optimal number of iterations is close to 1 (high confidence in obtaining the correct result)
    • The exact probability is sin2((2r+1)θ)\sin^2((2r+1)\theta), where rr is the number of iterations and θ\theta is the rotation angle
  • In practice, the optimal number of iterations is rounded to the nearest integer (discrete number of applications)


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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.