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Explanation via the Quantum Circuit Model

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The experiment can be generalized under the Quantum Circuit Model[1]. Assume that a box which potentially contains a bomb is defined to operate on a single probe qubit in the following way:

  • If there is no bomb, the qubit passes through unaffected.
  • If there is a bomb, the qubit gets measured:
    • If the measurement outcome is |0⟩, the box returns |0⟩.
    • If the measurement outcome is |1⟩, the bomb explodes.

The following quantum circuit can be used to test if a bomb is present:

Where:

  • B is the box/bomb system, which measures the qubit if a bomb is present
  • is the unitary matrix
  • is some small number

At the end of the circuit, the probe qubit is measured. If the outcome is |0⟩, there is a bomb, and if the outcome is |1⟩, there is no bomb.

Case 1: No bomb

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When there is no bomb, the qubit evolves prior to measurement as , which will measure as |0⟩ (the incorrect answer) with probability .

Case 2: Bomb

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When there is a bomb, the qubit will be transformed into the state , then measured by the box. The probability of measuring as |1⟩ and exploding is by the small-angle approximation. Otherwise, the qubit will collapse to |0⟩ and the circuit will continue iterating. The probability of measuring |1⟩ and detonating the bomb after any of the T iterations is at most approximately by the union bound. If the bomb doesn't explode, the box will return |0⟩, which will be the final measurement value.

In both cases, the circuit returns the incorrect answer(or detonates the bomb) with arbitrarily low probability, based on the decision of .

References

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"Applications of Quantum Search, Quantum Zeno Effect" (PDF). EECS Berkeley. 2005-11-13.

Notes

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