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Explanation via the Quantum Circuit Model

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
${\textstyle R_{\epsilon }}$ is the unitary matrix ${\textstyle {\begin{pmatrix}\cos {\epsilon }&-\sin {\epsilon }\\\sin {\epsilon }&\cos {\epsilon }\end{pmatrix}}}$
${\textstyle \epsilon }$ is some small number
${\textstyle T=\lceil {\frac {\pi }{2\epsilon }}\rceil }$

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

When there is no bomb, the qubit evolves prior to measurement as ${\textstyle R_{\epsilon }^{T}|0\rangle =\cos(T\epsilon )|0\rangle +\sin(T\epsilon )|1\rangle }$ , which will measure as $|0⟩$ (the incorrect answer) with probability ${\textstyle cos^{2}(T\epsilon )\approx O(\epsilon )}$ .

Case 2: Bomb

When there is a bomb, the qubit will be transformed into the state ${\textstyle \cos(\epsilon )|0\rangle +\sin(\epsilon )|1\rangle }$ , then measured by the box. The probability of measuring as $|1⟩$ and exploding is ${\textstyle \sin ^{2}(\epsilon )\approx \epsilon ^{2}}$ 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 ${\textstyle T\epsilon ^{2}\approx O(\epsilon )}$ 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 ${\textstyle \epsilon }$ .

References

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

Notes

^ EECS Berkeley 2005.

[FOOTNOTEEECS_Berkeley2005-1] EECS Berkeley 2005.

[1]