<p>A standard tool is the swap test, which receives two unentangled pure states |<i>φ</i><sub>1</sub>⟩ and |<i>φ</i><sub>2</sub>⟩ and answers “equal” with probability (1+|⟨<i>φ</i><sub>1</sub>|<i>φ</i><sub>2</sub>⟩|<sup>2</sup>)/2 and “not equal” with probability (1−|⟨<i>φ</i><sub>1</sub>|<i>φ</i><sub>2</sub>⟩|<sup>2</sup>)/2.  See [BCWW01] (Figure 1 shows the circuit).</p>

<p>[BCWW01] Harry Buhrman, Richard Cleve, John Watrous, and Ronald de Wolf.  Quantum fingerprinting.  <em>Physical Review Letters</em>, 87, article 167902, Sept. 2001.  DOI: <a href="http://link.aps.org/doi/10.1103/PhysRevLett.87.167902">10.1103/PhysRevLett.87.167902</a>.  A preprint version: <a href="http://arxiv.org/abs/quant-ph/0102001">arXiv:quant-ph/0102001</a>v1.</p>

CC BY-SA 3.0
2012-06-01T11:23:17.680Z
2012-06-01T11:23:17.680Z
Tsuyoshi Ito
<p>Consider this problem in quantum cryptography:</p>

<p>We have two pure states $\phi_1,\phi_2$ as input and constants $0 \leq \alpha &lt;\beta \leq 1 $, where "Yes instances" are those for which $$\left|\left&lt;\phi_1,\phi_2\right&gt;\right| \leq \alpha$$ and "No instances" are those for which $$\left|\left&lt;\phi_1,\phi_2\right&gt;\right|\geq \beta$$</p>

<p>I need to devise a quantum circuit that accepts "Yes instances" with probability at least $p$, and "No instances" with at most probability $q$ for some $q&lt;p$.</p>

<p>How can I do this? Is there a reference in which this problem is solved?</p>

<p>Thanks in advance.</p>

CC BY-SA 3.0
2012-06-01T11:25:05.840Z
Next time, please try to give an informative title.
Tsuyoshi Ito
<quantum-computing><quantum-information>
A promise problem to decide whether two given pure quantum states are close or far apart
MathematicalPhysicist
# A promise problem to decide whether two given pure quantum states are close or far apart

(Asked by: username_0 on 2012-06-01)

Consider this problem in quantum cryptography:

We have two pure states $\phi\_1,\phi\_2$ as input and constants $0 \leq \alpha <\beta \leq 1 $, where "Yes instances" are those for which $$\left|\left<\phi\_1,\phi\_2\right>\right| \leq \alpha$$ and "No instances" are those for which $$\left|\left<\phi\_1,\phi\_2\right>\right|\geq \beta$$

I need to devise a quantum circuit that accepts "Yes instances" with probability at least $p$, and "No instances" with at most probability $q$ for some $q

How can I do this? Is there a reference in which this problem is solved?

Thanks in advance.

**Tags:** quantum-computing, quantum-information

> **Comments:**
> **username_1**: Next time, please try to give an informative title.  

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## 1 answers

### Answer by username_1 on 2012-06-01. Score: 10 | Accepted answer

A standard tool is the swap test, which receives two unentangled pure states |*φ*1⟩ and |*φ*2⟩ and answers “equal” with probability (1+|⟨*φ*1|*φ*2⟩|2)/2 and “not equal” with probability (1−|⟨*φ*1|*φ*2⟩|2)/2. See [BCWW01] (Figure 1 shows the circuit).

[BCWW01] Harry Buhrman, Richard Cleve, John Watrous, and Ronald de Wolf. Quantum fingerprinting. *Physical Review Letters*, 87, article 167902, Sept. 2001. DOI: [10.1103/PhysRevLett.87.167902](http://link.aps.org/doi/10.1103/PhysRevLett.87.167902). A preprint version: [arXiv:quant-ph/0102001](http://arxiv.org/abs/quant-ph/0102001)v1.


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