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Suppose $x \in \{0,1\}^n$. The standard way to make a query is with an oracle $O_x$ that given an input $|i,b \rangle $ returns $|i,b \oplus x_i \rangle$. Via the phase kick-back trick, this can be used to make another type of query $O_{x}^{''}$ such that $O_{x}^{''}|i\rangle=(-1)^{x_i}|i\rangle$. This is easily seen by passing to the $|+\rangle, |- \rangle$ basis. How can it be seen that the converse is true i.e. that with a query of the latter type (and perhaps some ancillary qubit or transformation) one can implement a query of the former type?

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  • $\begingroup$ Are you sure that this is even possible? I have a vague memory from long ago that is was not clear how to go from a phase to a normal oracle. $\endgroup$ – Norbert Schuch Feb 16 at 19:34
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If we express the action of $O_x$ on the basis $\mid i, \pm \rangle$ instead of $\mid i , b \rangle$

\begin{eqnarray*} O_x \mid i , + \rangle = \mid i , + \rangle\\ O_x \mid i , - \rangle = (-1)^{x_i} \mid i , - \rangle\\ \end{eqnarray*}

Because we say ancillas must be started and ended with $0$,

\begin{eqnarray*} O_x'' &=& (1 \otimes S_x H) O_x (1 \otimes H S_x) \end{eqnarray*}

where the $1 \otimes H S_x$ conjugation takes care of turning the ancilla from $0$ to $-$ and back.

Be careful about what $O_x''$ does. It does the desired $(-1)^{x_i}$ only when the ancilla is in $0$ as it is supposed to be. If the ancilla is in $1$ instead, $O_x''$ acts like the identity.

\begin{eqnarray*} O_x'' \mid i , 0 \rangle &=& (-1)^{x_i} \mid i, 0 \rangle\\ O_x'' \mid i , 1 \rangle &=& \mid i, 1 \rangle\\ \end{eqnarray*}

Solve for $O_x$

\begin{eqnarray*} O_x &=& (1 \otimes H S_x) O_x'' (1 \otimes S_x H) \end{eqnarray*}

See what it does on $\mid i , + \rangle$ and $\mid i, - \rangle$

\begin{eqnarray*} (1 \otimes H S_x) O_x'' (1 \otimes S_x H) \mid i, + \rangle &=& (1 \otimes H S_x) O_x'' \mid i, 1 \rangle\\ &=& (1 \otimes H S_x) \mid i, 1 \rangle\\ &=& \mid i, + \rangle\\ (1 \otimes H S_x) O_x'' (1 \otimes S_x H) \mid i, - \rangle &=& (1 \otimes H S_x) O_x'' \mid i, 0 \rangle\\ &=& (1 \otimes H S_x) (-1)^{x_i} \mid i, 0 \rangle\\ &=& (-1)^{x_i} \mid i , - \rangle\\ \end{eqnarray*}

This matches with what we wanted for $O_x$ in the first two lines.

Edit:

\begin{eqnarray*} O_x \mid i, + \rangle &=& O_x \frac{1}{\sqrt{2}} (\mid i, 0 \rangle + \mid i, 1 \rangle)\\ &=& \frac{1}{\sqrt{2}} (O_x \mid i, 0 \rangle + O_x \mid i, 1 \rangle)\\ &=& \frac{1}{\sqrt{2}} (\mid i, 0+x_i \rangle + O_x \mid i, 1+x_i \rangle)\\ &=& \mid i, + \rangle \end{eqnarray*}

For the last equality, the two summands either swap or they don't depending on $x_i$. Similarly with -

\begin{eqnarray*} O_x \mid i, - \rangle &=& O_x \frac{1}{\sqrt{2}} (\mid i, 0 \rangle - \mid i, 1 \rangle)\\ &=& \frac{1}{\sqrt{2}} (O_x \mid i, 0 \rangle - O_x \mid i, 1 \rangle)\\ &=& \frac{1}{\sqrt{2}} (\mid i, 0+x_i \rangle - O_x \mid i, 1+x_i \rangle)\\ \end{eqnarray*}

so if $x_i=0$ there is no swap of terms and the result is $\mid i, - \rangle$. If $x_i=1$, there is a swap of terms and the result is $- \mid i, - \rangle$. This can be put together by saying $O_x \mid i, - \rangle = (-1)^{x_i} \mid i , - \rangle$

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  • $\begingroup$ How does this explain how to get from a phase oracle $|x\rangle \to (-1)^{f(x)} |x\rangle$ back to a normal oracle? Your $O_x$ is quite different! $\endgroup$ – Norbert Schuch Feb 16 at 19:34
  • $\begingroup$ Hi. Thanks for your reply. I don't understand how it matches what we expect from $O_x$ because $O_x$ is not supposed to ever multiply $|i\rangle$ by $-1$. $\endgroup$ – Karl Feb 17 at 11:06
  • $\begingroup$ Yes, but what I need instead is $O_x|i,b\rangle=|i,b\oplus x_i \rangle$. The content of your edit is basically the hypothesis. $\endgroup$ – Karl Feb 17 at 17:29
  • $\begingroup$ You can now see the rest of the answer. It was a this is what we want, write down a formula using O_x" and then check that it did the correct behavior on a basis. $\endgroup$ – AHusain Feb 17 at 17:36
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You are given a quantum circuit for $U$ compiled into the H/CNOT/T gateset. Derive a controlled version of $U$ by adding a control qubit $q$, replacing every H with a controlled H, every CNOT with a CCNOT, and every T with a controlled T. In all cases the new control goes on the $q$.

Recompile the modified gates down into the H/CNOT/T gate set. Prepend and append the circuit with a Hadamard on $q$. You now have a circuit that toggles $q$ when a -1 eigenstate of $U$ is input and leaves $q$ alone when a +1 eigenstate of $U$ is input.

In short:

  • Circuit $C$ implements $(-1)^{f(x)}$
  • Derive controlled $C$ implementing $(-1)^{q \cdot f(x)}$
  • Switch control basis from Z to X, achieving $q \mathrel{\oplus} = f(x)$
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  • $\begingroup$ Hi and thanks for your reply. Do you think the result is achievable also without using controlled $U$? $\endgroup$ – Karl Feb 17 at 11:11
  • $\begingroup$ And also: control basis Z is $|+\rangle,|-\rangle$ and X is $|0\rangle,|1\rangle$? $\endgroup$ – Karl Feb 17 at 13:07
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    $\begingroup$ @Karl You need the control in order to change the unobservable global phase into an observable relative phase. In practice you won't need to literally control every single operation; there's usually more efficient ways. This is just the simplest way to introduce a control. $\endgroup$ – Craig Gidney Feb 17 at 18:04

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