Quantum Oracles

An oracle $O$ is a "black box" operation that is used as input to another algorithm. Often, such operations are defined using a classical function $f : \{0, 1\}^n \to \{0, 1\}^m$ which takes an $n$-bit binary input and produces an $m$-bit binary output. To do so, consider a particular binary input $x = (x_{0}, x_{1}, \dots, x_{n-1})$. We can label qubit states as $\ket{\vec{x}} = \ket{x_{0}} \otimes \ket{x_{1}} \otimes \cdots \otimes \ket{x_{n-1}}$.

We may first attempt to define $O$ so that $O\ket{x} = \ket{f(x)}$, but this has a couple problems. First, $f$ may have a different size of input and output ($n \ne m$), such that applying $O$ would change the number of qubits in the register. Second, even if $n = m$, the function may not be invertible: if $f(x) = f(y)$ for some $x \ne y$, then $O\ket{x} = O\ket{y}$ but $O^\dagger O\ket{x} \ne O^\dagger O\ket{y}$. This means we won't be able to construct the adjoint operation $O^\dagger$, and oracles have to have an adjoint defined for them.

Defining an oracle by its effect on computational basis states

We can deal with both of these problems by introducing a second register of $m$ qubits to hold our answer. Then we will define the effect of the oracle on all computational basis states: for all $x \in \{0, 1\}^n$ and $y \in \{0, 1\}^m$,

$$ \begin{align} O(\ket{x} \otimes \ket{y}) = \ket{x} \otimes \ket{y \oplus f(x)}. \end{align} $$

Now $O = O^\dagger$ by construction, thus we have resolved both of the earlier problems.

Tip

To see that $O = O^{\dagger}$, note that $O^2 = \boldone$ since $a \oplus b \oplus b = a$ for all $a, b \in {0, 1}$. As a result, $O \ket{x} \ket{y \oplus f(x)} = \ket{x} \ket{y \oplus f(x) \oplus f(x)} = \ket{x} \ket{y}$.

Importantly, defining an oracle this way for each computational basis state $\ket{x}\ket{y}$ also defines how $O$ acts for any other state. This follows immediately from the fact that $O$, like all quantum operations, is linear in the state that it acts on. Consider the Hadamard operation, for instance, which is defined by $H \ket{0} = \ket{+}$ and $H \ket{1} = \ket{-}$. If we wish to know how $H$ acts on $\ket{+}$, we can use that $H$ is linear,

$$ \begin{align} H\ket{+} & = \frac{1}{\sqrt{2}} H(\ket{0} + \ket{1}) = \frac{1}{\sqrt{2}} (H\ket{0} + H\ket{1}) \\ & = \frac{1}{\sqrt{2}} (\ket{+} + \ket{-}) = \frac12 (\ket{0} + \ket{1} + \ket{0} - \ket{1}) = \ket{0}. \end{align} $$

In the case of defining our oracle $O$, we can similarly use that any state $\ket{\psi}$ on $n + m$ qubits can be written as

$$ \begin{align} \ket{\psi} & = \sum_{x \in \{0, 1\}^n, y \in \{0, 1\}^m} \alpha(x, y) \ket{x} \ket{y} \end{align} $$

where $\alpha : \{0, 1\}^n \times \{0, 1\}^m \to \mathbb{C}$ represents the coefficients of the state $\ket{\psi}$. Thus,

$$ \begin{align} O \ket{\psi} & = O \sum_{x \in \{0, 1\}^n, y \in \{0, 1\}^m} \alpha(x, y) \ket{x} \ket{y} \\ & = \sum_{x \in \{0, 1\}^n, y \in \{0, 1\}^m} \alpha(x, y) O \ket{x} \ket{y} \\ & = \sum_{x \in \{0, 1\}^n, y \in \{0, 1\}^m} \alpha(x, y) \ket{x} \ket{y \oplus f(x)}. \end{align} $$

Phase oracles

Alternatively, we can encode $f$ into an oracle $O$ by applying a phase based on the input to $O$. For instance, we might define $O$ such that $$ \begin{align} O \ket{x} = (-1)^{f(x)} \ket{x}. \end{align} $$ If a phase oracle acts on a register initially in a computational basis state $\ket{x}$, then this phase is a global phase and hence not observable. But such an oracle can be a very powerful resource if applied to a superposition or as a controlled operation. For example, consider a phase orcale $O_f$ for a single-qubit function $f$. Then, $$ \begin{align} O_f \ket{+} & = O_f (\ket{0} + \ket{1}) / \sqrt{2} \\ & = ((-1)^{f(0)} \ket{0} + (-1)^{f(1)} \ket{1}) / \sqrt{2} \\ & = (-1)^{f(0)} (\ket{0} + (-1)^{f(1) - f(0)} \ket{1}) / \sqrt{2} \\ & = (-1)^{f(0)} Z^{f(0) - f(1)} \ket{+}. \end{align} $$

More generally, both views of oracles can be broadened to represent classical functions which return real numbers instead of only a single bit.

Choosing the best way to implement an oracle depends heavily on how this oracle will be used within a given algorithm. For example, Deutsch-Jozsa algorithm relies on the oracle implemented in the first way, while Grover's algorithm relies on the oracle implemented in the second way.

For more details, we suggest the discussion in Gilyén et al. 1711.00465.