Exponentially cheaper coherent phase estimation via uncontrolled unitaries

Abstract

Phase kickback is a fundamental primitive that is used in many quantum algorithms, such as quantum phase estimation. Here we observe that by using information about the controlled unitary, we can replace the controlled unitary with an uncontrolled one at the cost of introducing controlled state preparations. We then show how this modified phase kickback can be used as part of the quantum phase estimation algorithm when the goal is to estimate the phase of an eigenstate whose preparation procedure is known. We prove that this yields an exponential reduction in the number of two-qubit gates for an m-bit phase estimation in the relevant limit. Examples of applications are also presented. Naturally, this can be adapted to any algorithm that uses the phase kickback phenomenon and satisfies the assumptions.

Figures (4)

• Figure 1: Circuit for coherent, uncontrolled phase kickback: a controlled-$W$, $U$, open-controlled-$W$ block that imprints phase $\theta$ on an ancilla qubit without an explicit controlled-$U$. Under the condition that $W\ket{\phi}_s = \ket{\psi}_s$ with $\ket{\phi}_s,\ket{\psi}_s$ eigenstates of $U$, the system (bottom line) ends up in $\ket{\psi}_s$ throughout, and the relative phase $e^{i 2 \pi (\theta - \phi)}$ accumulates on the ancilla.
• Figure 2: Circuit for the two-bit version of uncontrolled phase kickback. In the first block (ancilla $a_1$), a 1-controlled-$W^\dagger$ replaces the open-controlled-$W$ and uncontrolled $W^\dagger$, simultaneously disentangling $a_1$ and resetting the system to $\ket{\phi}_s$.
• Figure 3: Circuit for the m-bit version of uncontrolled phase kickback. Intermediate ancilla blocks use 1-controlled-$W^\dagger$ to reset the system, while the last block uses open-controlled-$W$.
• Figure 4: Circuit for phase kickback.