Fully homomorphic encryption (FHE) enables privacy-preserving cloud computation, but its efficiency is often limited by the cost of bootstrapping. In particular, existing functional bootstrapping techniques have complexity exponential in the plaintext size. In this work, we show that employing a single quantum server can reduce this dependence. We propose a quantum functional bootstrapping algorithm that allows to evaluate any efficiently computable function in time polynomial in the plaintext size. For general functional bootstrapping over $l$-bit plaintexts, we obtain a time--space tradeoff: poly($l$)-time evaluation can be achieved with O$(2^l)$ qubits, while reducing the space complexity increases the time complexity. Technically, we extend a key classical cryptographic operation, known as \emph{blind rotation}, to the quantum setting by replacing polynomial-exponent encoding with quantum phase encoding. Underlying our extension are insights for the quantum extension of polynomial-based cryptographic tools that may gain dramatic speedups.

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