Quantum Algorithm for the Parity Problem

Years and Authors of Summarized Original Work

• 1985; Deutsch

Problem Definition

The parity of n bits $x_{0},x_{1},\cdots \,,x_{n-1} \in \{0,1\}$ is

$$\displaystyle{x_{0} \oplus x_{1} \oplus \cdots \oplus x_{n-1} =\sum _{ i=0}^{n-1}x_{ i}\quad \mod 2.}$$

As an elementary Boolean function, parity is important not only as a building block of digital logic but also for its instrumental roles in several areas such as error correction, hashing, discrete Fourier analysis, pseudorandomness, communication complexity, and circuit complexity. The feature of parity that underlies its many applications is its maximum sensitivity to the input: flipping any bit in the input changes the output. The computation of parity from its input bits is quite straightforward in most computation models. However, two settings deserve attention.

The first is the circuit complexity of parity when the gates are restricted to AND, OR, and NOTgates. It is known that parity cannot be computed by such a circuit of a...