Limitation on Effective Degree of Quantum Parallelism

Consider a binary quantum channel with binary states |0> and |1> as an output channel of some quantum computation device and assume that, if the channel is used as a classical binary channel where |0> and |1> represent bit values of 0 and 1, respectively, the channel has small error probability of p. Then, |0> transmitted over the channel will typically be sqrt(1-p)|0> + exp(i*theta)|1> (0 ≤ theta and sqrt(1-p)|0> + exp(i*theta)|1> indistinguishable, which means different results of parallel execution of the device can’t be represented by |0> and sqrt(1-p)|0> + exp(i*theta)|1>. As representing N parallel binary results needs 2^N distinguishable states, effective degree of quantum parallelism of the device, which is defined as degree of parallelism of binary results with arbitrary small error probability by ideal encoding and ideal error correction, is limited by log2(π/2*sqrt(p) + 1). That is, in practice, quantum computers are only as powerful as classical ones. Then, a brief introduction on modern communication technology over photons is provided to show that capacity of a binary quantum channel is almost twice better than quantum physicists had thought, that a classical state representing an entangled state exists and that“qubit”is a bad idea. Finally, it is shown that, without error caused by noise, ideal classical computers can be arbitrary fast.