As I wrote:

Nanog News wrote:

Latest from ICANN: Quantum Computers are "Interesting"…

But Don't Lose your Head

Uselessness of quantum logic gate style quantum computers

will be discussed in a separate mail.

Quantum logic gate style quantum computers use qubits,

which have two orthogonal quantum state |0> and |1>

(which may be a horizontally and vertically polarized

photon, correspondingly, which is familiar for us,

communication engineers) used as bit values of 0 and

1. As they are orthogonal, we can consider |0>+|1>,

(which may be a diagonally polarized photon, easy

to understand classically). But such addition is

improperly called quantum superposition.

Two qubit states are represented as |00>, |11>

and |00>+|11>.

Then, if we can construct a linear circuit to

compute f, a 3 input and 3 output bits function,

f(a, b, c)=(d, e, f) as f(|abc>)=|def>. Then,

we can compute f(|000>)+f(|001>>+...+f(|111>) as

f(|000>+|001>+...+|111>), that is evaluating

f not 8 times but only once, which is quantum

parallelism. It can be extended for N qubit

cases for 2**N parallelism by 2**N terms.

However, though an N qubit state, like an N bit string,

can naturally distinguish 2**N cases, 2**N term

quantum superpositioned states require distinguishing

2**(2**N) cases, which is obviously impossible because

of noise/error (theory of Shannon).

So, QEC (Quantum Error Correction) was invented,

which was expected to reduce noise/error.

QEC certainly work for single term states. Small

non linear error on single term states is no different

from linear error.

The problem, however, is that, QEC is non-linear,

which means not applicable to 2**N term

superpositioned states.

But, it is implicitly and wrongly assumed that

all the 2**N terms will suffer from identical

error, in which case, QEC become linear.

That is single, so called, bit flip error on

the first qubit of:

>000>+|111>

occurs as:

>100>+|111>

or:

>000>+|011>

not (unless strong correlation exists, which

is an improper assumption):

>100>+|011>

which is the reason why quantum logic

gate quantum computer with practical

number of qubits is impossible.

A complication is that it is possible

to construct complex QEC scheme applicable

for 2 term states. So called Shor code

is such QEC. But Shor code itself use

states with more than 2 terms. Anyway,

once QEC scheme is fixed, it is not

applicable to 2**N term states for

large N.

It should also be noted that though

non-linear error on two or more term states

imply errors caused by interactions of multiple

terms, they are ignored.

Masataka Ohta