Re: Latest from ICANN: Quantum Computers + N85 Peering Forum

[Masataka Ohta <mohta () necom830 hpcl titech ac jp>]

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