Q1L3b.txt

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A good starting point for our journey into quantum error
correction is classical coding theory.
There are many similarities, but there are also
many important differences.
Let us illustrate classical coding
by looking at what is probably the simplest communication
channel for binary data.
In this channel, a zero is transmitted without error
with probability 1 minus p but is flipped with probability p.
Similarly, the 1 has the same 1 minus p and p.
The sender and receiver both have binary alphabets.
Thus, the probability of error for the channel is p,
and this channel is symmetric and known as the binary
symmetric channel.
The quantum analog of this will be the bit flip channel.
But first, let us begin with some definitions of codes.
We define a classical code with parameters n, k,
and d as being a set of 2 to the k n-bit strings.
The strings are constrained to have a minimum Hamming distance
from each other of value d.
The Hamming distance is a useful metric
of how much error has occurred.
And we define this distance between two strings, x and y,
as being the weight of x exclusive ORed
with y in a bitwise fashion.
This weight is the number of ones in that string.
Let us look at an example of such a classical code.
This code will have three bits and map 000 to being logical 0
and 111 to being logical 1.
It has three bits, encodes one logical bit,
and has distance of three.
Thus, the parameters are 3, 1, 3, for n, k,
and d for this code.
What do errors due to the state?
Well, let us apply the binary symmetric channel
with error probability p.
What the receiver sees as a result of this error
can be written as a stochastic mixture of a certain set
of received states.
000 is received with probability 1 minus p cubed.
Recall p is the error probability for a bit flip.
The receiver receives 001 with probability p times 1 minus p
squared, or, with the same probability, 010 and 100.
We may decode each of these receive states as a logical 0,
assuming, of course, a logical 0 was sent.
Two or three errors can also occur.
That happens with higher order probability.
And, in those cases, a 1 is decoded from the output,
and those are errors.
We may, therefore, compute the overall probability
of error of this error decoded result as being p cubed
plus p squared times 1 minus p.
To leading order this is p squared.
And that is better than the original channel which
had order of p for an error.
Thus, this is an effective method for correcting errors.