 # Review from last lecture: A Simple Quantum (3,1) Repetition Code

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Review from last lecture: A Simple Quantum (3,1) Repetition Code
Recovered state

Single Qubit errors Bit flip error: Do a bit flip using a operator.

Phase flip error: Do a phase flip using a operator.
Phase flip affects interference. Interference: when parallel computation are carried out by a quantum computer, these parallel computation can reinforce or cancel out each other.

Bit and phase flip error:
Do a bit and phase flip using a operator. X,Y,Z are pauli matrices. They are also called depolarisation errors

A review of a simple classical error correction encoding
3 bit repetition encoding: 0 encoded as 000 1 encoded as 111 Assuming only 1 bit error Decoding: Take majority vote of the 3 bits E.g. This scheme can correct 1 bit error and detect up to 2 bit errors. Classical linear error correcting codes involve encoding k bits to be protected into a n bit string, where n>k Recall also hypercubes

Why using classical error correction for correcting Qubits is not trivial?
First reason No cloning theorem Unable to encode as Measurement of qubits cause disturbance Need to do error correction without measuring the value of each qubit. Classical error correction takes for granted that bits can be measured as much as you want. Second reason

Why using classical error correction for correcting Qubits is not trivial?
Third reason Unable to correct phase errors Unable to correct small errors For , an error might change α and β by a small order. These small errors can accumulate. Classical methods only designed to correct large discrete errors (i.e. bit flips) Fourth reason Quantum errors are usually continuous. But we will solve all these problems

Quantum Error correcting codes
Correcting single bit flip error using 3 qubits Correcting single phase error using 3 qubits 9 qubits error correcting code 5 qubits error correcting code Concatenated code

Simple (3,1) repetition code circuit
This circuit can correct single bit flip and detect double bit flip.

Error Correction for 1 Bit Flip
This shows what happened If bit flip occurred in data bit than syndrome is 11, used for correction

Encoder for (3,1) Repetition Code
For encoding, use 2 extra qubits initially set to Encoding circuit: Calculated from in Dirac notation as xor of 1 and 0

We use slightly different notation to explain it even better
+

How decoder works? Assuming at most 1 bit will be flipped and the bit flip is just as likely to affect any qubit. Decoding circuit: Changes in second bit As usually red bits show change in our pictures Changes in third bit

The important idea of Syndrome
The last 2 qubits are called the syndrome and their values indicate the error type that occurred. All possible states at the end of decoding circuit: Syndrome Error 00 No error 01 3rd qubit flipped 10 2nd qubit flipped 11 1st qubit flipped Syndrom as a result of error that happened Only this is wrong good

In this case correction is trivial
Correction circuit: If syndrome bits are not ’00’, discard them and re-encode using new qubits.

Let us analyze one more time the Decoder for (3,1) Repetition Code using another notation
This are all possible signals with no error or with error from transmission This are all their counterpart final signals Results of correction. As we see this is majority

Correcting single phase flip in (3,1) circuits
Use Hadamard to convert a phase flip to bit flip Similarly: Pauli X Pauli Z This is another fundamental trick – convert one type of error to another which is easier to manipulate

Proof of the first of the above convertions
Now we will see how this idea is used

Correcting single phase flip
Complete Circuit for correcting single bit flip: Modified circuit to correct single phase flip. 1st one can correct bit flip but not phase flip. 2nd one can correct phase flip but not bit flip. To detect phase flip we add Hadamards at the end of encoder and at beginning of decoder If there is a phase flip, two hadamards will convert it to bit flip

Initial Problems Avoided
No cloning involved in encoding Able to diagnose the error without damaging the quantum information. Able to correct errors without knowing state of qubit. Able to correct bit flip or phase flip error depending on the circuit used. Few tricks solves many of problems listed earlier!!

Able to correct small errors
Few tricks solves many of problems listed earlier!! Able to correct small errors Example: Assume encoded qubit damaged such that: 0.7 probability of getting no errors 0.3 probability of getting 1st bit flipped

Step by step analysis of decoding and correction
After the circuit, 1st qubit will always be The decoding circuit maps the state into either one with no error, or one with an error which we know how to correct. Unique syndroms allow to correct if 11 Error correction is possible even if error is a superposition. Quantum error correction will digitalize the errors.

Shor’s 9 qubits error correcting code
The 2 codes earlier corrects either bit flips or phase flips. Shor’s 9 qubits error correcting code combines both codes. It can correct any arbitrary single qubit error 9 qubits used to encode 1 qubit.

Basic Idea of Shor Code Correction of bit & phase flip errors

Shor code:

Architecture of Shor Code
encoder decoder

First we explain the principle of Encoding in Shor code
Use 9 qubits to encode 1 qubit (9,1):

Thus for general qubit we have
Encoding circuit:

Shor code –Encoding Bell state |+> and |-> Entangled GHZ states

Now we show step by step how encoder works
Tensor product of results of Hadamards with zeros Xoring in second and third bits with 1 from first bit

Now we show step by step how DECODER works
Assuming at most 1 qubit error and the error is just as likely to affect any qubit. The decoding and correction circuit: Appreciate please the mirror like symmetry

Detailed analysis of an error
Example: Assume encoded qubit damaged such that: Send to line Received from line As we see the red error is in phase and bit flip of first qubit

Shor code –Decoding We can explain it using Bell and GHZ states quickly Or use full notation for analysis

From line Before correction

After correction

After Hadamards After bit flip decoding Xoring in bits 2 and 3 only in alpha part

And finally after the entire bit flip and phase flip corrections we get:
So we got what we wanted to get!

The (9,1) circuit – put all together
This paper published recently started a furry of results and great ideas

Now we explain in a different way how Shor’s [9,1,3] code works
Pauli Z in bit one

All nine bits used

Thus there are no small errors, only large errors which we can fix

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