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

Recovered state

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

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**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.

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**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

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**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

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**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

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**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

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**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

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**Simple (3,1) repetition code circuit**

This circuit can correct single bit flip and detect double bit flip.

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**Error Correction for 1 Bit Flip**

This shows what happened If bit flip occurred in data bit than syndrome is 11, used for correction

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**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

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**We use slightly different notation to explain it even better**

+

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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

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**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

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**In this case correction is trivial**

Correction circuit: If syndrome bits are not ’00’, discard them and re-encode using new qubits.

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**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

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**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

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**Proof of the first of the above convertions**

Now we will see how this idea is used

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**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

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**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!!

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**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

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**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.

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**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.

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

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Shor code:

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**Architecture of Shor Code**

encoder decoder

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**First we explain the principle of Encoding in Shor code**

Use 9 qubits to encode 1 qubit (9,1):

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**Thus for general qubit we have**

Encoding circuit:

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

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**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

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**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

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**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

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

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From line Before correction

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After correction

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**Before Hadamards, from previous slide**

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

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**And finally after the entire bit flip and phase flip corrections we get:**

So we got what we wanted to get!

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**The (9,1) circuit – put all together**

This paper published recently started a furry of results and great ideas

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**Now we explain in a different way how Shor’s [9,1,3] code works**

Pauli Z in bit one

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All nine bits used

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

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