Review from last lecture: A Simple Quantum (3,1) Repetition Code
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1 Review from last lecture: A Simple Quantum (3,1) Repetition Code Recovered state
2 Single Qubit errorsBit flip error:Do a bit flip using a operator.
3 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.
4 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
5 A review of a simple classical error correction encoding 3 bit repetition encoding:0 encoded as 0001 encoded as 111Assuming only 1 bit errorDecoding: Take majority vote of the 3 bitsE.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>kRecall also hypercubes
6 Why using classical error correction for correcting Qubits is not trivial? First reasonNo cloning theoremUnable to encode asMeasurement of qubits cause disturbanceNeed 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
7 Why using classical error correction for correcting Qubits is not trivial? Third reasonUnable to correct phase errorsUnable to correct small errorsFor , 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 reasonQuantum errors are usually continuous.But we will solve all these problems
8 Quantum Error correcting codes Correcting single bit flip error using 3 qubitsCorrecting single phase error using 3 qubits9 qubits error correcting code5 qubits error correcting codeConcatenated code
9 Simple (3,1) repetition code circuit This circuit can correct single bit flip and detect double bit flip.
10 Error Correction for 1 Bit Flip This shows what happenedIf bit flip occurred in data bit than syndrome is 11, used for correction
11 Encoder for (3,1) Repetition Code For encoding, use 2 extra qubits initially set toEncoding circuit:Calculated from in Dirac notation as xor of 1 and 0
12 We use slightly different notation to explain it even better +
13 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 bitAs usually red bits show change in our picturesChanges in third bit
14 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:SyndromeError00No error013rd qubit flipped102nd qubit flipped111st qubit flippedSyndrom as a result of error that happenedOnly this is wronggood
15 In this case correction is trivial Correction circuit:If syndrome bits are not ’00’, discard them and re-encode using new qubits.
16 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 transmissionThis are all their counterpart final signalsResults of correction. As we see this is majority
17 Correcting single phase flip in (3,1) circuits Use Hadamard to convert a phase flip to bit flipSimilarly:Pauli XPauli ZThis is another fundamental trick – convert one type of error to another which is easier to manipulate
18 Proof of the first of the above convertions Now we will see how this idea is used
19 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 decoderIf there is a phase flip, two hadamards will convert it to bit flip
20 Initial Problems Avoided No cloning involved in encodingAble 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!!
21 Able to correct small errors Few tricks solves many of problems listed earlier!!Able to correct small errorsExample: Assume encoded qubit damaged such that:0.7 probability of getting no errors0.3 probability of getting 1st bit flipped
22 Step by step analysis of decoding and correction After the circuit, 1st qubit will always beThe 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 11Error correction is possible even if error is a superposition.Quantum error correction will digitalize the errors.
23 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 error9 qubits used to encode 1 qubit.
24 Basic Idea of Shor CodeCorrection of bit & phase flip errors
27 First we explain the principle of Encoding in Shor code Use 9 qubits to encode 1 qubit (9,1):
28 Thus for general qubit we have Encoding circuit:
29 Shor code –EncodingBell state |+> and |->Entangled GHZ states
30 Now we show step by step how encoder works Tensor product of results of Hadamards with zerosXoring in second and third bits with 1 from first bit
31 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
32 Detailed analysis of an error Example: Assume encoded qubit damaged such that:Send to lineReceived from lineAs we see the red error is in phase and bit flip of first qubit
33 Shor code –DecodingWe can explain it using Bell and GHZ states quicklyOr use full notation for analysis