IntroductionPartial Cloning
Quantum Error Correction CodesReferences
CQIS Project Report
Abhineet Agarwal Palash Pandya
IIIT-Hyderabad
Abhineet Agarwal, Palash Pandya CQIS Project Report
IntroductionPartial Cloning
Quantum Error Correction CodesReferences
Contents
1 IntroductionBB-84 ProtocolNo cloning
2 Partial CloningWooters-ZurekBuzek-Hillery
3 Quantum Error Correction CodesIntroductionThree qubit bit flip codeThree qubit phase flip codeThe Shor code
4 References
Abhineet Agarwal, Palash Pandya CQIS Project Report
IntroductionPartial Cloning
Quantum Error Correction CodesReferences
BB-84 ProtocolNo cloning
BB-84 Protocol
Protocol to generate and share a secret key between two parties(Alice and Bob) for further communication in a secure manner usingthe key.
Alice uses two basis sets {| ↑〉, | →〉} and {| ↗〉, | ↖〉}.For sending 0 she can send: | ↑〉 or | ↗〉For sending 1 she can send: | →〉 or | ↖〉For each bit Alice sends Bob announces in which basis he measuredthe state in.
This is the raw key.
Abhineet Agarwal, Palash Pandya CQIS Project Report
IntroductionPartial Cloning
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BB-84 ProtocolNo cloning
BB-84 Protocol
Alice confirms if the basis sets are compatible.
If they are the bit is kept, otherwise discarded. Thus discarding 50%of the raw key.
Key obtained after basis reconciliation is called the shifted key.
Eve, can listen to the public channel but cannot observe thequantum channel.
If the error rate gets above a certain threshold then Alice and Bobdecide to abort.
Abhineet Agarwal, Palash Pandya CQIS Project Report
IntroductionPartial Cloning
Quantum Error Correction CodesReferences
BB-84 ProtocolNo cloning
No Cloning Theorom
The no-cloning theorem states that it is not possible to perfectlyclone an unknown quantum state, or a state drawn from a set of two(or more) nonorthogonal states.
Proof - can be proved by contradaction
Assuming there exists an ideal cloner Uc and it clones twonon-orthogonal states ideally, |ψ〉 and |ϕ〉.Uc(|ψ〉 |ε〉 |A〉) = |ψ〉 |ψ〉 |Aψ〉Uc(|ϕ〉 |ε〉 |A〉) = |ϕ〉 |ϕ〉 |Aϕ〉
Abhineet Agarwal, Palash Pandya CQIS Project Report
IntroductionPartial Cloning
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BB-84 ProtocolNo cloning
No Cloning Theorom
Inner product of the above two equations L.H.S with L.H.S must beequal to R.H.S with R.H.S, that is
〈ψ|ϕ〉 = 〈ψ|ϕ〉2 〈Aψ|Aϕ〉Inner product of the ancilla states and the input states must be lessthan or equal to 1, that is 〈Aψ|Aϕ〉 ≤ 1 and also 〈ψ|ϕ〉 ≤ 1.
But from the inner product of the two equations above, we get therelation
〈ψ|ϕ〉 = 1〈Aψ|Aϕ〉
Both of them cannot be less than or equal to 1 as they are inverselypropotional. Hence, the assumption we took is wrong.
No ideal quantum cloner is possible.
Abhineet Agarwal, Palash Pandya CQIS Project Report
IntroductionPartial Cloning
Quantum Error Correction CodesReferences
Wooters-ZurekBuzek-Hillery
Partial Cloning
By non-cloning theorom, we know ideal quantum cloners are notpossible and we have to look for approximate cloning.
The quality of partial cloning can be either dependent on input stateor not, that is two types of partial cloners possible
State-independent Quantum Cloning Machines - Good for somestatesState-dependent or Universal Quantum Cloning Machines - Identicalcloning for all states
Abhineet Agarwal, Palash Pandya CQIS Project Report
IntroductionPartial Cloning
Quantum Error Correction CodesReferences
Wooters-ZurekBuzek-Hillery
Wooters-Zurek
This is an example of state-dependent cloning Machine.
Eve wants to partially clone Alice’s qubit using the defined relations
|0〉a |ε〉e |Q〉x → |0〉a |0〉e |Q0〉x|1〉a |ε〉e |Q〉x → |1〉a |1〉e |Q1〉x
In the above relations, we have taken the tensor product of the threeHilbert spaces, Alice’s, Eve’s and the ancilla’s.
From unitarity of quantum operations and orthonormality of basisstates, we can say that
〈Q0|Q0〉x = 1 and 〈Q0|Q0〉x = 1.
〈Q0|Q1〉x = 0 (assuming orthogonality of ancilla states)
Now, we clone a superposition state
|s〉a = α |0〉a + β |1〉a
Abhineet Agarwal, Palash Pandya CQIS Project Report
IntroductionPartial Cloning
Quantum Error Correction CodesReferences
Wooters-ZurekBuzek-Hillery
Wooters-Zurek
Using the above relations, the state after cloning is|s〉a |ε〉e |Q〉x → α |0〉a |0〉e |Q0〉x + β |1〉a |1〉e |Q1〉x = |ψ〉outabx .
The reduced state density matrix for combined system of Alice andEve is
ρoutae = α2 |00〉ae 〈00|+ β2 |11〉ae 〈11|Reduced state densite matrix for Alice and Eve is same, which is
ρouta = Tre [ρoutae ] = α2 |0〉a 〈0|+ β2 |1〉a 〈1|The copying quality can be measured by calculating Hilbert-Schmidtnorm
Da = Tr [ρina − ρouta ]2
Da = 2α2β2 = 2α2(1− α2)
This distance shows that it is dependent on paramter α. It is bestfor basis state and worst for pure superposition states.
Abhineet Agarwal, Palash Pandya CQIS Project Report
IntroductionPartial Cloning
Quantum Error Correction CodesReferences
Wooters-ZurekBuzek-Hillery
Buzek-Hillery
This is an universal or state-independent quantum cloning machinewhich clones all states equally well.
The transformation rules in Buzek-Hillery cloning machine are givenby
|0〉a |ε〉e |Q〉x → |0〉a |0〉e |Q0〉x + [|0〉 |1〉ae + |1〉 |0〉ae ] |Y0〉x|1〉a |ε〉e |Q〉x → |1〉a |1〉e |Q1〉x + [|0〉 |1〉ae + |1〉 |0〉ae ] |Y1〉x
Due to unitarity of quantum transformations and assuming that |Qi 〉and |Yi 〉 are mutually orthogonal, we can say
x 〈Qi |Qi 〉x + x 〈Yi |Yi 〉x = 1, i = 0, 1
x 〈Y0|Y1〉x = x 〈Y1|Y0〉x = 0
x 〈Q0|Q1〉x = x 〈Q1|Q0〉x = 0
x 〈Q0|Y0〉x = x 〈Q1|Y1〉x = 0
Abhineet Agarwal, Palash Pandya CQIS Project Report
IntroductionPartial Cloning
Quantum Error Correction CodesReferences
Wooters-ZurekBuzek-Hillery
Buzek-Hillery
Applying BH cloning machine to a superposition state |s〉 and weobtain|s〉a |ε〉e |Q〉x →|0〉a |0〉e |Q0〉x + |1〉a |1〉e |Q1〉x + [|0〉 |1〉ae + |1〉 |0〉ae ][|Y0〉x + |Y1〉x ]Using the assumptions and also the orthonormality conditions, thereduced state density matrix for combined system of Alice and Eve is
ρoutae = α2 |00〉ae 〈00| 〈Q0|Q0〉x +√
2αβ |00〉ae 〈+| 〈Y1|Q0〉x +√2αβ |+〉ae 〈00| 〈Q0|Y1〉x + |+〉ae 〈+| [2α2 〈Y0|Y0〉x +
2β2 〈Y1|Y1〉x ] +√
2αβ |+〉ae 〈11| 〈Q1|Y0〉x +√2αβ |11〉ae 〈+| 〈Y0|Q1〉x + β2 |11〉ae 〈11| 〈Q1|Q1〉x
The reduced state density matrix Alice and Eve will be the same andwe can obtain them by tracing the density matrix for combinedsystem over Eve or Alice respectively. We will get the reduced statefor Alice as
ρouta = |0〉a 〈0| [α2 + β2 〈Y1|Y1〉x − α2 〈Y0|Y0〉x ] +|0〉a 〈1|αβ[〈Q1|Y0〉x + 〈Y1|Q0〉x ] + |1〉a 〈0|αβ[〈Q0|Y1〉x +〈Y0|Q1〉x ] + |0〉a 〈0| [β2 + α2 〈Y0|Y0〉x − β2 〈Y1|Y1〉x ]
Abhineet Agarwal, Palash Pandya CQIS Project Report
IntroductionPartial Cloning
Quantum Error Correction CodesReferences
Wooters-ZurekBuzek-Hillery
Buzek-Hillery
To simplify the equation, we wil be using these substitutions
〈Y0|Y0〉x = 〈Y1|Y1〉x = ξ
〈Y0|Q1〉x = 〈Q0|Y1〉x = 〈Q1|Y0〉x = 〈Y1|Q0〉x = η2
Substituting the inner products with the variables, we get
ρouta = |0〉a 〈0| [α2 + ξ(β2 − α2)] + |0〉a 〈1|αβη + |1〉a 〈0|αβη +|0〉a 〈0| [β2 + ξ(α2 − β2)]
The Hilbert-Schmidt norm for BH-cloning machine is
Da = 2ξ2(4α4 − 4α2 + 1) + 2α2β2(η − 1)2
As BH-cloning machine is state-independent dDa
dα2 = 0, thus resultingto a relation
η = 2ξ − 1
The norm Da = 2ξ2
Considering dDab
dα2 = 0 then the cloning machine is state-independentfor
ξ = 16 .
Abhineet Agarwal, Palash Pandya CQIS Project Report
IntroductionPartial Cloning
Quantum Error Correction CodesReferences
Wooters-ZurekBuzek-Hillery
Applyign partial Cloning to BB84
Alice sends some information to Bob via quantum channal.
Eve is also having access to this quantum channal and it tries toread the infromation so that Alice and Bob does not get to knowabout her presence.
As noise is also present, as noise is also some unitary transformationfrom a set of unitary transformations, the infromation Alice sendsgets disturbed or distorted due to it and then Eve applies parialcloning to it and finally this is measured by Bob in one of the basisset (horizontal-vertical or oblique).
Abhineet Agarwal, Palash Pandya CQIS Project Report
IntroductionPartial Cloning
Quantum Error Correction CodesReferences
IntroductionThree qubit bit flip codeThree qubit phase flip codeThe Shor code
Introduction
Key idea: to protect from the effects of noise using enoughredundancy to recover the corrupt part of the message.Suppose sending a bit through a noisy classical channel, bit flipchannel.The channel flips the bit from 0 to 1 or 1 to 0, with a probabilityp > 0. Such a channel is called binary symmetric channel(Fig 1)
Figure : Binary Symmetric Channel
Abhineet Agarwal, Palash Pandya CQIS Project Report
IntroductionPartial Cloning
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IntroductionThree qubit bit flip codeThree qubit phase flip codeThe Shor code
Introduction
Introduce redundancy might be to replace the original bit by 3copies of itself.
0→ 000
1→ 111
000 and 111 are referred to as logical 0 and logical 1 respectively.
If 001 was received, it is most likely that the third bit was flipped,and 0 was the bit that was sent. This is called majority voting.
The probability that more than one bit is flipped is 3p2(1− p) + p3,thus the probability of error is pe = 3p2 − 2p3.
Encoding makes the transmission more reliable if pe < p, whichhappens when p < 1/2.
Abhineet Agarwal, Palash Pandya CQIS Project Report
IntroductionPartial Cloning
Quantum Error Correction CodesReferences
IntroductionThree qubit bit flip codeThree qubit phase flip codeThe Shor code
Introduction
For quantum error correction codes we have to take into accountthe differences in the classical and quantum information,
No Cloning : The duplication of quantum states is forbidden by theNo cloning theorem.Continuous Errors: A errors that might affect a qubit occur in acontinuum. Determining which error to correct requires infiniteprecision and thus infinite resources.Measurement destroys Quantum Information: Observation generallycauses the state to collapse into some other state making therecovery impossible.
Abhineet Agarwal, Palash Pandya CQIS Project Report
IntroductionPartial Cloning
Quantum Error Correction CodesReferences
IntroductionThree qubit bit flip codeThree qubit phase flip codeThe Shor code
Three qubit bit flip code
A bit flip channel, which leaves the bits untouched with probability1− p, and flips with the probability p
X , is the bit flip operator. For this kind of noise, the bit flip code isdeveloped.
State a|0〉+ b|1〉 is encoded as a|000〉+ b|111〉.
|0〉 → |0L〉 ≡ |000〉|1〉 → |1L〉 ≡ |111〉
Abhineet Agarwal, Palash Pandya CQIS Project Report
IntroductionPartial Cloning
Quantum Error Correction CodesReferences
IntroductionThree qubit bit flip codeThree qubit phase flip codeThe Shor code
Three qubit bit flip code
All the three qubits are sent through the bit flip channel. If bit flipoccurs on one or fewer qubits, then a two stage error correctionprocedure is used.
Error correction or syndrome diagnosis: By the help of measurementwe find out if an error has occurred or not. Have four projectionoperators for the bit flip channel:
P0 ≡ |000〉〈000|+ |111〉〈111| no error
P1 ≡ |100〉〈100|+ |011〉〈011| bit flip on qubit one
P2 ≡ |010〉〈010|+ |101〉〈101| bit flip on qubit two
P3 ≡ |001〉〈001|+ |110〉〈110| bit flip on qubit three
Abhineet Agarwal, Palash Pandya CQIS Project Report
IntroductionPartial Cloning
Quantum Error Correction CodesReferences
IntroductionThree qubit bit flip codeThree qubit phase flip codeThe Shor code
Three qubit bit flip code
Suppose bit flip occurs on the first qubit: |ψ〉 = a|100〉+ b|011〉.Then the measurement 〈ψ|P1|ψ〉 = 1, else 0.
The operator does not change the state, and does not reveal anyinformation on the possible values of a or b.
Recovery : According to the measurement that results in value 1, weknow which qubit is flipped and accordingly perform the recoveryprocedure by flipping the appropriate bit and decode the state withperfect accuracy.
The probability that the error remains uncorrected is 3p2 − 2p3. Thereliability of the communication is increased only if p < 1/2.
Abhineet Agarwal, Palash Pandya CQIS Project Report
IntroductionPartial Cloning
Quantum Error Correction CodesReferences
IntroductionThree qubit bit flip codeThree qubit phase flip codeThe Shor code
Three qubit bit flip code
Just two measurements of the observables Z1Z2 (Z ⊗ Z ⊗ I ) andZ2Z3 (I ⊗ Z ⊗ Z ) can be used instead of four.
Z1Z2 = (|00〉〈00|+ |11〉〈11|)⊗ I − (|01〉〈01|+ |10〉〈10|)⊗ I
Like comparing the first and second or the second and third qubits.If they are same we get 1 else −1.
From the measurement results, we can conclude:
Z1Z2 = Z2Z3 = 1, then no error.Z1Z2 = 1 and Z2Z3 = −1, then third qubit flipped.Z1Z2 = −1 and Z2Z3 = −1, then second qubit flipped.Z1Z2 = −1 and Z2Z3 = 1, then first qubit flipped.
Application of X operator to the erroneous qubit recovers theoriginal state.
Abhineet Agarwal, Palash Pandya CQIS Project Report
IntroductionPartial Cloning
Quantum Error Correction CodesReferences
IntroductionThree qubit bit flip codeThree qubit phase flip codeThe Shor code
Three qubit phase flip code
Phase flip channel : With probability p the relative phases of thequbits is flipped. The state a|0〉+ b|1〉 changes to a|0〉 − b|1〉.The result of application of the phase flip operator Z .
The qubit basis {|+〉, |−〉}, and with respect to this basis the Zoperator acts as the bit flip operator.
States |0L〉 ≡ |+ ++〉 and |1L〉 ≡ | − −−〉 are used.
Encoding is done in two steps, first create 3 qubits exactly as theinput state and then apply Hadamard gates to each qubit.
Error detection similarly done using projection operators conjugatedby Hadamard gates, H⊗3PjH
⊗3.
Abhineet Agarwal, Palash Pandya CQIS Project Report
IntroductionPartial Cloning
Quantum Error Correction CodesReferences
IntroductionThree qubit bit flip codeThree qubit phase flip codeThe Shor code
Three qubit phase flip code
Can just use the two projection operators H⊗3Z1Z2H⊗3 = X1X2 andH⊗3Z2Z3H⊗3 = X2X3.
This is also like comparing the qubits, from X1X2,
+1 for states |+〉|+〉 ⊗ (·) or |−〉|−〉 ⊗ (·)−1 for states |+〉|−〉 ⊗ (·) or |−〉|+〉 ⊗ (·).
For error recovery, the appropriate bit is recovered using theHadamard conjugated X operator, HXH = Z .
The bit flip and phase flip models are unitarily equivalent, as theyare connected by a unitary operator, H.
Abhineet Agarwal, Palash Pandya CQIS Project Report
IntroductionPartial Cloning
Quantum Error Correction CodesReferences
IntroductionThree qubit bit flip codeThree qubit phase flip codeThe Shor code
The Shor code
Shor code is a combination of the bit flip and phase flip codes, andit allows recovery from arbitrary errors on a single qubit.
First encode the qubit using the phase flip code and then each ofthose bits is encoded using the bit flip code, this resulting in ninequbit code.
|0〉 → |0L〉 ≡(|000〉+ |111〉)(|000〉+ |111〉)(|000〉+ |111〉)
2√
2
|1〉 → |1L〉 ≡(|000〉 − |111〉)(|000〉 − |111〉)(|000〉 − |111〉)
2√
2
Abhineet Agarwal, Palash Pandya CQIS Project Report
IntroductionPartial Cloning
Quantum Error Correction CodesReferences
IntroductionThree qubit bit flip codeThree qubit phase flip codeThe Shor code
The Shor code
Consider the bit flip error occurring in the first qubit. First measureZ1Z2 and then Z2Z3, according to the results described before, bitflip on the first qubit is confirmed.
Tf there is a phase flip in any of the first three qubits it will change|000〉+ |111〉 to |000〉1|111〉 or vice versa.
The syndrome measurement is done by comparing the signs of thefirst and second blocks and then of the second and third blocks.
We change the sign in the affected block by applying Z1Z2Z3,similarly for all the nine qubits.
Even if the errors combined, i.e. the operator ZX was applied onsome qubit, the same methods of recovery recover from thecombined error.
Abhineet Agarwal, Palash Pandya CQIS Project Report
IntroductionPartial Cloning
Quantum Error Correction CodesReferences
References
Quantum Information and Quantum Computation, Nielsen andChuang.
V. Buzek and M. Hillery, Phys. Rev. A 54, 1844 (1996).
W. K. Wootters and W. H. Zurek, Nature 299, 802 (1982).
Abhineet Agarwal, Palash Pandya CQIS Project Report