Introduction to Quantum Error Correction & Fault-Tolerant Quantum Logic

1. Introduction to Quantum Error Correction& Fault-Tolerant Quantum Logic Cherrie Huang

2. Why Quantum Error Correction?[6] Cause: circuit interacts with the surroundings • decoherence • decay of the quantum information stored in the device Solution: Quantum Error Correcting Codes • protect quantum information against errors. • perform operations fault-tolerantly on encoded states.

3. Major Difficulties [9] 1. No cloning theorem: impossible to duplicate an arbitrary unknown qubit • Solution: Fight Entanglement with entanglement(encode the information that we want to protect in entanglement). impossible possible Explain why “no cloning” is important in this context

4. Major Difficulties [9] 2. Errors are continuous: continuous errorsrequires infinite resources and infinite precision • Solution: Digitalize the errors that circuit makes. 3. Measurement destroys quantum information : recovery is impossible if quantum information state is destroyed. • Solution: Measure the errors without measuring the data.

5. Central Idea of QEC • A small subspace of the Hilbert space of the device is designated as the code subspace. • This space is carefully chosen so that all of the errors that we want to correct move the code space to mutually orthogonal error subspaces. • We can make a measurement after our system has interacted with the environment that tells us in which of these mutually orthogonal spaces the system resides, and hence infer exactly what type of error occurred. • The error can then be repaired by applying an appropriate unitary transformation.

6. Key Ideas of QEC • Encode the message with redundant information • Redundancy in the encoded message allows to recover the information in the original message. • Measure the errors, not the data.

7. General Model of QEC [1] • Deal with errors: • Error detection • Error correction • Errors also in encoding and recovery (they are themselves complex quantum computations) But, fault-tolerant recovery possible if error rate is not high (Peter Shor, 1996). • Problems to store an unknown quantum state with high fidelity for an indefinitely long time and problems to do quantum computationBut, possible if error rate is below threshold (Manny Knill and Raymond Laflamme, 1996).

8. Application of what? • Storage: CDs, DVDs, “hard drives” • Wireless: Cell phones, wireless links • Satelite and Space: TV, Mars rover • Digital Television: DVD, MPEGS layover • High Speed Modems: ADSL, DSL

9. Classical Error Correction • Hierarchy linear cyclic BCH Bose-Chaudhuri-Hochquenghem Hamming Reed-Solomon

10. Classical Repetition Code • Transmission: Sending one bit of information across the channel. • Noise: flips the bit with the probability p • Encoding: triple each bit : 0000, 1111, C={000,111} • Decoding: majority voting • Example:10111000 101001 10 • Limitation: not possible to recover the information correctly if more than one bit is flipped.

11. Classical Repetition Code • Analysis What is this? Explain better this and the whole table

12. QEC: The Three QubitBit Flip Code • Example: sending one qubit through a channel. • Noise: flips the qubit with the probability p. • In other words, the state |ψ> is taken to state X\ψ>with the probability p, where X (bit flip matrix) • Encoding: • Decoding: Majority Logic • Limitation: may be unable to recover the information correctly if more than one bit is flipped in some cases.

13. QEC: The Three Qubit Bit Flip Code • Encoding : |0>+|1>  |000>+|111> • Encoding Circuit • Measuring the ancilla bits reveals the error but not the information qubit.[2] Explain Why?

14. QEC: The Three Qubit Bit Flip Code Pre-assumption of the errors: One or none error occurs • Transfer the stored information to the output qubit. • Limited if more than one error. • We don’t have enough info of the location of errors.

15. Add more detailed captions to the table Analysis OF WHAT? The error probability can depend significantly on the initial state.

16. Fault-Tolerant Computation General Stages: Preparation/ Encode Verify Computation of Error Syndrome Recovery Explain better what each block does, especially verify

17. Fault-Tolerant Computation[6] Rules: • Implement gates that can process encoded information. • Control propagation of errors. • Ensure that recovery from errors is performed reliably.

18. Fault-Tolerant Computation[6] • 1st Law: Don’t use the same bit twice. Bad: Error propagates, so infection spreads. Good!

19. Fault-Tolerant Computation[6] • 2nd Law: Copy/measure the errors, not the data. • Copy the information from the data to the ancilla. • Measure the ancilla to find an error syndrome. • Based on the error syndrome, we perform the required recovery.

20. Fault-Tolerant Computation[6] • 3rd Law: Verify when you encode a known quantum state. A nondestructive measurement is performed (twice performed above) to verify that the encoding was successful. More explanation needed

21. Fault-Tolerant Computation • 4th Law: Repeat the operations More explanation needed

22. Fault-Tolerant Computation • 5th Law: Use the right code More explanation needed

23. Error Correction in The Three Qubit Code • |0>|000>, |1>|111> • Error Correction: More explanation needed

24. Example: The Shor Code • Also known as the 9-qubit code • Combination of the three qubit phase flip codes and bit flip codes. • Seen as a two-levelconcatenated code.[3] • One qubit is encoded into 9 qubits: • The data is no longer stored in a single qubit, but instead spread out among nine of them.[8] • Correction of bit flips: majority voting.

25. Assumptions of the Shor Code • For simplicity, we assume that any qubit error consists in the application of bit flip error, phase flip error, and/or combination of these two. • X (bit flip error) • Z (phase flip error) • Y = iXZ (combination of bit flip and phase flip error)

26. Preparation in The Shor Code Block #1

27. Majority Logic in The Shor Code[3] Explain decoding and recovery, how majority works, may be you need more slides for this

28. Bit Flip Correction • Bit flip : switch |0> and |1> • Describe the error as bit flip matrix X • Correction: • For a block, compare the first two qubits, and compare the first with the third. • If the first was flipped, it will disagree with the third. • If the second was flipped, the first and third will agree.

29. Phase Flip Correction • Example : • Describe the error as phase flip matrix Z • Correction : • By comparing the sign of the first block of three with the second block of three, we can see that a sign error has occurred in one of those blocks. • Then, by comparing the signs of the first and third blocks of three, we can narrow down the location of phase error and flip it back.

30. SimultaneousBit and Phase Flip Error • Describe the error as Y=iXZ • Correction: We can fix the bit flip first, and then fix the phase flip for the simultaneous bit and phase flip error, even if they are on different qubits.

31. Stabilizer Coding in The Shor Code[8] • Bit Flip Error: • Equivalent to measure the eigenvalues of Z1Z2 and Z1Z3. • For example, if the first two qubits are the same, the eigenvalue of Z1Z2 is +1; otherwise, the value is –1. • Phase Flip Error: • Equivalent to measure the eigenvalues of X1X2X3X4X5X6 and X1X2X3X7X8X9. • If the signs agree, the eigenvalues will be +1; otherwise, the values is –1. Remind on an example what are eigenvalues , define them

32. Stabilizer Coding in The Shor Code • In order to totally correct the code, we must measure the eigenvalues of a total of eight operators. Explain what we see here

33. Another Phase Error Correction in The Shor Code • Hadamard Transformation on each qubit. • The qubits taken : 1, 4, and 7 (or 2,5,8 or 3,6,9).

34. Stabilizer Code • Many quantum states can be more easily described by working with the operators that stabilize them than by working explicitly with the state itself. • |ψ> • X1X2|ψ> = |ψ> and Z1Z2|ψ> = |ψ> • |ψ> is stabilized by the operators X1X2 and Z1Z2. • |ψ> is the unique quantum state which is stabilized by these operators X1X2 and Z1X2.

35. Stabilizer Code • In making continuous weak measurements on our system, we would like to choose the measurements in such a manner that we gather as much information about the errors as possible while disturbing the logical qubits as little as possiblequantum error correcting code. • Stabilizer formalism provides a way to easily characterize many of the error correcting codes. • Pauli group Pn= {1, i}{I,X,Y,Z}n Give examples of Pauli group operators

36. Stabilizer Code[4] • There exist a set of operators in Pn, called the stabilizer generatorsand denoted by g1, g2, ..., gr. • They are such that every state in C is an eigenstate with eigenvalue +1 of all the stabilizer generators. • That is, gi|ψ>= |ψ>for all i and for all states |ψ>in C. • Moreover, these stabilizer generators are all mutually commuting.

37. Stabilizer Code[4a] • The stabilizer code error correction procedure involves: • 1) simultaneously measuring all the stabilizer generators and then • 2) inferring what correction to applyfrom the measurement results. • The formalism states that the stabilizer measurement results indicate a unique correction operation.

38. Example:The Three Qubit Bit Flip Code

39. The Classical [7,4,3] Hamming Code • Transmit the block 0011 0 0 1 1 Parity bits = comes from the rule that the total number of 1’s contained in each circle should be even. 0 1 1 0 1 1 0

40. Steane’s Code • One qubit is encoded into seven qubits. • Logic 0= those with even number of 1’s • Logic 1= those with odd number of 1’s

41. Encoder for the Steane’s Code

42. Computation of Bit Flip Syndrome

43. Computation of Phase Errors R = Hadamard Rotation =

44. Fault-Tolerant Logic Gate • The fault-tolerant quantum gates and measurements must prevent a single error from propagating to more than one error in any code block. • Therefore the small correctable errorswill not grow to exceed the correction capability of the code. [7]

45. One-Bit Teleportation • One-teleportation is based on Swap gate I do not understand . This is not swap Explain why we need one-bit teleportation

46. Several Facts to derive one-bit teleportation • Fact 1 : X = HZH where

47. Several Facts to derive one-bit teleportation • Fact 2: When the control qubit is measured, a quantum-controlled gate can be replaced by a classical controlled operation. U is performed if the measurement result is 1. Why it is so?

48. Z-teleportation • The two bits are disentangled before the second Hadamard gate. • Therefore the second qubit can be measured before the second Hadamard gate without affecting the unknown state in the first qubit. H|> Why it is so?

49. X-teleportation Why it is so?

50. |0> |x > H X |0> |y > H X |0> |z + xy > Z |x > |y > |z > H Fault-Tolerant Toffoli Using One-bit Teleportation Non-FT Gate