Quantum Error Correction

1. Quantum Error Correction Joshua Kretchmer Gautam Wilkins Eric Zhou

2. Error Correction • Physical devices are imperfect • Interactions with the environment • Error must be controlled or compensated for • One step has probability to succeed = p • t steps has probability to succeed = pt

3. Classical Error Correction • Error Model • Channels provide description of the type of error • Encoding • Extra bits added to protect logical bit • String of bits  codeword • Redundancy • Error Recovery • Recovery operation • Measure bits and re-set all values to majority vote

4. b b 0 b 0 b Classical 3-Bit Code: Bit Flip Error • Bit flip channel: bit is flipped with prob. = p < 1/2 • Encoding: • Error Recovery: 000  { (000, (1-p)3), (001, p(1-p)2), (010, p(1-p)2), (100, p(1-p)2) (011, p2(1-p)), (110, p2(1-p)), (101, p2(1-p)) (111, p3) } • Prob(unrecoverable error) = 3p2(1-p)+p3 =3p2-2p3

5. Problems with QEC • No cloning theorem • Can’t copy an arbitrary quantum state • Entanglement • Measurement • Cannot directly measure a qubit • Error syndrome • Quantum evolution is continuous

6. Quantum 3-Bit Code: Bit Flip Error • Encoding  a|000>+b|111> • Error channel • Noise acts on each qubit independently • Probability noise does nothing = 1 - p • Probability noise applies x = p < 1/2 |> |>=a|0>+b|1> |0> |0> M |0> |0> Encoding Decode X Error Channel M Diagnose and Correct

7. Quantum 3-Bit Code: Bit Flip Error • After channel  8 possible results State: Probability: a|000>+b|111> (1-p)3 a|100>+b|011> p(1-p)2 a|010>+b|101> p(1-p)2 a|001>+b|110> p(1-p)2 a|110>+b|001> p2(1-p) a|101>+b|010> p2(1-p) a|011>+b|100> p2(1-p) a|111>+b|000> p3

8. Quantum 3-Bit Code: Bit Flip Error • After CNOT’s  4 possible results State: Probability: a|000>+b|111>|00> (1-p)3 a|100>+b|011>|10> p(1-p)2 a|010>+b|101>|01> p(1-p)2 a|001>+b|110>|11> p(1-p)2 a|110>+b|001>|01> p2(1-p) a|101>+b|010>|10> p2(1-p) a|011>+b|100>|11> p2(1-p) a|111>+b|000>|00> p3

9. Quantum 3-Bit Code: Bit Flip Error • Measure 2 ancilla qubits  error syndrome Measured syndrome action 00 do nothing 01 apply x to 3rd qubit 10 apply x to 2nd qubit 11 apply x to 1st qubit • Designed to correct if there’s an error in 1 or no qubits • Error in 2 or 3 qubits is an uncontrollable error

10. Quantum 3-Bit Code: Bit Flip Error • Failing probability  pu = 3p2(1-p)+p3 =3p2-2p3 = O(p2) • Fidelity  success probability = 1- pu = 1- 3p2 • Without error correction pu = O(p)

11. Quantum 3-Bit Code: Phase Error • Random rotation of qubits about z-axis • Continuous error • P() = ei 0 = cos()I +isin()z 0 e-i •  - fixed quantity stating typical size of rotation •  - random angle

12. Quantum 3-Bit Code: Phase Error • Apply H to each qubit at either end of the channel • HIH = HH = I; HzH = x •  HPH = cos()I +isin()x • Same result from bit flip code • Fidelity = 1 - 3p2 • p = <sin2()>  (2)2/3 for <<1

13. General Quantum Error • Errors occur due to interaction with environment • |0>|E>  1|0>|E1> + 2|1>|E2> • |1>|E>  3|1>|E3> + 4|0>|E4> • (0|0> + 1|1>)|E>  01|0>|E1> +02|1>|E2> + 13|1>|E3> +14|0>|E4>

14. General Quantum Error • (0|0> + 1|1>)|E>  1/2(0|0> + 1|1>)(1|E1> + 3|E3>) + 1/2(0|0> - 1|1>)(1|E1> - 3|E3>) + 1/2(0|1> + 1|0>)(2|E2> + 4|E4>) + 1/2(0|1> - 1|0>)(2|E1> - 4|E4>) • 0|0> + 1|1> = |> 0|0> - 1|1> = Z|> 0|1> + 1|0> = X|> 0|1> - 1|0> = XZ|>

15. General Quantum Error • (0|0> + 1|1>)|E>  1/2(|>)(1|E1> + 3|E3>) + 1/2(Z|>)(1|E1> - 3|E3>) + 1/2(X|>)(2|E2> + 4|E4>) + 1/2(XZ|>)(2|E1> - 4|E4>) • Error basis = I, X, Z, XZ • |>L|>e  (i|>L)|i>e • |>L  general superposition of quantum codewords • i  error operator = tensor product of pauli operators

16. Correction of General Errors • |>L|>e  (i|>L)|i>e • |>L -orthonormal set of n qubit states • To extract syndrome attach an n-k qubit ancilla “a” to system  perform operations to get syndrome  |si>a  |0>a(i|>L)|i>e  |si>a(i|>L)|i>e • Measure si to determine i-1  correct for error  |si>a(i|>L)|i>e  |si>a(|>L)|i>e

17. Shor’s Algorithm • Each qubit is encoded as nine qubits

18. Shor’s Algorithm Assume decoherence on first bit of first triple, becomes:

19. Shor’s Algorithm

20. Shor’s Algorithm • No error • Z error • X error • ZX = Y error

21. Shor’s Algorithm • Success Rate: • Works if only one qubit decoheres • If probability of a qubit decohering is p • Probability of 2 or more out of 9 decohering is1-(1+8p)(1-p)836p2 • Therefore probability that 9*k qubits can be decoded is (1-36p2)k

22. Shor’s Algorithm • More on decoherence • Decoherence probability increases with time • Use watchdog effect to periodically reset quantum state • Unfortunately, each reset introduces small amount of extra error • Therefore cannot store indefinitely

23. Steane’s Algorithm • Basis 1 is |0 , |1  • Also called basis F, or “flip” basis • Basis 2 is |0 + |1 , |0 - |1  • Also called basis P, or “phase” basis

24. Steane’s Algorithm • The word |000…0  consisting of all zeroes in basis 1 is equal to a superposition of all 2n possible words in basis 2, with equal coefficients. • If the jth bit of each word is complemented in basis 1, then all words in basis 2 in which the jth bit is a 1 change sign. • Hamming Distance • The number of places two words of the same length differ • Minimum Distance • Smallest Hamming distance between any two code words in a code

25. Steane’s Algorithm • A code of minimum distance d allows [(d-1)/2] to be corrected • If less than d/2 errors occur, the correct original code word that gave rise of the erroneous word can be identified as the only code word at a distance of less than d/2 from the received word. • [n,k,d] is a linear set of 2k code words each of length n, with minimum distance d

26. Steane’s Algorithm • Parity Check Matrix • Matrix H of dimensions (n-k) by n, where Hv = 0 iff v is in the code C • Generator Matrix • Matrix G of dimensions n by k, basis for a linear code • w = cG, where w is a unique codeword of linear code C, and c is a unique row vector • For a linear code C in basis 1, a superposition with equal coefficients, then in basis 2 the words of the superposition form the dual code of C • The Parity Check Matrix of C is the Generator Matrix for its dual code

27. Steane’s Algorithm • Let |a  and |b  be expressed as [7, 3, 4] in basis 1:

28. Steane’s Algorithm • |a  and |b  are non-overlapping, and have distance of 3 • Find bit flip with parity check • Switch to basis 2: • |c  =|a  +|b  • Contains only even parity words of a [7,4,3] code • |d  =|a  -|b  • Contains only odd parity words • Distance between |c  and |d  is at least 3 • Phase error can be found with a parity check

29. Implications for Physical Realizations of Quantum Computers

30. Why Do We Need It? • Quantum computers are very delicate. • External interactions result in decoherence and introduction of errors.

31. Fault-tolerance • Especially important when considering physical implementations. • Must consider errors introduced by all parts, including gates. • Incorrect syndromes introduce errors.

32. Impact on Physical Systems • Increased size • Level of coherence determines increase

33. Alternative to Error Correction • Topological Quantum Computing • Involves particles called anyons that form braids, whose topology determines quantum state.

34. Topological Quantum Computing Slight perturbations to system cause braids to be deformed, but only large disturbances result in them being cut or joined.

35. Summary • Error correction is vital for physical realizations of trapped particle quantum computers. • Allows reliable quantum computation without requiring extremely high levels of coherence.