Quantum Information Theory Graduate Course Spring 2005

1. 1 Quantum Information TheoryGraduate Course � Spring 2005 Marco Lanzagorta & Jeff Tollaksen George Mason University

2. 2 Quantum Error-Correction High-Fidelity Transmission and Manipulation of Quantum Information

3. 3 Why error correction is important? Bowman: Open the pod bay doors, HAL. HAL 9000: I�m sorry Dave, I�m afraid I cannot do that.

4. 4 All you need to know� An [n,k,d] quantum error correction code C(S) is the vector space VS stabilized by a subgroup S of Gn such that and S has n-k independent and commuting generators: and logical states stabilized by: which can correct a set of correctable error operators {Ei} in Gn such that, for all j and k:

5. 5 Structure The best way to arrive to this complicated result, while fully understanding its meaning and implementation, is through an iterative structure: We start by looking at error correction in classical computing, then we extend these ideas into quantum computing, then we formalize these results, and then we will further formalize, and so on� Some concepts and examples will be seen more than once, but each time we will add a new layer of detail, complexity and abstraction. To avoid getting lost, it is important to be sure that you understand each step of the way!

6. 6 Outline Classical Error Correction (L5) Quantum Error Correction (L5) Symmetrisation Procedures (L6) Quantum Error Correction Codes (L6) Formal Theory of Quantum Error Correction (L7) The Stabilizer Code Formalism (L7) The Stabilizer Group (L8) Constructing Quantum Codes (L8) Conclusions (L8)

7. 7 1. Classical Error Correction �This kind of thing has cropped up before, and it has always been attributable to human error� HAL 9000

8. 8 Classical Error Correction Error Correction is not unique to quantum computing, it has its origins in classical computing. Classical Error Correction makes classical communications fault tolerant. In reality, there is no errorless communication of information. CEC allows communications over noisy and erroneous data channels. Of fundamental importance for telecommunications, internet, fax�

9. 9 Classical Errors Suppose c binary sequences (or codewords) of length n: w1, w2� wc. During transmission or storage, external noise produces random flips (the only type of error in CC). If the channel is a binary symmetric memory-less channel, then the set of possible received sequences is the set of all the 2n binary sequences of length n: v1, v2� v2n.

10. 10 Binary Symmetric Channel In a binary symmetric channel, each bit is transmitted with an error probability of e. 0 0 1 1

11. 11 The Hamming Distance The task of the receiver is, given v0, to identify the most likely codeword wi sent by the transmitter. That is, the wi closest to v0. The distance between two binary sequences, d(wi, v0), is knows as the Hamming distance. It is measured by the number of digits in which the two strings differ. For a binary symmetric memoryless channel, the codeword with the smallest Hamming distance is also the most likely.

12. 12 The Hamming Bound The larger the distance between codewords, the easier to distinguish them in the presence of errors. The code is more robust against the effects of noise. If: then up to h errors can be corrected. Upper bound to the number of codewords c able to correct up to h errors.

13. 13 The Hamming Bound: Geometric View The Hamming bound can be visualized as a sphere of radius h around each codeword. If the code can correct the errors, then these spheres must be disjoint. The number of sequences in each sphere, times the number of spheres, has to be smaller than the total number of sequences of length n:

14. 14 Parity Check Codes The codewords w are chosen in such a way that they satisfy a set of linear equations, characterized by the parity check matrix M: Mw = 0 The receiver then tests if v satisfies the equation. If v fails, then the receiver corrects its value.

15. 15 Error Pattern and Error Syndrome Error Pattern: z = w � v Error syndrome: s = -Mv = -M(w-z) = Mz The receiver has to detect an error syndrome s, and then try to determine the error pattern z that might have produced s.

16. 16 Hamming Bound for Parity Check Codes If m=rank(M), then k=n-m bits can be specified arbitrarily, while the remaining m are parity check digits. Then, the number of linearly independent codewords is c = 2n-m = 2k then, the lower bound for the number of check digits m is:

17. 17 Simple Example If we wish to encode 1 bit using n bits, in such a way that 1 bit errors are tolerated, then the Hamming bound implies that we need to have n>2. Therefore, we need at least 3 bits to encode 1 bit in a code that protects against 1 bit errors.

18. 18 Data Redundancy The major idea behind more sophisticated CECs is to encode k bits of data in n bits (n>k). The n-k bits introduce additional information, encoding the original data in a redundant way. Such redundancy can be used to detect bit errors during transmission. An error correcting code with data redundancy that repeats the original information a number of times is also known as a repetition code.

19. 19 Classical Linear Codes A code that encodes k bits in n bits (n>k) is called an [n,k] code. A general code encoding k bits in n bits requires 2k codewords of length n to specify the encoding. A linear code C encodes k bits of information into an n bit code space specified by an n by k generator matrix G with elements in Z2. A linear code only requires kn bits to specify the encode.

20. 20 Linear Encoding We encode a k bits codeword x, into a n bits codeword c using a [k by n] generator matrix G as follows: Error correction for linear codes is done using a [(n-k) by n] parity matrix.

21. 21 Parity Check Equations Parity check matrix H is such that: H c = 0 and H GT= 0 The receiver gets the codeword r, which incorporates an error e: r = c + e Then, the syndrome s is given by: s = H r = H e

22. 22 Error Correction & Recovery Once we detect the syndrome s, we can find the error that occurred e. Now we can correct the error as: c = r � e And finally we can recover the original codevector with: xT = cT G-1

23. 23 General Steps for Classical Error Correction Encoding The sender encodes the codeword bits Error-Detection or Syndrome Diagnosis The receiver determines what type of error has occurred Recovery The receiver transform the defective codeword to obtain the most likely error-free codeword. Decoding - The receiver transforms the encoded codeword to retrieve the original codeword.

24. 24 Logical Codewords The most general error correction codes are of the form [n,1] and work only with the two logical codewords: So a general binary message is transformed as:

25. 25 Dual Construction Suppose C is a [n,k] code with generator matrix G and parity check matrix H. Then: H GT= 0 and therefore G HT = 0 The dual of C, denoted by , is the code with generator matrix H and parity check matrix GT. The dual of C consists of all the codewords y such that y is orthogonal to all the codewords of C.

26. 26 Self-Dual Codes A code is said to be weakly self-dual if: A code is said to be strictly self-dual if:

27. 27 2. Quantum Error Correction �To make errors is part of human nature, but if you really want to mess up things, then you need a quantum computer.� Popular Proverb

28. 28 The Quantum Channel A quantum channel is a connection that transmits qubits. Ideally, coherence is preserved in a quantum channel, but this is not the case in real systems. Quantum errors are usually more severe and varied than their classical counterparts.

29. 29 The Problem of Quantum Noise Environment quantum noise affects the stability of the quantum registers. Quantum registers require to be stable for relatively long periods of time, so computations can be performed or information transmitted to other systems. Imposes severe limitations to practical quantum computing.

30. 30 Limits of Quantum Computing Example, an ion trap quantum computer (qubit stored in a meta-stable optical transition) running Shor�s algorithm for n < 2L. Then, because of spontaneous emissions, L is bounded by: 2.2 (Ca+), 1.6 (Hg+) and 4.5 (Ba+). Completely unacceptable bounds for most practical applications.

31. 31 Quantum Error-Correction Quantum noise leads to the corruption of the information in the quantum register, and therefore it is unacceptable for the reliable transmission, manipulation and processing of quantum information. Quantum Error-Correction protocols encode quantum states in a special way that makes them resilient against the effects of noise, and then decoding when it is wished to recover the original state.

32. 32 Error-Free Gates In the context of quantum error correction codes, we assume that the errors occur while a quantum state is being transmitted or seated on a register. That is, we assume that the quantum gates are error-free. Obviously this is not a realistic scenario, but �Fault Tolerant Quantum Computing� is the technique that deal with gate errors.

33. 33 Illustration

34. 34 Error Correction Codes A quantum correction code usually allows correction of a particular set S={E} of correctable errors. The task of code construction consists of finding codes whose correctable errors include the most likely to appear in the physical device.

35. 35 General Steps for Quantum Error Correction Encoding The sender encodes the codeword Error-Detection or Syndrome Diagnosis The receiver determines what type of error has occurred Recovery The receiver transform the defective codeword to obtain the most likely error-free codeword. Decoding - The receiver transforms the encoded codeword to retrieve the original codeword.

36. 36 Quantum Errors A quantum error may include any of the following: A qubit may flip its value. A qubit may change its phase. A qubit may decohere (become entangled with the environment).

37. 37 Difficulties for QEC Non-Cloning Theorem We cannot copy quantum states. Continuous Errors We require infinite precision to determine the error in a single qubit. Measurement destroys quantum information We cannot read the state of the quantum register before the end of the computation.

38. 38 Quantum Codevectors In QEC we use quantum codevectors |w>, which are entangled states of n qubits. The information we want to protect is then �spread� by the entanglement over all the n qubits. Reading or decohering a few qubits will not necessarily lead to an irreversible loss of quantum information.

39. 39 Error Measurement As we cannot measure the qubit to determine what error has occurred, we need a more subtle syndrome diagnosis procedure. Codevectors are chosen in such a way that an error will move the |w> into mutually orthogonal subspaces. Measurement of the syndrome will therefore reveal only which subspace |w> has moved to.

40. 40 A Neat Measuring Trick Suppose we have an operator M, which is Hermitian and unitary, with eigenvalues +1 and �1. Then, the following circuit can be used to measure the eigenvalues of M, and leaving the qubit in the corresponding eigenvector:

41. 41 Error Probabilities We presume that each qubit can undergo an error with probability e. We also assume that the errors on different qubits are independent. Then, the probability of errors on two qubits is of O(e2). So, if e is very small, then we can assume that only one error has occurred, and the probability of success is (1-e). A QEC that corrects single errors can increase the probability of success to (1-O(e2)). A QEC that corrects t errors can increase the probability of success to (1-O(et+1)).

42. 42 3. Symmetrisation Procedures �I have to stay functional until my mission is complete. Then it does not matter.� The Terminator

43. 43 Symmetrisation Procedures Suppose that we can prepare R copies of a quantum state that we need to manipulate. Then, we project the state of the combined system onto the symmetric subspace (the space that contains all states which are invariant under any permutation of the sub-systems). The error free evolution of the R independent copies starts and finishes in the symmetric subspace. Then, frequent projections on the symmetric subspace will reduce errors induced by the environment.

44. 44 Symmetrisation Operator In the sake of simplicity, consider a two qubit state. The symmetrisation operator is: S = (1/2)(P12 + P21) where: then:

45. 45 Density Matrices We perform the symmetrisation of the density matrix: In particular: with: So rS is purer than r.

46. 46 Environment Interaction To model the interaction with the environment, let us also suppose that: where z is a traceless Hermitean matrix that represents the interaction with the environment

47. 47 Purity of the States Taking the first order in perturbation: We can calculate the average purity of the two copies before symmetrisation by calculating the average trace of the squared states:

48. 48 Purity of the Symmetrised States After symmetrisation each qubit is in the state: and has purity: Since this purity is closer than 1 than with the original states, the symmetrised state is left in a purer state.

49. 49 Fidelity Before symmetrisation: After symmetrisation:

50. 50 R Copies Purity: Fidelity: Thus, by choosing R very large, the residual error can in principle be controlled to lie within a small tolerance.

51. 51 4. Quantum Error Correction Codes �They will fix you. They fix everything.� Robocop

52. 52 Quantum Error Correction Codes General types of QECC, according to the type of quantum error: Qubit bit flip codes Qubit phase flip codes Qubit bit flip and phase flip codes

53. 53 Quantum Bit Flip Errors In this type of errors, a noisy channel may flip a qubit (aka bit flip channel). Similar to classical error. One possible solution: data redundancy.

54. 54 The Three Qubit Bit Flip Code (1) Encoding (repetition code): Projectors for Syndrome Diagnosis:

55. 55 The Three Qubit Bit Flip Code (2) Syndrome Diagnosis: If <P0> = 1, then, no error. If <P1> = 1, then, bit flip error on qubit one If <P2> = 1, then, bit flip error on qubit two If <P3> = 1, then, bit flip error on qubit three Measurement in this basis does not change the state because they are orthogonal. Measurement does not yield any information about the values of a and b.

56. 56 The Three Qubit Bit Flip Code (3) Recovery: Flip the bits accordingly. This procedure works perfectly as long as a flip error occurs on at most 1 qubit (probability of error is: 3e2 � 2e3). Measurement of the projector operators is theoretically viable, but it may be hard to implement using a small set of quantum gates.

57. 57 Alternative Bit Flip Error Correction Code (1) Instead of using the projectors describe, we could perform two measurements on the observables Z1Z2I3 and I1Z2 Z3. Remember that: Then:

58. 58 Alternative Bit Flip Error Correction Code (2) Each of these has eigenvalues +1 and �1, and the eigenvectors are the two valid codewords |000> and |111>. Syndrome table:

59. 59 Error Analysis (1) Quantum error correction should increase the fidelity of the transmission channel. Before error correction: with Fidelity:

60. 60 Error Analysis (2) After using the three qubit bit flip code we have: with fidelity: Therefore, the fidelity bound is improved only if p<1/2.

61. 61 Quantum Phase Flip Errors In this type of errors, a noisy channel may flip the phase of a qubit (aka phase flip channel). Without equivalent in classical error theory. One possible solution: data redundancy.

62. 62 Three Qubit Phase Flip Code (1) A very easy solution, when encoding the qubit, change the basis to: Then, the phase flip error is equivalent to: Which is a bit flip error! Then correct for bit flip errors in the {+,-} basis.

63. 63 Three Qubit Phase Flip Code (2)

64. 64 Three Qubit Phase Flip Code (3) Observables for syndrome measurement: Syndrome table:

65. 65 Quantum Bit Flip and Phase Flip Errors A more general error may combine a bit flip (X) with a phase flip (Z). XZ=Y To protect against both errors, first we encode the qubit using the phase flip code, and then encode them again using the bit flip code. Concatenation: a common technique used to build better codes by applying one after the other.

66. 66 Shor�s Nine Qubit Code (1) For the 3-qubit examples we have seen, to encode for phase flip and then for bit flip means that we require to encode 1 qubit of quantum information using 9 qubits:

67. 67 Shor�s Nine Qubit Code (2) Then:

68. 68 Shor�s Nine Qubit Code (3) Operators for Bit Flip Syndrome diagnosis: Z1Z2 Z2Z3 Z4Z5 Z5Z6 Z7Z8 Z8Z9 Operators for Phase Flip Syndrome diagnosis: X1X2 X3X4 X5X6 X4X5 X6X7 X8X9

69. 69 More General Types of Errors So far we have seen discrete errors: No Error (I) Bit Flip Error (X) Phase Flip Error (Z) Bit and Phase Flip Error (Y=XZ) However, a more general type of error can occur:

70. 70 Arbitrary Errors Because the 4 Pauli matrices (I,X,Y,Z) form a basis for 2x2 matrices, then every general error E can be written as: E = e1 I + e2 X + e3 Z + e4 Y Therefore, Shor�s nine qubit code can be used as a protection against completely arbitrary errors, as long as they affect a single qubit.

71. 71 Discretization of Errors (1) A Continuum of errors can be reduced to a discrete set of errors. If we measure a syndrome in the (I,X,Y,Z) basis, the state collapses the superposition of errors into one of the four states. Then, we reduce an arbitrary error to one of the four basic ones.

72. 72 Discretization of Errors (2) Suppose we have a noisy quantum channel. We describe the noise with a trace-preserving quantum operator E, which is represented by {Ei}. Then: If error occurs only in the first qubit:

73. 73 Discretization of Errors (3) Measurement of the error syndrome then collapses the state into: And recovery process is performed only for one type of error (X, Z, or Y). By correcting a discrete set of errors, the code automatically corrects for a much larger class of errors.

74. 74 5. Formal Theory of Quantum Error Correction �Quantum Computers are like the Gods in Nordic Mythology (Odin, Thor,�): they impose lots of rules and show no mercy!� Popular Proverb

75. 75 A Zoology of Quantum Codes Quantum Error Correction Codes are characterized by the triplet [n,k,d], where: n is the length of the resulting codeword. k is the number of qubits to be encoded. d is the minimum distance. Notice that data redundancy implies n>k A code with minimal distance d=2t+1 is able to correct errors on up to t bits.

76. 76 Weight and Distance The weight t of an error is the number of qubits acted on by non-trivial Pauli Matrices (X,Y,Z). A code with minimal distance d can correct all errors with weight up to (d-1)/2. Thus, error correction means d at least 2t+1 (recall classical theory). Three-qubit bit flip code: [3,1,1] Shor�s nine qubit code: [9,1,3].

77. 77 Encoding as Space Mapping A quantum error correction encoding can be viewed as a mapping of k qubits in a 2k dimensional Hilbert space, into n qubits in a 2n dimensional Hilbert space. The additional n-k qubits provide the redundancy.

78. 78 Quantum Codes Quantum states are encoded by a unitary operation into a quantum error-correcting code, a subspace C of a much larger Hilbert space. An operator P projects a quantum state onto the code space C. These subspaces have to be orthogonal for reliable syndrome measurement. Errors mapping to different subspaces must take the orthogonal codewords to orthogonal states.

79. 79 Logical States A general state in the C space is called an encoded or logical state: Remember, the first step in constructing error correction codes is to determine the most suitable 2k logical states that form a basis for C.

80. 80 Example (1) Three qubit bit flip code. The encoding creates two logical states: The error correction code subspace C is spanned by these two logical states. The projector P is given by:

81. 81 Example (2) Consider the set of errors: Clearly, they take the logic states to orthogonal spaces: Then, there is no ambiguity about the error syndrome that has occurred.

82. 82 Example (3) A really bad choice for our quantum error correction code would be, for instance: Because: And we cannot distinguish what error really happened.

83. 83 General Assumptions The quantum noise is described by a quantum operator E. The complete error-correction procedure is effected by a trace-preserving quantum operator R, which we call the error-correction operation.

84. 84 Successful Quantum Error Correction If the quantum error correction code is successful, then, for any state r we have: Sometimes we may be interested in E being a non-trace-preserving error operation, such as a measurement, so we cannot write �=�.

85. 85 Quantum Error Correction Conditions Theorem: Let C be a quantum code, and let P be the projector onto C. Suppose E is a quantum operation with operation elements {Ei}. A necessary and sufficient condition for the existence of R correcting E on C is that: for some Hermitean matrix a of complex numbers.

86. 86 Correctable Errors If the Quantum Error Correction Conditions are satisfied, then we call the {Ei} elements the noise E errors and: If such an error correcting code R exists, then we say that {Ei} constitutes a correctable set of errors.

87. 87 Example For the 3 qubit bit flip code, we have: And also: Then, for instance:

88. 88 Distinguishable Errors (1) A sufficient condition for a code to correct two errors Ea and Eb, is that it must be able to distinguish them when acting on two different logical codewords. For Ea and Eb to be distinguishable, they have to be orthogonal: It is sufficient, but not necessary!

89. 89 Distinguishable Errors (2) A necessary condition for error correction, however, is that: That is, the �corrupted codevectors� have to be orthogonal. Otherwise, we cannot tell them apart.

90. 90 Distinguishable Errors (3) Therefore, the necessary and sufficient condition for error recovery is: where: is an arbitrary Hermitian matrix independent of the i-states. This is exactly the theorem we just saw!

91. 91 Degenerate Codes When two or more different types of errors lead to the same codewords, we say we have a degenerate code. Example: Z1 and Z2 have the same effect on both logical codewords of Shor code. Degeneracy is a quantum effect with no classical counterpart. Makes very difficult to establish bounds on code performance. On the good news, they �pack more information�.

92. 92 Discretization of Errors Theorem: Suppose C is a quantum code and R is the error-correction operation for E with {Ei}. Suppose F is a quantum operation with operator elements {Fi} which are linear combinations of {Ei}. Then, R also corrects for the effects of the noise process F on the code C.

93. 93 Advantages of the Discretization of Errors Therefore, any code that corrects the depolarization channel automatically implies the ability to correct any arbitrary single qubit quantum operation. This is in strong contrast to classical error correction for analog systems, which is very complex because we cannot perform such a reduction.

94. 94 Independent Error Models (1) Single-qubit errors may occur in more than 1 qubit. This problem is simplified if we suppose that the errors are independent. If the noise is sufficiently weak, then we can protect the information. Note: this is different than:

95. 95 Independent Error Models (2) Consider the depolarization channel: with fidelity: The depolarization channel for many qubits is:

96. 96 Errors in more than 1 qubit What happens if an error affects more than one qubit? If the error is small and the noise acts on the qubits independently, then we can correct the code. We make an expansion on the power of the error. No-error and 1-qubit-error will dominate the expansion.

97. 97 Quantum Bounds Three most important bounds to quantum error correction codes are: Quantum Hamming Bound Quantum Gilbert-Barshamov Bound Quantum Singleton / Knill-Laflamme Bound The are helpful to understand the theoretical limitations of building quantum error correction codes.

98. 98 Quantum Hamming Bound (1) Suppose a non-degenerate quantum error correction code that encodes k qubits in n qubits in such a way that can protect up to h errors. Then: Compared to the classical case, we have an extra 3j factor due to the three possible errors we can have (X,Y,Z) in the quantum case.

99. 99 Quantum Hamming Bound (2) Note that the quantum Hamming bound only applies to non-degenerate codes but gives some insight regarding the degenerate cases. Consider the case in which we wish to encode 1 qubit in n qubits in such a way that errors on 1 qubit are tolerated (k=1, h=1). In this case, n>4.

100. 100 Quantum Gilbert-Varshamov Bound Valid for non-degenerative [n,k,d] codes. A quantum code encoding k qubits in n qubits correcting errors on t qubits satisfy: where H is the Shannon entropy:

101. 101 Quantum Singleton / Knill-Laflamme Bound For degenerate [n,k,d] quantum codes. To correct errors on any t qubits, the code has to satisfy: This limit the minimal size of a quantum code resolving any arbitrary single error to n greater or equal than 5 qubits.

102. 102 Good Quantum Codes A family of [n,k,d] quantum error correction codes is said to be good if: The Gilbert-Varshamov bound tells us that such codes do exist.

103. 103 Noise Reduction (1) As we have seen, the interaction between system and environment is given by: Quantum error correction returns all terms of this sum having correctable errors to r0. Therefore, the fidelity of the corrected state, compared to the noise-free state is the sum of all coefficients ai associated with the uncorrectable errors.

104. 104 Noise Reduction (2) Noise is typically a continuous process that affects all qubits all the time. In error correction, the syndrome is extracted by a projective measurement. The probability that an error occurs is equivalent to the probability that the syndrome extraction projects the state onto one which differs from the noise-free state by an error operator E.

105. 105 Noise Reduction (3) It is convenient to rewrite HI as follows: If only terms of weight=1 appear, the environment acts individually on each qubit. It does not directly produce correlated errors across two or more qubits. In this case, errors of all weights still appear in the density matrix, but they are suppressed by a term of O(e2wt(E)), where e is the coupling strength between system and environment.

106. 106 Noise Reduction (4) In this model, the fidelity of the corrected state can be estimated as: F = 1 � P(t+1) Where P is the probability of error with weight (t+1) given by: when all single-qubit error amplitudes can add coherently (the qubits share a common environment).

107. 107 Noise Reduction (5) And: when the errors act incoherently (either separate environments, or a common environment with couplings of randomly changing phase).

108. 108 Noise Reduction (6) Notice that for a good code, t tends to infinity while t/n and k/n remain fixed. Therefore, good codes exist when t is large and e2< t/3n. Our uncorrelated noise hypothesis is a reasonable approximation to many physical systems. But we have to be extremely careful regarding the order of the small coupling constants.

109. 109 Error Avoiding A different case of quantum error correction codes can be used when a set of correlated errors, called burst errors, dominate the system-environment coupling. In principle, we could find a code whose stabilizer includes all these uncorrelated errors. This is called error avoiding, as in this case, the errors do not affect the logical states. In general, the more we know about the environment and its coupling to the physical system, the better we can find an error correction code.

110. 110 6. The Stabilizer Code Formalism �I may be synthetic, but I�m not stupid� Bishop

111. 111 Digitization of Noise Any interaction between a set of qubits and another system can be expressed by: where Ei is a tensor product of Pauli operators acting on the qubits, and the environment states are not required to be orthogonal or normalized. Then, we express decoherence and noise in terms of Pauli operators.

112. 112 Tensor Products of Pauli Operators We introduce the notation XuZv for an arbitrary error operator, where u and v are binary vectors of length n. The non zero coordinates of u and v indicate where X and Z appear in the tensor product. Example: Remember that Y = XZ and thus we only need to correct X and Z errors.

113. 113 Error Correction Error correction then takes place when: If there are n qubits in the quantum system, then error operators will be of length n. The weight of an operator is the number of terms not equal to I. Then, X10011Z00110 from the previous example has length 5 and weight 4.

114. 114 Stabilizer Let H={M} be a set of commuting error operators. Because they commute, let C={|u>} be the orthonormal set of simultaneous eigenstates with eigenvalue +1: The set C is the quantum error correcting code, and H is the stabilizer. The states |u> are the code vectors or quantum codewords.

115. 115 Example For Shor�s nine qubit code, we have that C has 2 quantum codewords and : Then, these are indeed quantum codevectors of the code, and Z1Z2 is part of their stabilizer.

116. 116 Stabilizer Group We restrict our attention to the case where H is a group. For a [n,k,d] code, C has 2k members and the size of the stabilizer is 2n-k. C spans a 2k dimensional Hilbert space inside a 2n dimensional Hilbert space. The group H is spanned by n-k linearly independent members of H. Stay tuned: more about the stabilizer group in the next section.

117. 117 Decoherence Free Subspaces Error operators in the stabilizer are correctable as these operators have no effect on the logical state. If these are the only errors, then the quantum error correction code is a noise-free subspace or decoherence-free subspace.

118. 118 Correctable Errors The set of correctable errors S can be any set of errors {Ei} such that the product E1E2 is member of H, or anti-commutes with any member of H. But this is the same as before: However, the stabilizer formulation is based only on operators and is completely independent of states.

119. 119 Stabilizer Construction We construct the stabilizer using X and Z Pauli operators. Suppose M and M� are members of the stabilizer. Then:

120. 120 Matrix Construction The stabilizer is specified by writing down the n-k linearly independent error operators that span it. We put the binary strings u and v which indicate the X and Z parts in the form of two (n-k) by n binary matrices HX and HZ.

121. 121 Stabilizer Matrix We then specify the stabilizer as the (n-k) by 2n binary matrix: H = (HX|HZ) and the requirement that all the operators commute (i.e. H is an abelian matrix): H is the quantum analogue to the parity check matrix in classical error correction.

122. 122 Generator Matrix The quantum analogue to the classical generator matrix is: G = (GX|GZ) which is (n+k) by 2n and satisfies: Therefore, H and G are duals with respect to the inner product defined by: uu�+vv�=0

123. 123 Dual Operators Because of: then G contains H. Let G be the set of error operators generated by G, then also G contains H. We can directly obtain H from its dual G.

124. 124 Detectable Errors If all members of G (other than the identity) have weight at least d, then all error operations (other than the identity) of weight less than d anticommute with a member of H, and so are detectable. Then, such a code can correct all error operators of weight less than (d-1)/2, where d is the minimum distance of the code. Same result as before!

125. 125 Code Construction The problem of code construction is reduced to a problem of finding binary matrices H which satisfy: and whose duals G, defined by: have large weights.

126. 126 7. The Stabilizer Group �We are not computers, Sebastian, we are physical!� Roy Batty

127. 127 Stabilizer Codes (1) Stabilizer codes are also known as Additive Codes. Consider the EPR state: It is easy to verify that: So we say that the state is stabilized by the operators X1X2 and Z1Z2.

128. 128 Stabilizer Codes (2) Such a state is unique, as it is the only one (up to a global phase) to be stabilized by both X1X2 and Z1Z2. The basic idea of using the stabilizer group is to work with the stabilizer operators as group generators rather than with the states. The group theoretical formalism of the stabilizer codes offers a more compact description of the quantum error correction codes.

129. 129 The Pauli Group The Pauli Group G1 on 1 qubit is given by: This set of matrices generates a group under the operation of matrix multiplication. The Pauli Group Gn on n qubits is given by the n-fold tensor product of Pauli matrices.

130. 130 Stabilized Vector Spaces Suppose S is a subgroup of Gn, and define VS to be the set of n qubit states which are fixed by every element of S. VS is the vector space stabilized by S, and S is the stabilizer of the space VS, since every element of VS is stable under the action of elements of S.

131. 131 Simple Example Consider n=3 and S={I, Z1Z2, Z2Z3, Z1Z3} a subgroup of Gn. Then, the states stabilized by each member of the subgroup are: Therefore VS is spanned by |000> and |111>. That is, S is the stabilizer of VS.

132. 132 Trivial Vector Spaces Consider S = {+I,-I,+X,-X}, then: So, S is the stabilizer of the trivial vector space. Two conditions are necessary for S to stabilize a non-trivial vector space: The elements of S must commute. (-I) is not an element of S.

133. 133 Error Detection In order to perform error-detection operations using a stabilizer code, all we need to do is measure the eigenvalue of each generator of the stabilizer. This measurement uniquely identifies the syndrome from which we can conclude on the error occurred.

134. 134 Shor�s Nine Qubit Stabilizer Code (1) Shor�s nine qubit code is an stabilizer code! Recalling the syndrome detection operators:

135. 135 Shor�s Nine Qubit Stabilizer Code (2) We can check that the two valid codewords we defined before are indeed eigenvectors of all eight operators M1 to M8 with eigenvalue +1. The Mi�s are the generators gi�s of the stabilizer group S.

136. 136 Shor�s Nine Qubit Stabilizer Code (3) Remember that measuring the eigenvalue of M1 is used to determine if a bit flip error has occurred on either qubit 1 or qubit 2. These bit flip errors are represented by X1 and X2 operators. Then:

137. 137 Syndrome Detection So, in general, Mi anticommutes with the errors it can detect, while commutes with the errors it cannot detect: We can use these relationships to uniquely identify what error has occurred.

138. 138 Group Generators In general a [n,k,d] quantum error correction code will have an stabilizer group S generated by n-k independent and commuting elements from Gn. If the stabilizer has n-k generators, then we can prove that VS is a 2k-dimensional space. Then, a [n,1,d] code requires n-1 generators and will only have two logic states.

139. 139 Logical States (1) In principle, given n-k generators for the stabilizer S we can choose any 2k orthonormal vectors in the codes to act as our logical computational basis states. A more systematic method is to choose operators such that forms an independent and commuting set.

140. 140 Logical States (2) The Zj operators play the role of a logical Pauli z operator on logical qubit number j. We can also define Xj operators that take Zj into �Zj under conjugation, and leaves all other Zi and gi alone. Clearly, Xj has the effect of a NOT gate on the jth encoded qubit. Thus, Xj commutes with all the other members of the stabilizer, except for Zj, with which it anticommutes.

141. 141 Logical States (3) The logical computational basis state is therefore defined to be the state with the stabilizer

142. 142 Example Consider the three qubit bit flip code logical states: Then, it is clear that they are stabilized by

143. 143 Unitary Gates (1) The stabilizer formalism can also be used to describe the dynamics of the vector spaces under a variety of quantum operations. Suppose we apply a unitary operation U to a vector space VS stabilized by S. Then, for each element of VS we have:

144. 144 Unitary Gates (2) Thus the state U|Y> is stabilized by UgUt, from which we deduce that the vector space UVS is stabilized by the group: Thus, to compute the change in the stabilizer, we need only to compute how it affects the generators of the stabilizer.

145. 145 Compact Description of Entanglement (1) Imagine n qubits in a state stabilized by: Applying the Hadamard gate to each qubit we arrive to a state stabilized by: This state is the uniform superposition of all computational basis states.

146. 146 Compact Description of Entanglement (2) Then, the description of the state vector requires 2n amplitudes. But we only require n parameters to describe it using its stabilizer! In general this compact description using the stabilizer is possible whenever we use Hadamard, phase and controlled-NOT operations.

147. 147 The Normalizer The set of U unitary operators such that UGnUt = Gn is the normalizer of Gn and is denoted as N(Gn). Theorem: Suppose U is any unitary operator on n qubits with the property that: Then, up to a global phase U may be composed from O(n2) Hadamard, phase and controlled-NOT gates (the normalizer gates).

148. 148 Stabilizer Group Transformations

149. 149 Measurement in the Stabilizer Formalism (1) Measurements in the computational basis may also be easily described within the stabilizer formalism. Consider the measurement of a g from Gn. As g is a Hermitian operator in can be regarded as an observable. There are two possibilities: g commutes with all the generators of S g anti-commutes with one or more generators of S.

150. 150 Measurement in the Stabilizer Formalism (2) If g commutes with the generators, then g or �g is an element of S. In this case, a measurement of g yields +1 or �1 with probability one, and the measurement does not disturb the state of the system, leaving the stabilizer invariant.

151. 151 Measurement in the Stabilizer Formalism (3) If g anticommutes with one or more members of S, then we have equal probabilities (=1/2) that we will obtain +1 or -1, and the new state of the system (and its stabilizer) is:

152. 152 The Gottesman-Knill Theorem Theorem: Suppose a quantum computation is performed which involves only the following elements: state preparation in the computational basis, Hadamard gates, phase gates, controlled-NOT gates, Pauli gates, and measurements of observables in the Pauli group (which includes measurements in the computational basis as a special case), together with the possibility of classical control conditioned on the outcome of such measurements. Such a computation may be efficiently simulated on a classical computer.

153. 153 Quantum Computing Simulations The Gottesman-Knill theorem shows that some quantum computations involving highly entangled states may be simulated efficiently (in polynomial time complexity) on classical computers. These computations include quantum teleportation and superdense coding. However, not all types of entanglement can be described efficiently with the stabilizer formalism.

154. 154 Error Correction Suppose C(S) is a stabilizer code corrupted by an error E in Gn. If E takes C(S) to an orthogonal subspace, then the error can in principle be detected and corrected. If E is part of S the error does not corrupt the space at all. Potential problem: if E commutes with all the elements of S, but it is not in S. That is:

155. 155 The Centralizer The centralizer of S in Gn, denoted by Z(S) is the set of E in Gn such that E commutes with all the generators of S. For the stabilizer groups of concern to error correction, the centralizer is identical to the normalizer of S.

156. 156 Error Correction Conditions Theorem: Let S be the stabilizer of the stabilizer code C(S). Suppose {Ei} is a set of operators in Gn such that: for all j and k. Then, {Ei} is a correctable set of errors for the code C(S).

157. 157 Error Detection Suppose g1,�,gn-k is the set of generators for the stabilizer of an [n,k] stabilizer code, and that {Ej} is the set of correctable errors for the code. Error detection is performed by measuring the generators of the stabilizer to obtain the error syndrome, which consists of the results b1,�,bn-k. If the error E occurred then the error syndrome is given by bm such that:

158. 158 Recovery (1) In the case where E is the unique error operator having this syndrome, recovery is done by applying Et. In the case where there there are two distinct errors E and E� giving rise to the same error syndrome, it follows that: EPEt = E�PE�t and then EtE�PE�tE = P and therefore EtE� is part of S.

159. 159 Recovery (2) Thus applying Et after the error E� has occurred results in a successful recovery. Thus, for each possible error syndrome we simply pick out a single error E with that syndrome, and apply Et to achieve recovery when that syndrome is observed.

160. 160 Quantum Error Correction Without Measurement (1) As discussed, for error correction, we measure an operator M and then execute conditionally an operator U. It is possible to do this correction by unitary operations and ancillas without any measurement. This is useful when measurement is undesirable.

161. 161 Quantum Error Correction Without Measurement (2) We prepare an ancilla system with the basis state |i> corresponding to the possible error syndromes. The ancilla start in a standard pure state |0> before error correction. We define the operator U over the principal system and ancilla as:

162. 162 8. Constructing Quantum Codes �It�s all in the reflexes�� Jack Burton

163. 163 Now we know how to build a quantum error correction code! An [n,k,d] quantum error correction code C(S) is the vector space VS stabilized by a subgroup S of Gn such that and S has n-k independent and commuting generators: and logical states stabilized by: which can correct a set of correctable error operators {Ei} in Gn such that, for all j and k:

164. 164 The Three Qubit Bit Flip Code [3,1,1]

165. 165 The Nine Qubit Shor Code [9,1,3]

166. 166 The Five Qubit Code [5,1,3]

167. 167 Calderbank-Shor-Steane Codes (1) CSS codes are a subclass of stabilizer codes. They construct quantum error correction codes from classical linear codes. As a general rule, to detect X errors, CSS take a classical parity check matrix P, replaces 1 by Z and I�s elsewhere. To detect Z errors, replace X�s instead of Z�s in the matrix.

168. 168 Calderbank-Shor-Steane Codes (2) If P and Q are orthogonal then we can combine these two codes. This means that the dual code of each code must be a subset of the other code. Combining a P[n,k,d] with a Q[n,k�,d�] yields a CCS(P,Q)[n,|k-k�|,d3] with d3 = min{d,d�}.

169. 169 Calderbank-Shor-Steane Codes (3) Remember that to construct a code we need to find H and G such that: Then, we can write where Hi is the parity check matrix of the classical code Ci generated by Gi.

170. 170 Calderbank-Shor-Steane Codes (4) Clearly: and we force which means that:

171. 171 Calderbank-Shor-Steane Codes (5) In this case, the quantum logical states are given by: where u is a k-bit binary word, x is an n-bit binary word, and D is a k by n matrix of coset leaders.

172. 172 The Steane Seven Qubit CSS Code [7,1,3] The 7-qubit Steane code is the most popular CSS code. It is created with a classical Hamming code [7,4,3] which is self dual. The matrix P is taken as the classical parity check matrix H. The matrix Q is taken as the transposed of its generator GT.

173. 173 The Steane Seven Qubit CSS Code [7,1,3]

174. 174 Conclusions �Let there be light!� Dark Star Bomb Computer

175. 175 On the positive side� Quantum error correction can be formalized in terms of quantum states and projectors, stabilizer subspaces or the stabilizer group. All these formalizations are equivalent. Getting through the mathematical language, the theory of quantum error correction is quite elegant and simple. We saw examples of several quantum error correction codes, and discussed how they are used to correct wide classes of errors on 1 or more qubits.

176. 176 On the negative side� While the theory is simple, the implementation may be very difficult. Construction of quantum error correction codes is not an easy task. Implementation of these quantum error correction protocols using quantum gates may also be a nontrivial task. Not so much for the complexity of the code, but for the large number of gates involved.