Entanglement-Assisted Quantum Error Correction

CHAPTER

Entanglement-Assisted Quantum Error Correction

9

CHAPTER OUTLINE 9.1 Entanglement-Assisted Quantum Error Correction Principles ............................... 338 9.2 Entanglement-Assisted Canonical Quantum Codes .............................................. 339 9.3 General Entanglement-Assisted Quantum Codes.................................................. 341 9.3.1 EA-QECCs Derived from Classical Quaternary and Binary Codes.......... 343 9.4 Encoding and Decoding for Entanglement-Assisted Quantum Codes ..................... 347 9.5 Operator Quantum Error Correction Codes (Subsystem Codes) ............................. 349 9.6 Entanglement-Assisted Operator Quantum Error Correction Coding (EA-OQECC)..... 352 9.6.1 EA-OQECCs Derived from Classical Binary and Quaternary Codes ....... 355 9.7 Summary ......................................................................................................... 356 9.8 Problems ......................................................................................................... 357 References ............................................................................................................. 360

In this chapter, the entanglement-assisted (EA) quantum error correction codes (QECCs) [1e13] are described, which make use of pre-existing entanglement between transmitter and receiver to improve the reliability of transmission. In the previous chapter, the concept of EA quantum codes was briefly introduced. EA quantum codes can be considered as a generalization of superdense coding [14,15]. They can also be considered as particular instances of subsystem codes [16e18], described in Chapter 8. A key advantage of EA quantum codes compared to CSS codes is that EA quantum codes do not require the corresponding classical codes, from which they are derived, to be dual containing. That is, arbitrary classical code can be used to design EA quantum codes, providing that there exists a pre-entanglement between source and destination. A general description of entanglement-assisted quantum error correction is provided in Section 9.1. In Section 9.2, the entanglement-assisted canonical code is studied, along with its error correction capability. Further, in Section 9.3 the concept of EA canonical code is generalized to arbitrary EA code. Also in this section, the design of EA codes from classical codes, in particular classical quaternary codes, is described. In Section 9.4, encoding and decoding for EA quantum codes are discussed. In Section 9.5, the concept of operator quantum error correction is introduced. Entanglement-assisted operator quantum error correction is considered in Section 9.6. In this chapter, the notation due to Brun, Devetak, Hsieh, and Wilde [1e6], who invented this class of quantum codes, is used. Quantum Information Processing and Quantum Error Correction. DOI: 10.1016/B978-0-12-385491-9.00009-5 Copyright
2012 Elsevier Inc. All rights reserved.

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9.1 ENTANGLEMENT-ASSISTED QUANTUM ERROR CORRECTION PRINCIPLES The block scheme of entanglement-assisted quantum code, which requires a certain number of entangled qubits (ebits) to be shared between the source and destination, is shown in Figure 9.1. The source encodes quantum information in K-qubit state jji with the help of local ancillary qubits j0i and source-half of shared ebits (e ebits) into N qubits, and then sends the encoded qubits over a noisy quantum channel (such as a free-space or fiber-optic channel). The receiver performs decoding on all qubits (N þ e qubits) to diagnose the channel error and performs a recovery unitary operation to reverse the action of the channel. Notice that the channel does not affect the receiver’s half of shared ebits at all. By omitting the ebits, the conventional quantum coding scheme is obtained. It was discussed in previous chapters that both noise N and recovery R processes can be described as quantum operations N ,R : L(H) / L(H), where L(H) is the space of linear operators on Hilbert space H. These mappings can be represented in terms of the operator-sum representation: y

N ðrÞ ¼ Si Ei rEi , Ei ˛ LðHÞ, N ¼ fEi g:

(9.1)

Given the quantum code CQ that is subspace of H, we say that the set of errors E are correctable if there exists a recovery operation R such that R N (r) ¼ r for any state r from L(CQ). Further, each error operator Ei can be expanded in terms of Pauli operators: Ei ¼

S

2

½a jb  ˛ F 2N

ai, ½ajb E½ajb , E½ajb ¼ jl XðaÞZðbÞ; a ¼ a1 .aN ; b ¼ b1 .bN ;

ai , bi ¼ 0, 1; l ¼ 0, 1, 2, 3

XðaÞ h X1a1 5.5XNaN ; ZðbÞhZ1b1 5.5ZNbN : (9.2)

An [N,K,e;D] EA-QECC consists of: (i) an encoding map, U^enc : L5K 5L5e /L5N ,

(9.3)

and (ii) decoding trace-preserving mapping, D : L5N 5L5e /L5K ,

K information qubits ψ〉 Quantum Entangled states e qubits encoder preparation circuit

Quantum channel

(9.4)

Quantum decoder

e qubits

FIGURE 9.1 A generic entanglement-assisted quantum error correction scheme.

K information qubits

9.2 Entanglement-Assisted Canonical Quantum Codes

such that the composition of encoding mapping, channel transformation, and decoding mapping is the identity mapping:
(9.5) D +N +Uenc +Uapp ¼ I 5K ; I: L/L; Uapp jji ¼ jji
F5e i, where Uapp is an operation that simply appends the transmitter half of the maximum entangled state jF5e i:

9.2 ENTANGLEMENT-ASSISTED CANONICAL QUANTUM CODES The simplest EA quantum code, the EA canonical code, performs the following trivial encoding operation:
(9.6) U^c : jji/j0il 5
F5e i5jjiK ; l ¼ N  K  e: The canonical EA code, shown in Figure 9.2, is therefore an extension of the canonical code presented in Chapter 7, in which e ancillary qubits are replaced by e maximally entangled kets shared between transmitter and receiver. Since the number of information qubits is K and codeword length is N, the number of remaining ancillary qubits in the j0i state is l ¼ N  K  e. The EA canonical code can correct the following set of errors: N

canonical

 ¼ XðaÞZðbÞ5Zðbeb ÞXðaeb Þ5Xðaða, beb , aeb ÞÞZðbða, beb , aeb ÞÞ: a, b˛F2l ;beb , aeb ˛F2e



a, b: F2l  F2e  F2e /F2K ,

(9.7)

where the Pauli operators X(c) (c ˛ {a,aeb,a(a,beb,aeb)}) and Z(d) (d ˛ {b,beb,b (a,beb,aeb)}) are introduced in (9.2). In order to prove claim (9.7), let us observe one particular error Ec from the set of errors (9.7) and study its action on the transmitted codeword j0il 5jF5e i5jjiK :
XðaÞZðbÞj0il 5ðZðbeb ÞXðaeb Þ5I Rx Þ
F5e i5Xðaða, beb , aeb ÞÞZðbða, beb , aeb ÞÞjji: (9.8) The action of operator X(a)Z(b) on state j0il is given by XðaÞZðbÞj0il

ZðbÞj0il ¼j0il

¼ XðaÞj0il ¼ jai: On the other hand, the action of Z(beb)X(aeb) on the maximum entangled state is ðZðbeb ÞXðaeb Þ5I Rx ÞjF5e i ¼ jbeb , aeb i, where I Rx denotes the identity operators applied on the receiver half of the maximum entangled state. Finally, the action of channel on information portion of the codeword is given by Xðaða, beb , aeb ÞÞZðbða, beb , aeb ÞÞjji ¼ jj0 i: Clearly, the vector (a b beb aeb) uniquely specifies the error operator Ec, and can be called the syndrome vector. Since the state j0il is invariant on the action of Z(b), the vector b can be omitted from

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CHAPTER 9 Entanglement-Assisted Quantum Error Correction

Encoder ψ〉

K qubits

0〉

l qubits e qubits

Φ⊗e 〉

Quantum channel X(α)Z( β) X(a)Z(b) Z(be )X(ae )

X-1(α)Z-1( β) a〉 〈a } be ae 〉 〈be ae }

e qubits

K qubits ψ〉

a

be ae Decoder

FIGURE 9.2 An entanglement-assisted canonical code.

the syndrome vector. To obtain the a portion of the syndrome, we have to perform simultaneous measurements on Z(di), where di ¼ (0.01i0.0). On the other hand, in order to obtain the (beb aeb) portion of the syndrome vector, we have to perform measurements on Z(d1)Z(d1),.,Z(de)Z(de) for beb and X(d1)X(d1),.,X(de)X(de) for aeb. Once the syndrome vector is determined, we apply X 1 ðaða, be , ae ÞÞ Z 1 ðbða, be , ae ÞÞ to undo the action of the channel. From syndrome measurements it is clear that we applied exactly the same measurements as in superdense coding, indicating that EA codes can indeed be interpreted as a generalization of superdense coding (see Section 8.8 or Refs [4,15]). In order to avoid the need for measurement we can perform the following controlled unitary decoding: Ucanocnical, dec ¼

X a ; beb ; aeb

jaihaj5jbeb , aeb ihbeb , aeb j5X 1

 ðaða, beb , aeb ÞÞZ 1 ðbða, beb , aeb ÞÞ:

(9.9)

EA error correction can also be described using the stabilizer formalism interpretation. Let Scanonical be the non-Abelian group generated by Scanonical ¼ hScanonical, I , Scanonical, E i; Scanonical, I ¼ hZ1 , ., Zl i, Scanonical, E ¼ hZlþ1 , ., Zlþe , Xlþ1 , ., Xlþe i,

(9.10)

where Scanonical,I denotes the isotropic subgroup and Scanonical,E the sympletic subgroup of anticommuting operators. The representation (9.10) has certain similarities with subsystem codes from Section 8.7 of the previous chapter. Namely, by interpreting Scanonical,I as the stabilizer group and Scanonical,E as the gauge group, it turns out that EA codes represent a generalization of the subsystem codes from Section 8.7. Moreover, the subsystem codes do not require entanglement to be established between transmitter and receiver. On the other hand, EA codes with few ebits have reasonable complexity of encoder and decoder, compared to dualcontaining codes, and as such represent promising candidates for fault-tolerant quantum computing and quantum teleportation applications.

9.3 General Entanglement-Assisted Quantum Codes

We perform an Abelian extension of the non-Abelian group Scanonical, denoted as Scanonical,ext, which now acts on N þ e qubits, as follows: Z1 5I, ., Zl 5I Zlþ1 5Z1 , Xlþ1 5X1 , ., Zlþe 5Ze , Xlþe 5Xe :

(9.11)

Out of N þ e qubits, the first N correspond to the transmitter and an additional e qubits correspond to the receiver. The operators on the right-hand side of (9.11) act on receiver qubits. This extended group is Abelian and can be used as the stabilizer group that fixes the code subspace Ccanonical,EA. The obtained stabilizer group can be called the entanglement-assisted stabilizer group. The number of information qubits of this stabilizer can be determined by K ¼ N  l  e, so that the code can be denoted as [N,K,e;D], where D is the distance of the code. Before concluding this section, it will be useful to identify the correctable set of errors. In analogy with the stabilizer codes described in Chapter 8, we expect the correctable set of errors of EA-QECC, defined by Scanonical ¼ , to be given by N

canonical

n
o
y ¼ Em
cE1 , E2 0E2 E1 ˛Scanonical, I WðGN  CðScanonical ÞÞ ,

(9.12)

where C(Scanonical) is the centralizer of Scanonical (i.e. the set of errors that commute with all elements from Scanonical). We have already shown above (see Eq. (9.7) and corresponding text) that every error can be specified by the syndrome (a beb aeb). If two errors E1 and E2 have the same syndrome (a beb y aeb), then E2 E1 will have all-zero syndrome and clearly such an error belongs to y Scanonical,I. Such an error E2 E1 is trivial, as it fixes all codewords. If two errors y have different syndromes, then the syndrome for E2 E1 , which has the form (a beb aeb), indicates that such an error can be corrected provided that it does not belong to C(Scanonical). That is, when an error belongs to C(Scanonical), then its action results in another codeword different from the original codeword, so that such an error is undetectable.

9.3 GENERAL ENTANGLEMENT-ASSISTED QUANTUM CODES From group theory it is known that if V is an arbitrary subgroup of GN of order 2M then there exists a set of generators fX pþ1 , ., X pþq ;Z 1 , ., Z pþq g satisfying similar commutation properties to Eq. (8.28) of the previous chapter, namely: ½X m , X n  ¼ ½Z m , Z n  ¼ 0 cm, n; ½X m , Z n  ¼ 0 cmsn; fX m , Z n g ¼ 0 cm ¼ n: (9.13)

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CHAPTER 9 Entanglement-Assisted Quantum Error Correction

Encoder ψ〉

K qubits

0〉

l qubits e qubits

Φ⊗e 〉

Quantum channel X(α)Z( β)

U

U-1

X(a)Z(b) Z(beb)X(aeb)

X -1(α)Z -1( β) U

U-1

a〉 〈a }

K qubits ψ〉

a

beb aeb 〉 〈beb a eb }

e qubits

beb a eb Decoder

FIGURE 9.3 An entanglement-assisted code obtained as a generalization of the canonical EA code.

We also know from the theorem in Section 2.7 that the unitary equivalent observables, A and UAU1, have identical spectra. This means that we can establish bijection between two groups V and S that will preserve their commutation relations, so that c v ˛ V there exists corresponding s ˛ S such that v ¼ UsU1 (up to a general phase constant). By using these results, we can establish bijection between canonical entanglement-assisted quantum code described by Scanonical ¼ hScanonical, I , Scanonical, E i and a general EA code characterized by S ¼ hSI , SE i with S ¼ UScanonicalU1 (or equivalently Scanonical ¼ U1SU), where U is a corresponding unitary operator. Based on the discussion above, we can generalize the results from the previous section as follows. Given a general group S ¼ (jSIj ¼ 2NKe and jSEj ¼ 22e), there exists an [N,K,e;D] EA-QECC code, illustrated in Figure 9.3, denoted as CEA, defined by the encodingedecoding pair (E,D), with the following properties: 1. The EA code CEA can correct any error from the following set of errors N : o n

y N ¼ Em
cE1 , E2 0E2 E1 ˛SI WðGN  CðSÞÞ :

(9.14)

2. The code space CEA is a simultaneous eigenspace of the Abelian extension of S, denoted as Sext. 3. To decode, the error syndrome is obtained by simultaneously measuring the observables from Sext. Because the commutation relations of S are the same as that of EA canonical code Ccanonical,EA, we can find a unitary operator U such that Scanonical ¼ USU1. Based on Figure 9.3 it is clear that the encoding mapping can be represented as a composition of unitary operator U and encoding mapping of canonical code E canonical, i.e. E ¼ U+E canonical : In similar fashion, the decoding mapping can be represented by D ¼ Dcanonical +U^1 , U^ ¼ extensionðUÞ. Further, the

9.3 General Entanglement-Assisted Quantum Codes

composition of encoding, error, and decoding mapping would result in the identity operator: ^ D+N +E ¼ Dcanonical +U^1 +UN

^ ¼ I 5K : ^1 +UE

(9.15)

canonical U

The set of errors given by (9.14) is the correctable set of errors because Ccanonical,EA is simultaneous eigen-space of Scanonical,ext, and Sext ¼ UScanonical,extU1, and by definition CEA ¼ U(Ccanonical,EA), so that we conclude that Cext is a simultaneous eigenspace of Sext. We learned in the previous section that the decoding operation of Dcanonical involves: (i) measuring the set of generators of Scanonical,ext, yielding the error syndrome according to the error Ec; and (ii) performing the recovery operation to undo the action of the channel error Ec. The decoding action of CEA is equivalent to the measurement of Sext ¼ UScanonical,extU1, followed by the recovery operation UEcU1 and U1 to undo the encoding. The distance D of EA code can be introduced in similar fashion to that of a stabilizer code. The distance D of EA code is defined as the minimum weight among undetectable errors. In other words, we say that EA quantum code has distance D if it can detect all errors of weight less than D, but none of weight D. The error correction capability t of EA quantum code is related to minimum distance D by t ¼ P(D  1)/2R. The weight of a qubit error [ujv] (u,v ˛ F2N ) can be defined, in similar fashion to that for stabilizer codes, as the number of components i for which (ui,bi) s (0,0) or equivalently wt([ujv]) ¼ wt(u þ v), where “þ” denotes bit-wise addition mod 2. EA quantum bounds are similar to the quantum coding bounds presented in Chapter 7. For example, the EA quantum Singleton bound is given by N þ e  K  2ðD  1Þ:

9.3.1 EA-QECCs Derived from Classical Quaternary and Binary Codes We turn our attention now to establishing the connection between EA codes and quaternary classical codes, which are defined over F4 ¼ GF(4) ¼ {0,1,u,u2 ¼ 6}. As a reminder, the addition and multiplication tables for F4 are given below:

þ 0 6 1 u

0 0 6 1 u

6 6 0 u 1

1 1 u 0 6

u u 1 6 0

 0 6 1 u

0 0 0 0 0

6 0 u 6 1

1 0 6 1 u

u 0 1 u 6

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On the other hand, the addition table for binary two-tuples, F22 ¼ {00,01,10,11}, is: þ 00 01 11 10

00 00 01 11 10

01 01 00 10 11

11 11 10 00 01

10 10 11 01 00

Further, from Chapter 7 we know that the multiplication table of Pauli matrices is given by:  I X Y Z

I I X Y Z

X X I jZ jY

Y Y jZ I jX

Z Z jY jX I

If we ignore the phase factor in Pauli matrix multiplication, we can establish the following correspondences between G1, F22 , and F4: G

F22

F4

I X Y Z

00 10 11 01

0 u 1 6

It can easily be shown that the mapping f: F4 / F22 is an isomorphism. Therefore, if a classical code (n,k,d) C4 over GF(4) exists, then an [N,K,e,D] ¼ [n,2k  n þ e,e;D] EA-QECC exists for some non-negative integer e, which can be described by the following quantum-check matrix:   ~ 4 Þ; H ~ 4 ¼ uH4 , (9.16) AEA ¼ f ðH uH4 where H4 denotes the parity-check matrix of classical code (n,k,d) over F4. The sympletic product of binary vectors from (9.16) is equal to the trace of the product of their GF(4) representations as follows:   hm 1hn ¼ tr f 1 ðhm Þf  1 ðhn Þ , trðwÞ ¼ w þ w,

(9.17)

9.3 General Entanglement-Assisted Quantum Codes

where hm (hn) is the mth (nth) row of AEA and the overbar denotes the conjugation. Based on Chapter 7, the sympletic product of AEA is given by  tr

uH4 uH4 

¼ tr  ¼

1 u



uH4 uH4 y

H4 H4 y uH4 H4

y

 ¼ tr y

uH4 H4 y H4 H4

 uH4 y ð uH4 uH4



  u 1 y 5H4 H4 þ u 1

 ¼ tr

1 u

y uH4 Þ



 u y 5H4 H4 1

 u 5H4 HT4 1

(9.18)

~ y will not change if we perform the similarity trans~ 4H The rank of trace of H 4   y 1 0 1 u y y ~ ~ 5I, B ¼ 5I, to obtain: formation BAtrðH4 H4 ÞA B , where A ¼ u 1 1 1  T 0 y T ~ 4H ~ y Ay By ¼ H4 H4 BAH (9.19) y ¼ H4 H4 þ H4 H4 : 4 0 H4 H4 ~ y ÞAy By is then obtained as ~ 4H The rank of BAtrðH 4 ~ 4H ~ y ÞAy By Þ ¼ rank ðH4 HT þ H4 Hy Þ ¼ rank ðH4 HT Þ þ rank ðH4 Hy Þ rank ðBAtrðH 4 4 4 4 4 y y ¼ 2rank ðH4 H4 Þ ¼ 2e, e ¼ rank ðH4 H4 Þ: (9.20) Equation (9.20) indicates that the number of required ebits is e ¼ rankðH4 Hy4 Þ, so that the parameters of this EA code, based on Eq. (9.16), are [N,K,e] ¼ [n,n þ e  2(n  k),e] ¼ [n,2k  n þ e,e]. Example. The EA code derived from (4,2,3) 4-ary classical code described by parity-check matrix   1 1 0 1 H4 ¼ 1 u 1 0 has parameters [4,1,1,3], and based on (9.16) the corresponding quantum-check matrix is given by   ~4 ¼ f AEA ¼ f H



uH4 uH4



02

u B6 u 6 ¼ fB @4 u u

u u u 1

0 u 0 u

31 0 u X C BX 07 7C ¼ B u 5A @ Z 0 Z

X Z Z Y

I X I Z

1 X IC C: ZA I

In the remainder of this section, we describe how to relate EA quantum codes to binary classical codes. Let H be a binary parity-check matrix of dimensions (n  k)  n. We will show that the corresponding EA-QECC has the parameters

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[n,2k  n þ e;e], where e ¼ rank(HHT). The quantum-check matrix of CSS-like EA quantum code can be represented as follows: 
 H

0 : (9.21) AEA ¼ 0
H The rank of the sympletic product of AAE, based on (9.17), is given by    

  H 0 T T þ rankfAEA 1ATEA g ¼ rank 0 H H 0 0 H 

   0 0 0 HHT þ ¼ rank HHT 0 0 0     ¼ rank HHT þ HHT ¼ 2rank HHT   ¼ 2e, e ¼ rank HHT :

(9.22)

From (9.22) it is clear that the number of required qubits is e ¼ rankðHHT Þ, so that the parameters of this EA code are [N,K,e] ¼ [n,n þ e  2(n  k),e] ¼ [n,2k  n þ e,e]. Dual-containing codes can therefore be considered as EA codes for which e ¼ 0. The EA code given by (9.21) can be generalized as follows: (9.23) AEA ¼ ðAx jAz Þ, where the (n  k)  n submatrix Ax (Az) contains only X operators (Z operators). The rank of the sympletic product of AAE can be obtained as rankfAEA 1ATEA g ¼ rankfAx ATz þ Az ATx g ¼ 2e, e ¼ rankfAx ATz þ Az ATx g=2: (9.24) The parameters of this EA code are [N,K,e] ¼ [n,n þ e  (n  k),e] ¼ [n,k þ e,e]. Another generalization of (9.21) is given by
  Hx

0 , (9.25) AEA ¼ 0
Hz where the (n  kx)  n submatrix Hx contains only X operators, while the (n  kz)  n submatrix Hz contains only Z operators. The rank of the sympletic product of AAE is given by 

   Hx 0 T T T rankfAEA 1AEA g ¼ rank ð 0 Hz Þ þ H ð Hx 0 Þ 0 z  ¼ rank

0 0

Hx HTz 0



 þ

0 Hz HTx

0 0



¼ rankfHx HTz þ Hz HTx g ¼ 2e, e ¼ rankfHx HTz g:

(9.26)

9.4 Encoding and Decoding for Entanglement-Assisted Quantum Codes

The EA code parameters are [N,K,e;D] ¼ [n,n þ e  (n  kx)  (n  kz),e; min(dx,dz)] ¼ [n,kx þ kz  n þ e,e;min(dx,dz)].

9.4 ENCODING AND DECODING FOR ENTANGLEMENTASSISTED QUANTUM CODES Encoders and decoders for entanglement-assisted codes can be implemented in similar fashion as for subsystem codes, described in the previous chapter: (i) using standard format formalism [21,22] and (ii) by using the conjugation method due to Grassl et al. [23] (see also Refs [7,24]). In the previous section, we defined an EA [N,K,e] code by using a non-Abelian subgroup S of GN (multiplicative Pauli group on N qubits) with 2e þ l generators, where l ¼ N  K  e. The subgroup S can be described by the set of independent generators fX eþ1 , ., X eþl ;Z 1 , ., Z eþl g, which satisfy the commutation relations (9.13). The non-Abelian subgroup S can be decomposed into two subgroups: the commuting isotropic group SI ¼ fZ 1 , ., Z l g and the entanglement subgroup with anticommuting pairs fX lþ1 , ., X lþe ;Z lþ1 , ., Z lþe g. This decomposition allows us to determine the EA code parameters. Because the isotropic subgroup is a commuting group, it corresponds to ancillary qubits. On the other hand, since the entanglement subgroup is composed of anticommuting pairs, its elements correspond to the transmitter and receiver halves of ebits. This decomposition can be performed using the sympletic GrameSchmidt procedure [7]. Let us assume that the following generators create the subgroup S: g1,.,gM. We start the procedure with generator g1 and check if it commutes with all other generators. If it commutes, we remove it from further consideration. If g1 anticommutes with a generator gi, we relabel g2 as gi and vice versa. For the remaining generators we perform the following manipulation: 0, ½a, b ¼ 0 f ðg2 , gm Þ f ðg1 , gm Þ g2 , f ða, bÞ ¼ g m ¼ gm g 1 ; m ¼ 3, 4, ., M: (9.27) 1, fa, bg ¼ 0 The generators g1 and g2 are then removed from further discussion, and the same algorithm is applied on the remaining generators. At the end of this procedure, the generators removed from consideration create the code generators satisfying the commutation relations (9.13). We can perform an extension of the non-Abelian subgroup S into Abelian and then apply the standard form procedure for encoding and decoding, which was fully explained in the previous chapter. Here, instead, we will concentrate to the conjugation method for EA encoder implementation. The key idea is very similar to that of subsystem codes, described in Section 8.7. That is, we represent the non-Abelian group S using finite geometry interpretation. We then perform Gauss elimination to transform the EA code into canonical EA code:


(9.28) jci/
F5e ij0i5.5j0i
F5e ijjiK ¼
F5e ij0il jjiK , |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} l¼N K e

times

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CHAPTER 9 Entanglement-Assisted Quantum Error Correction

where jci is the EA codeword, jF5e i is the ebit state, and j0il are l ancillary states prepared into the j0i state. In other words, we have to apply the sequence of gates {U} to perform the following transformation of the quantum-check matrix of EA code:
3 2 e ︷l ︷K
︷ 6 0 0 0
Ie 0 0 7 fUg
7 (9.29) AEA ¼ ½Ax jAz  / AEA, canonical ¼ 6 4 Ie 0 0
0 0 0 5:
0 Il 0 0 0 0 The right-hand side of Eq. (9.29) is obtained from Eq. (9.11). We apply the same set of rules as for subsystem codes. The action of the Hadamard gate on the ith qubit is to change the position of ai and bi as follows, where we underline the affected qubits for convenience: Hi

ða1 . ai .aN jb1 . bi .bN Þ /ða1 . bi .aN jb1 . ai .bN Þ:

(9.30)

The action of the CNOT gate on generator g, where the ith qubit is the control qubit and the jth qubit is the target qubit, the action denoted as CNOTi,j, can be described as

CNOTi; j ða1 . aj .aN
b1 . bi .bN Þ / ða1 .aj1 aj þ ai ajþ1 .aN
b1 .bi1 bi þ bj biþ1 .bN Þ: (9.31)

Therefore, the jth entry in the X portion and the ith entry in the Z portion are affected. The action of phase gate P on the ith qubit is to perform the following mapping: P

ða1 . ai .aN jb1 . bi .bN Þ/ða1 . ai .aN jb1 . ai þ bi .bN Þ:

(9.32)

Finally, the application of a CNOT gate between the ith and jth qubits three times in an alternative fashion leads to swapping of the ith and jth columns in both the X and Z portions of the quantum-check matrix, SWAPi, j ¼ CNOTi, j CNOTj, i CNOTi, j , as shown in Chapter 3:

SWAPi; j ða1 . ai . aj .aN
b1 . bi . bj .bN Þ / ða1 . aj . ai .aN
b1 . bj . bi .bN Þ: (9.33)

Let us denote the sequence of gates being applied during this transformation to canonical EA code as {U}. By applying this sequence of gates in the opposite order we obtain the encoding circuit of the corresponding EA code. Notice that adding the nth row to the mth row maps gm / gmgn so that codewords are invariant to this change. Since this method is very similar to that used for subsystem codes in the previous chapter, we turn our attention now to the sympletic GrameSchmidt method due to Wilde [7], which has already been discussed above at the operator level. Here we

9.5 Operator Quantum Error Correction Codes (Subsystem Codes)

describe this method using a finite geometry interpretation, based on Eqs (9.30)e(9.33). Let us observe the following example due to Wilde [7]:

Clearly, we have transformed the initial EA into canonical form (9.29). The corresponding encoding circuits can be obtained by applying the set of gates to the canonical codeword, used to transform the quantum-check matrix into canonical form, but now in the opposite order. The encoding circuit for this EA code is shown in Figure 9.4.

9.5 OPERATOR QUANTUM ERROR CORRECTION CODES (SUBSYSTEM CODES) The operator quantum error correction codes (OQECCs) [2,4,16e18,24e28] represent a very important class of quantum codes, which are also known as subsystem codes. This class of codes has already been discussed in the previous chapter. Nevertheless, the following interpretation due to Hsieh, Devetak, and Brun [2,4] is useful for a deeper understanding of the concept. We start our description with canonical OQECCs. An [N,K,R] canonical OQECC can be represented by the following mapping: jfijji/ j0i5.5j0i jfiR jjiK ¼ j0is jfiR jjiK , |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} s¼N K R

times

(9.34)

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CHAPTER 9 Entanglement-Assisted Quantum Error Correction

Φ ⊗e 〉

x

H 〉 〉

H

H

P

H

H

H

ψ〉

H

x

H

H

FIGURE 9.4 The encoding circuit for the EA code from the example above.

where jji represents a K-qubit information state, jfi represents the R-qubit gauge state, and j0is represent s ancillary qubits. So the key difference with respect to canonical stabilizer code is that R ancillary j0i qubits are converted into gauge qubits. This canonical OQECC code can correct the following set of errors: N

canonical;OQECC

 ¼ XðaÞZðbÞ5XðcÞZðdÞ5XðaðaÞÞZðbðaÞÞ:  a, b˛F2s ; c, d˛F2R ; a, b: F2s /F2K :

(9.35)

In order to prove claim (9.35), let us observe one particular error Ec from the set of errors (9.35) and study its action on the transmitted codeword j0is jfiR jjiK : XðaÞZðbÞj0is 5ðXðcÞZðdÞÞjfi5XðaðaÞÞZðbðaÞÞjji:

(9.36) ZðbÞj0i ¼j0i

¼s s The action of operator X(a)Z(b) on state j0is is given by XðaÞZðbÞj0is XðaÞj0is ¼ jai: On the other hand, the action of X(c)Z(d) on the gauge state is ðXðcÞZðdÞÞjfi ¼ jf0 i: Finally, the action of a channel on the information portion of the codeword is given by XðaðaÞÞZðbðaÞÞjji ¼ jj0 i: Clearly, the vector (a b c d) uniquely specifies the error operator Ec, and can be called the syndrome vector. Since the state j0is is invariant on action of Z(b), the vector b can be omitted from the syndrome vector. Also, since the final state of the gauge subsystem is irrelevant to the information qubit states, the vectors c and d can also be omitted from consideration. To obtain the a portion of the syndrome, we have to perform simultaneous measurements on Z(di), where di ¼ (0.01i0.0) on ancillary qubits. Once the syndrome vector a is determined, we apply the operator X 1 ðaðaÞÞZ 1 ðbðaÞÞ to the information portion of the codeword to undo the action of the channel, as illustrated

9.5 Operator Quantum Error Correction Codes (Subsystem Codes)

Quantum channel ψ〉

K information qubits

0〉s

s ancillary qubits

φ〉

R gauge qubits

X(α)Z( β) X(a)Z(b )

X -1(α)Z -1( β) a〉 〈a }

K qubits

a

X(c )Z(d )

FIGURE 9.5 Canonical OQECC encoder and decoder principles.

in Figure 9.5. In order to avoid the need for measurement we can perform the following controlled unitary decoding operation: X Ucanocnical, OECC, dec ¼ (9.37) jaihaj5I5X 1 ðaðaÞÞZ 1 ðbðaÞÞ, a

and discard the unimportant portion of the received quantum word. The OQECCs can also be described by using the stabilizer formalism interpretation as follows: Scanonical, OQECC ¼ hScanonical, OQECC, I , Scanonical, OQECC, E i Scanonical, OQECC, I ¼ hZ1 , ., Zs i, Scanonical, OQECC, E ¼ hZsþ1 , ., ZsþR , Xsþ1 , ., XsþR i, (9.38)

where Scanonical, OQECC,I denotes the isotropic subgroup of size 2s and Scanonical, OQECC,E denotes the sympletic subgroup of size 22R. The isotropic subgroup defines a 2KþR-dimensional code space, while the sympletic subgroup defines all possible operations on gauge qubits. The OQECC can correct the following set of errors: o n

y N ¼ Em
cE1 , E2 0E2 E1 ˛Scanonical, OQECC WðGN  CðScanonical, OQECC, I ÞÞ : (9.39) Using a similar approach as in Section 9.3, we can establish bijection between canonical OQECC described by Scanonical, OQECC ¼ hScanonical, OQECC, I , Scanonical, OQECC, E i and a general OQECC characterized by SOQECC ¼ hSOQECC;I , SOQECC;E i with SOQECC ¼ UScanonical,OQECCU1 (or equivalently Scanonical,OQECC ¼ U1SOQECCU), where U is a corresponding unitary operator. This [N,K,R;D] OQECC code, illustrated in Figure 9.6, denoted as COQECC, is defined by the encodingedecoding pair (E,D) with the following properties: (i) The OQECC code COQECC can correct any error for the following set of errors: n
o
y N ¼ Em
cE1 , E2 0E2 E1 ˛SOQECC WðGN  CðSOQECC, I ÞÞ :

(9.40)

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CHAPTER 9 Entanglement-Assisted Quantum Error Correction

Encoder ψ〉

X(α)Z( β)

0〉s s ancillary qubits φ〉

Decoder

Quantum channel

K information qubits U

U-1

R gauge qubits

X-1(α)Z-1( β)

X(a)Z(b)

U

U-1

a〉 〈a }

K qubits

a

X(c)Z(d)

FIGURE 9.6 EA-OQECC encoder and decoder principles.

(ii) The code space COQECC is a simultaneous eigenspace of SOQECC,I. (iii) To decode, the error syndrome is obtained by simultaneously measuring the observables from SOQECC,I. The encoding and decoding of OQECC (subsystem) codes has already been described in the previous chapter.

9.6 ENTANGLEMENT-ASSISTED OPERATOR QUANTUM ERROR CORRECTION CODING (EA-OQECC) The entanglement-assisted OQECCs represent the generalization of EA codes and OQECCs (also known as subsystem codes). That is, in addition to information and gauge qubits used in OQECCs, several entanglement qubits (ebits) are shared

(a)

Encoder

ψ〉

K information qubits

0〉s

s ancillary qubits

φ〉

R gauge qubits

Decoder

Quantum channel X(α)Z( β)

X-1(α)Z-1( β) a〉 〈a }

X(a)Z(b)

a

X(c)Z(d)

e qubits

X(ae)Z(be)

Φ ⊗e 〉

aebe〉 〈aebe }

e qubits

(b) ψ〉

Quantum channel

Encoder K information qubits

R gauge qubits e qubits

Φ ⊗e 〉

X-1(α)Z-1( β) a〉 〈a }

X(a)Z(b) U

U-1

X(c)Z(d)

ae be Decoder

X(α)Z( β)

0〉s s ancillary qubits φ〉

K qubits ψ〉

U

X(ae)Z(be)

e qubits

FIGURE 9.7 EA-OQECCs: (a) canonical code; (b) general code.

a

U-1 aebe〉 〈aebe }

ae be

K qubits ψ〉

9.6 Entanglement-Assisted Operator Quantum Error Correction Coding (EA-OQECC)

between the source and destination. The simplest EA-QECC, the canonical code, performs the following trivial encoding operation:
(9.41) U^c : jjiK /jjiK 5j0is 5jfiR 5
F5e i; s ¼ N  K  R  e: The codeword in the canonical [N,K,R,e] EA-OQECC, shown in Figure 9.7a, is composed of K information qubits, R gauge qubits, and e maximally entangled kets shared between transmitter and receiver. Since the number of information qubits is K and codeword length is N, the number of ancillary qubits remaining in the j0i state is s ¼ N  K  R  e. The canonical EA-OQECC can correct the following set of errors: N canonical ¼ fXðaða, aeb ; beb ÞÞZðbða, aeb ; bebÞÞ5XðaÞZðbÞ5XðcÞZðdÞ5Xðaeb ÞZðbeb Þ: a, b˛F2s ; c, d˛F2R ; beb , aeb ˛F2e ; a, b: F2s  F2e  F2e /F2K , (9.42) where the Pauli operators X(e) (e ˛ {a,aeb,c,a(a,beb,aeb)}) and Z(e) (e ˛ {b,beb,d,b (a,beb,aeb)}) are introduced by (9.2). In order to prove claim (9.42), let us observe one particular error Ec from the set of errors (9.42) and study its action on the transmitted codeword jjiK 5j0is 5jfiR 5jF5e i : Xðaða, aeb , beb ÞÞZðbða, aeb , beb ÞÞjji5XðaÞZðbÞj0is 5XðcÞZðdÞjfiR 5
ðXðaeb ÞZðbeb Þ5I Rx Þ
F5e i:

(9.43)

ZðbÞj0i ¼j0i

¼s s The action of operator X(a)Z(b) on state j0is is given by XðaÞZðbÞj0is XðaÞj0is ¼ jai: On the other hand, the action of X(aeb)Z(beb) on the maximum entangled state is ðXðaeb ÞZðbeb Þ5I Rx ÞjF5e i ¼ jaeb ; beb i, where I Rx denotes the identity operators applied on the receiver half of the maximum entangled state. The action of a channel on gauge qubits is described by ðXðcÞZðdÞÞjfi ¼ jf0 i: Finally, the action of a channel on the information portion of the codeword is given by Xðaða, aeb ; beb ÞÞ Zðbða, aeb ; beb ÞÞjji ¼ jj0 i: Clearly, the vector (a b c d aeb beb) uniquely specifies the error operator Ec, and can be called the syndrome vector. Since the state j0is is invariant on the action of Z(b), the vector b can be omitted from the syndrome vector. Similarly, since the state of gauge qubits is irrelevant to the information qubits state, the vectors c and d can also be dropped from the syndrome vector. To obtain the a portion of the syndrome, we have to perform simultaneous measurements on Z(di), where di ¼ (0.01i0.0). On the other hand, in order to obtain the (aeb beb) portion of the syndrome vector, we have to perform measurements on Z(d1)Z(d1), ., Z(de)Z(de) for beb and X(d1)X(d1), ., X(de) X(de) for aeb. Once the syndrome vector is determined, we apply X 1 ðaða, ae ; be ÞÞ Z 1 ðbða, be , ae ÞÞ to undo the action of the channel. In order to avoid the need for measurement, we can perform the following controlled unitary decoding: X  X 1 ðaða, aeb ; beb ÞÞZ 1 ðbða, aeb ; beb ÞÞ Ucanocnical, dec ¼ a ; beb ; aeb

5jaihaj5I5jaeb ; beb ihaeb ; beb jg:

(9.44)

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The canonical EA-OQECC can also be described by using the stabilizer formalism interpretation. Let Scanonical be the non-Abelian group generated by Scanonical ¼ hScanonical, I , Scanonical, E , Scanonical, G i,

(9.45)

where Scanonical,I denotes the isotropic subgroup, Scanonical,E the entanglement subgroup, and Scanonical,G the gauge subgroup, given respectively by Scanonical, I ¼ hZ1 , ., Zs i, Scanonical, E ¼ hZsþ1 , ., Zsþe , Xsþ1 , ., Xsþe i, Scanonical, G ¼ hZsþeþ1 , ., ZsþeþR , Xsþeþ1 , ., XsþeþR i:

(9.46)

By using the finite geometry representation, the generators of (9.45) and (9.46) can be arranged as follows:
2︷ K ︷ s ︷ R ︷ e

︷ K ︷s ︷ R ︷ e3 0 0 0 0 0 I 0 07
s 6 6 0 0 0 0

0 0 I R 0 7 7 (9.47) AEA-OQECC;canonical ¼ 6 6 0 0 I R 0
0 0 0 0 7: 4 5
0 0 0 0
0 0 0 Ie 0 0 0 Ie
0 0 0 0 The EA-OQECC can correct the following set of errors: n

y N ¼ Em
cE1 , E2 0E2 E1 ˛hScanonical, I , Scanonical, G i o WðGN  CðhScanonical, I , Scanonical, G iÞÞ :

(9.48)

Using a similar approach as in Section 9.3, we can establish the following bijection between canonical EA-OQECC described by Scanonical ¼ hScanonical, I , Scanonical, E , Scanonical, G i, and a general EA-OQECC characterized by S ¼ hSI , SE , SG i, with S ¼ UScanonicalU1 (or equivalently Scanonical ¼ U1SU), where U is a corresponding unitary operator. This [N,K,R,e] EA-OQECC, illustrated in Figure 9.7b, is defined by the encodingedecoding pair (E,D) with the property that it can correct any error from the following set of errors: o n

y (9.49) N ¼ Em
cE1 , E2 0E2 E1 ˛hSI , SG iWðGN  CðhSI , SG iÞÞ : The EA-OQECC is very flexible, as it can be transformed into another code by adding or removing certain stabilizer elements. It can be shown that an [N,K,R,e;D1] can be transformed into an [N,K þ R,0,e;D2] or an [N,K,0,e;D3] code. The minimum distance of EQ-OQECCs is defined in fashion similar to that in previous sections. For encoder implementation, we can use the conjugation method, in which we employ the gates described in Section 9.4 to transform the EA-OQECC into the corresponding canonical form (9.47). By applying this sequence of gates in the opposite order we obtain the encoding circuit of the corresponding EA-OQECC.

9.6 Entanglement-Assisted Operator Quantum Error Correction Coding (EA-OQECC)

9.6.1 EA-OQECCs Derived from Classical Binary and Quaternary Codes The EA-OQECCs can be derived from the EA codes described in Section 9.3, by properly rearranging and combining the stabilizers. Example. Let us observe the [8,1,1;3] EA code, described by the following group: SEA ¼ hSI , SE i, SI ¼ hZ1 Z2 , Z1 Z3 , Z4 Z5 , Z4 Z6 , Z7 Z8 , X1 X2 X3 X4 X5 X6 i, SE ¼ hZ8 , X1 X2 X3 X7 X8 i: The corresponding Pauli encoded operators are given by {Z1Z4Z8,X4X5X6}. By using this EA code we can derive the following [8,1,R ¼ 2,e ¼ 1;3] EA-OQECC: SEA ¼ hSI , SE , SG i, SI ¼ hZ1 Z2 Z4 Z5 , Z1 Z3 Z4 Z6 , Z7 Z8 , X1 X2 X3 X4 X5 X6 i, SE ¼ hZ8 , X1 X2 X3 X7 X8 i, SG ¼ hZ1 Z2 , X2 X5 , Z4 Z6 , X3 X6 i: For CSS-like codes, the number of ebits can be determined by e ¼ rank(HHT), where H is the parity-check matrix of classical code. For example, the BCH codes described in Chapter 6 can be used in the design of EA codes. The BCH code over GF(2m) has codeword length n ¼ 2m  1, minimum distance d  2t þ 1, number of parity bits n  k  mt, and its parity-check matrix is given by 3 2 an1 an2 . a 1 6 a3ðn1Þ a3ðn2Þ . a3 1 7 7 6 5ðn1Þ 5ðn2Þ 5 (9.50) H¼6 a . a 1 7 7: 6 a 5 4 . . . . . að2t1Þðn1Þ að2t1Þðn2Þ . a2t1 1 In Chapter 6, we described how to obtain a binary representation of this parity-check matrix. For example, for m ¼ 6, the number of ebits required is e ¼ rank(HHT) ¼ 6. This BCH (63,39;9) code, therefore, can be used to design [n ¼ 63,2k  n þ e ¼ 21,e ¼ 6] EA code. By inspection we can find that the last six rows of the binary matrix H are sympletic pairs that form an entanglement group. The gauge group can be formed by removing one sympletic pair at a time from the entanglement group and adding it to the gauge group. Clearly, the following set of quantum codes can be obtained using this approach: {[63,21,R ¼ 0,e ¼ 6], [63,21,R ¼ 1, e ¼ 5], [63,21,R ¼ 2,e ¼ 4], [63,21,R ¼ 3,e ¼ 3], [63,21,R ¼ 4,e ¼ 2], [63,21,R ¼ 5, e ¼ 1], [63,21,R ¼ 6,e ¼ 0]}. The first code from the set is a pure EA code, and the last code from the set is a pure subsystem code, while the quantum codes in between are EA-OQECCs. The EA-OQECC codes can also be derived from classical low-density paritycheck (LDPC) codes. Chapter 10 considers the different classes of quantum codes derived from classical LDPC codes [3,29e33]. Quaternary classical codes can also be used in the design of EA-QEECCs. The first step is to start with the corresponding EA code, using the concepts outlined in

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Section 9.3. The second step is to carefully move some of the sympletic pairs from the entanglement group to the gauge group so that the minimum distance is not affected that much. Let us consider the following (15,10,4) quaternary code [4]: 2 3 1 0 0 0 1 1 a2 0 1 a2 0 a a2 1 0 6 0 1 0 0 1 0 a a2 1 a 0 0 1 a 1 7 6 7 2 6 H4 ¼ 6 0 0 1 0 a a 1 a 1 0 0 a 1 a2 a 7 7, 4 0 0 0 1 1 a2 0 1 a2 a 0 a2 1 0 a2 5 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 (9.51) where the elements of GF(4) are {0,1,a,a2}, generated by the primitive polynomial p(x) ¼ x2 þ x þ 1. By using Eq. (9.16) we obtain the following quantum-check matrix of [15,9,e ¼ 4;4] EA code:     aH4 ~4 ¼ f AEA ¼ f H a2 H4 2 3 X I I I X X Y I Z Y I Z Y X I 6 I X I I X I Z Y X Z I I X Z X7 6 7 6 I I X I Z Y X Z X I I Z X Y Z 7 6 7 6 I I I X X Y I X Y Z I Y X I Y 7 6 7 6 I I I I I I I I I I X I I I I 7 6 7: (9.52) ¼6 7 6 Z I I I Z Z X I Z X I Y X Z I 7 6 I Z I I Z I Y X Z Y I I Z Y Z 7 6 7 6 I I Z I Y X Z Y Z I I Y Z X Y 7 6 7 4 I I I Z Z X I Z X Y I X Z I X 5 I I I I I I I I I I Z I I I I From Eq. (9.52) it is clear that the stabilizers in the third and fourth rows commute, and therefore they represent the isotropic subgroup. The remaining eight anticommute and represent the entanglement subgroup. By removing any two anticommuting pairs from the entanglement subgroup, say the first two, and moving them to the gauge subgroup, we obtain the [15,9,e ¼ 3,R ¼ 1] EA-OQECC. By using the MAGMA approach [34], we can identify an EA-QECC derived from (9.52) having the largest possible minimum distance.

9.7 SUMMARY This chapter has considered entanglement-assisted quantum error correction codes, which use pre-existing entanglement between transmitter and receiver to improve the reliability of transmission. A key advantage of EA quantum codes compared to CSS codes is that EA quantum codes do not impose the dual-containing constraint. Therefore, arbitrary classical codes can be used to design EA quantum codes. The number of required ebits has been determined by e ¼ rank(HHT), where H is the

9.8 Problems

parity-check matrix of the corresponding classical code. A general description of entanglement-assisted quantum error correction was provided in Section 9.1. In Section 9.2, we studied entanglement-assisted canonical code and its error correction capability. In Section 9.3, the concept of EA canonical code to arbitrary EA code was generalized. We also described how to design EA codes from classical codes, in particular classical quaternary codes. In Section 9.4, encoding for EA quantum codes was discussed. In Section 9.5, the concept of operator quantum error correction, also known as subsystem codes, was introduced. Entanglement-assisted operator quantum error correction was discussed in Section 9.6. In the following section, we provide a set of problems that will help the reader gain better understanding of the material in this chapter.

9.8 PROBLEMS 1. If V is an arbitrary subgroup of GN of order 2M, then there exists a set of generators fX pþ1 , ., X pþq ;Z 1 , ., Z pþq g satisfying similar commutation properties to Eq. (8.28) of the previous chapter: ½X m , X n  ¼ ½Z m , Z n  ¼ 0 cm, n; ½X m , Z n  ¼ 0 cmsn; fX m , Z n g ¼ 0 cm ¼ n: Prove this claim. 2. If there exists bijection between two groups V and S that preserves their commutation relations, prove that c v ˛ V there exists corresponding s ˛ S such that v ¼ UsU1 (up to a general phase constant). 3. As a generalization of the EA quantum codes represented above, we can design a continuous-variable EA quantum code given by AEA ¼ ðAx jAz Þ, where AEA is an (n  k)  2n real matrix, and both Ax and Az are (n  k)  n real matrices. The number of required entangled states is e ¼ rankfAz ATx  Ax ATz g=2: Prove this claim. 4. A formula similar to that from the previous example can be derived for EA qudit codes, by replacing the subtraction operation by subtraction mod p (where p is a prime) as follows: e ¼ rankfðAz ATx  Ax ATz Þmod pg=2: The quantum-check matrix of this qudit code is given by AEA ¼ ðAx jAz Þ, with matrix elements being from finite field Fp ¼ {0,1,.,p  1}. Notice that error

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operators need to be redefined as it was done in the previous The Pp1chapter. p ffiffiffi p: corresponding entanglement qudits (edits) have the form jmijmi= m¼0 Prove the claims. Can you generalize the EA qudit code design over GF(q) (q ¼ pm, m > 1)? 5. This problem concerns extended EA codes. If an EA [N,K,e;D] code exists prove that an extended [N þ 1,K  1,e0 ;D0 ] also exists for some e0 and D0  D. Provide the quantum-check matrix of the extended EA code, assuming that the quantum-check matrix of the original EA code is known. 6. In certain quantum technologies, the CNOT gate is challenging to implement. For EA code given by the quantum-check matrix below, determine the encoding circuit without using any SWAP operators because they require the use of three CNOT gates. 2

AEA

0 60 ¼6 41 1

1 0 1 1


0

1 0

1 0

0 1
0

0 0 1 0

0 1 1 0

1 0 0 0

3 0 17 7: 05 0

7. By using the standard form method, conjugation method, and sympletic GrameSchmidt method for EA code encoder implementation, described by the quantum-check matrix below, provide the corresponding encoding circuits. Discuss the complexity of the corresponding realizations. 2

AEA

0 61 6 60 6 ¼6 61 60 6 41 0

0 0 1 0 0 1 1

1 0 0 0 1 1 0

0 1 1 0 1 0 0

1 1 0 0 0 0 1

0 0 0 1 1 0 1

3 1 07 7 17 7 17 7: 07 7 05 0

8. Prove that the canonical OQECC can correct the following set of errors: o n

y N ¼ Em
cE1 , E2 0E2 E1 ˛Scanonical, OQECC WðGN  C ðScanonical, OQECC, I ÞÞ : 9. Prove that [N,K,R,D] OQECC code has the following properties: (i) It can correct any error for the following set of errors: o n

y N ¼ Em
cE1 , E2 0E2 E1 ˛SOQECC WðGN  C ðSOQECC, I ÞÞ : (ii) The code space COQECC is a simultaneous eigenspace of SOQECC,I. (iii) To decode, the error syndrome is obtained by simultaneously measuring the observables from SOQECC,I.

9.8 Problems

10. Prove that canonical [N,K,R,e] EA-OQECC can correct the following set of errors: n

y N ¼ Em
cE1 , E2 0E2 E1 ˛hScanonical, I , Scanonical, G i o WðGN  CðhScanonical, I , Scanonical, G iÞÞ : Prove also that [N,K,R,e] EA-OQECC can correct the following set of errors: n
o
y N ¼ Em
cE1 , E2 0E2 E1 ˛hSI , SG iWðGN  CðhSI , SG iÞÞ : 11. Prove that an [N,K,R,e;D1] EA-OQECC can be transformed into an [N,K þ R,0,e;D2] or an [N,K,0,e;D3] code. Show that the minimum distances satisfy the inequalities D2  D1  D3 : 12. The (15,7) BCH code has the following parity-check matrix: 2

1 60 6 60 6 61 H¼6 61 6 61 6 41 1

1 1 0 1 1 0 1 0

1 1 1 1 1 1 0 0

1 1 1 0 1 0 0 0

0 1 1 1 0 0 0 1

1 0 1 0 1 1 1 1

0 1 0 1 1 0 1 0

1 0 1 1 1 1 0 0

1 1 0 0 1 0 0 0

0 1 1 0 0 0 0 1

0 0 1 1 1 1 1 1

1 0 0 0 1 0 1 0

0 1 0 0 1 1 0 0

0 0 1 0 1 0 0 0

3 0 07 7 07 7 17 7: 07 7 07 7 05 1

Determine the parameters of the corresponding EA code. Describe all possible EA-OQECCs that can be derived from this code. 13. Let us observe an EA code described by the following quantum-check matrix: 2 3 X I I I X X Y I Z Y I Z Y X I 6 I X I I X I Z Y X Z I I X Z X7 6 7 6 I I X I Z Y X Z X I I Z X Y Z 7 6 7 6 I I I X X Y I X Y Z I Y X I Y 7 6 7 6 I I I I I I I I I I X I I I I 7 6 7: AEA ¼ 6 7 6 Z I I I Z Z X I Z X I Y X Z I 7 6 I Z I I Z I Y X Z Y I I Z Y Z 7 6 7 6 I I Z I Y X Z Y Z I I Y Z X Y 7 6 7 4 I I I Z Z X I Z X Y I X Z I X 5 I I I I I I I I I I Z I I I I By using this quantum-check matrix determine an EA-OQECC of largest possible minimum distance. What are the parameters of this code? By using the conjugation method provide the corresponding encoding circuit.

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CHAPTER 9 Entanglement-Assisted Quantum Error Correction

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