Quantum BCH and Reed-Solomon Entanglement-Assisted Codes Francisco - - PowerPoint PPT Presentation

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Quantum BCH and Reed-Solomon Entanglement-Assisted Codes

Francisco R. F. Pereira

Joint work with Ruud Pellikaan

TU/e, the Netherlands

40th WIC SITB, Belgium May 28, 2019

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Content

Introduction Super Dense Coding Quantum Error Correction Entanglement-Assisted Quantum Error Correcting Code New QUENTA codes BCH and Reed Solomon Codes Entanglement Assisted Quantum Cyclic Codes

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Classical and Quantum Information

◮ Classical information

• ften represented by a finite alphabet, e.g., bits 0 and 1

◮ Quantum-bit (qubit) basis states: |0 = 1

• ∈ C2

|1 = 1

• ∈ C2

general pure state |ψ = α |0 + β |1 where α, β ∈ C, |α|2 + |β|2 = 1 measurement (read-out): “0” with probability |α|2 “1” with probability |β|2

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Classical and Quantum Information

◮ Bit strings larger set of messages represented by bit strings of length n, i.e., x ∈ {0, 1}n ◮ Quantum register basis states: |b1 ⊗ · · · ⊗ |bn = |b1 . . . bn = |b where bi ∈ {0, 1} general pure state: |ψ =

• x∈{0,1}n

cx |x where

• x∈{0,1}n

|cx|2 = 1

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Classical and Quantum Information

◮ Bit strings larger set of messages represented by bit strings of length n, i.e., x ∈ {0, 1}n ◮ Quantum register basis states: |b1 ⊗ · · · ⊗ |bn = |b1 . . . bn = |b where bi ∈ {0, 1} general pure state: |ψ =

• x∈{0,1}n

cx |x where

• x∈{0,1}n

|cx|2 = 1 For example, |Φ =

1 √ 2(|00 + |11)

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Error Basis

◮ Pauli Matrices X = 1 1

• ,

Y = −i i

• ,

Z = 1 −1

• ,

I = 1 1

• vector space basis of all 2 × 2 matrices

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Content

Introduction Super Dense Coding Quantum Error Correction Entanglement-Assisted Quantum Error Correcting Code New QUENTA codes BCH and Reed Solomon Codes Entanglement Assisted Quantum Cyclic Codes

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Super Dense Coding

How does it work?

|Φ

sender receiver

H

• Z

id

b0 b1

• ˜

b0 ˜ b1

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Super Dense Coding

How does it work?

|Φ

sender receiver

H

• Z

id

b0 b1

• ˜

b0 ˜ b1

◮ Two classical bits with one use of a quantum channel ◮ Proposed by Bennett and Weisner in 1992

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Content

Introduction Super Dense Coding Quantum Error Correction Entanglement-Assisted Quantum Error Correcting Code New QUENTA codes BCH and Reed Solomon Codes Entanglement Assisted Quantum Cyclic Codes

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Attempt of Repetition Code

◮ Good candidate for Bit-flip channels |ψ = α |0 + β |1 → |ψenc = α |000 + β |111

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Attempt of Repetition Code

◮ Good candidate for Bit-flip channels |ψ = α |0 + β |1 → |ψenc = α |000 + β |111 ◮ Possibles 1-qubit error in our setting |000 |111

I⊗I⊗I

− → |000 |111

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Attempt of Repetition Code

◮ Good candidate for Bit-flip channels |ψ = α |0 + β |1 → |ψenc = α |000 + β |111 ◮ Possibles 1-qubit error in our setting |000 |111

I⊗I⊗I

− → |000 |111 |000 |111

X⊗I⊗I

− → |100 |011

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Attempt of Repetition Code

◮ Good candidate for Bit-flip channels |ψ = α |0 + β |1 → |ψenc = α |000 + β |111 ◮ Possibles 1-qubit error in our setting |000 |111

I⊗I⊗I

− → |000 |111 |000 |111

X⊗I⊗I

− → |100 |011 |000 |111

I⊗X⊗I

− → |010 |101

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Attempt of Repetition Code

◮ Good candidate for Bit-flip channels |ψ = α |0 + β |1 → |ψenc = α |000 + β |111 ◮ Possibles 1-qubit error in our setting |000 |111

I⊗I⊗I

− → |000 |111 |000 |111

X⊗I⊗I

− → |100 |011 |000 |111

I⊗X⊗I

− → |010 |101 |000 |111

I⊗I⊗X

− → |001 |110

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Content

Introduction Super Dense Coding Quantum Error Correction Entanglement-Assisted Quantum Error Correcting Code New QUENTA codes BCH and Reed Solomon Codes Entanglement Assisted Quantum Cyclic Codes

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Entanglement-Assisted Quantum Error Correcting Code

◮ The first QUENTA code was proposed by Bowen1 ◮ The stabilizer formalism for qubits QUENTA code was done by Brun et al.2 ◮ This class of codes can violate the quantum Hamming bound3

1Bowen, G.: Entanglement required in achieving entanglement-assisted channel capacities. Physical Review A 66, 052313–1-052313–8 (2006) 2Brun, T., Devetak, I., Hsieh, M.H.: Correcting quantum errors with entanglement. Science 314(5798), 436–439 (2006) 3Li, R., Guo, L., Xu, Z.: Entanglement-assisted quantum codes achieving the quantum Singleton bound but violating the quantum hamming bound. Quantum Information & Computation 14(13), 1107–1116 (2014)

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Entanglement-Assisted Quantum Error Correcting Code

◮ The first QUENTA code was proposed by Bowen1 ◮ The stabilizer formalism for qubits QUENTA code was done by Brun et al.2 ◮ This class of codes can violate the quantum Hamming bound3 Entanglement-Assisted Quantum Error Correcting Code Quantum Error Correction Super Dense Coding

1Bowen, G.: Entanglement required in achieving entanglement-assisted channel capacities. Physical Review A 66, 052313–1-052313–8 (2006) 2Brun, T., Devetak, I., Hsieh, M.H.: Correcting quantum errors with entanglement. Science 314(5798), 436–439 (2006) 3Li, R., Guo, L., Xu, Z.: Entanglement-assisted quantum codes achieving the quantum Singleton bound but violating the quantum hamming bound. Quantum Information & Computation 14(13), 1107–1116 (2014)

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QUENTA Code Scheme

|0n−k−c |ψ |Φ⊗c E N

sender receiver

id⊗c D |ψ′

n − k − c qudits k qudits c qudits c qudits k qudits n qudits

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Euclidean Construction Method

Proposition

Let C1 and C2 be two linear codes with parameters [n, k1, d1]q and [n, k2, d2]q and parity check matrices H1 and H2, respectively. Then there is an QUENTA code with parameters [[n, k1 + k2 − n + c, d ≥ min{d1, d2}; c]]q that requires c = rank(H1HT

2 )

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Euclidean Construction Method

Proposition

Let C1 and C2 be two linear codes with parameters [n, k1, d1]q and [n, k2, d2]q and parity check matrices H1 and H2, respectively. Then there is an QUENTA code with parameters [[n, k1 + k2 − n + c, d ≥ min{d1, d2}; c]]q that requires c = rank(H1HT

2 ) = dim(C⊥ 1 ) − dim(C⊥ 1 ∩ C2)

maximally entangled states.

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Euclidean Construction Method

Proposition

Let C1 and C2 be two linear codes with parameters [n, k1, d1]q and [n, k2, d2]q and parity check matrices H1 and H2, respectively. Then there is an QUENTA code with parameters [[n, k1 + k2 − n + c, d ≥ min{d1, d2}; c]]q that requires c = rank(H1HT

2 ) = dim(C⊥ 1 ) − dim(C⊥ 1 ∩ C2)

maximally entangled states. A entanglement-assisted quantum code is ◮ MDS if d = (n − k + c)/2 + 1 ◮ Maximal entanglement if c = n − k

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Content

Introduction Super Dense Coding Quantum Error Correction Entanglement-Assisted Quantum Error Correcting Code New QUENTA codes BCH and Reed Solomon Codes Entanglement Assisted Quantum Cyclic Codes

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Content

Introduction Super Dense Coding Quantum Error Correction Entanglement-Assisted Quantum Error Correcting Code New QUENTA codes BCH and Reed Solomon Codes Entanglement Assisted Quantum Cyclic Codes

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Notation

◮ Let q = 2 be a prime power and Fq be a finite field ◮ n denotes the code length, with gcd(n, q) = 1, and m =

• rdn(q)

◮ Z(C) denotes the defining set of a cyclic code C ◮ Lastly, g(x) is the generator polynomial of C

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BCH Codes

Definition

Let b ≥ 0, δ ≥ 1, and α ∈ Fqm. A cyclic code C of length n over Fq is a BCH code with designed distance δ if g(x) = lcm{mb(x), mb+1(x), . . . , mb+δ−2(x)} where mi(x) is the minimal polynomial of αi over Fq. In particular, Z(C) = {b, b + 1, . . . , b + δ − 2}. If n = qm − 1 then the BCH code is called primitive, and if b = 1 it is called narrow sense.

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BCH Codes

Definition

Let b ≥ 0, δ ≥ 1, and α ∈ Fqm. A cyclic code C of length n over Fq is a BCH code with designed distance δ if g(x) = lcm{mb(x), mb+1(x), . . . , mb+δ−2(x)} where mi(x) is the minimal polynomial of αi over Fq. In particular, Z(C) = {b, b + 1, . . . , b + δ − 2}. If n = qm − 1 then the BCH code is called primitive, and if b = 1 it is called narrow sense. It is possible to show that the dimension is equal to n − |Z(C)| and the minimal distance of C is at least δ

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Euclidian Dual BCH Codes

Proposition

Let C be a BCH code of length n and defining set Z(C). Then the defining set of C⊥ is given by Z(C⊥) = Zn \ {−i|i ∈ Z(C)} and the generator polynomial is given by the lcm between the min- imal polynomials over Fq of the elements αj such that j ∈ Z(C⊥).

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Reed-Solomon Codes

Definition

Let b ≥ 0, n = q − 1, and 1 ≤ k ≤ n. A cyclic code RSk(n, b)

• f length n over Fq is a Reed-Solomon code with minimal distance

n − k + 1 if g(x) = (x − αb)(x − αb+1) · · · (x − αb+n−k−1), where α is a primitive element of Fq.

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Reed-Solomon Codes

Definition

Let b ≥ 0, n = q − 1, and 1 ≤ k ≤ n. A cyclic code RSk(n, b)

• f length n over Fq is a Reed-Solomon code with minimal distance

n − k + 1 if g(x) = (x − αb)(x − αb+1) · · · (x − αb+n−k−1), where α is a primitive element of Fq. Defining set: Z(RSk(n, b)) = {b, b + 1, . . . , b + n − k − 1}

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Euclidean Dual Reed-Solomon Codes

Proposition

Let RSk(n, b) be a Reed-Solomon code. Then its Euclidian dual can be described as RSk(n, b)⊥ = RSn−k(n, n − b + 1) In particular, the defining the of RSk(n, b)⊥ is given by Z(RSk(n, b)⊥) = {n − b + 1, n − b + 2, . . . , n − b + k}.

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Content

Introduction Super Dense Coding Quantum Error Correction Entanglement-Assisted Quantum Error Correcting Code New QUENTA codes BCH and Reed Solomon Codes Entanglement Assisted Quantum Cyclic Codes

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The General Cyclic Case

Lemma

Let C1 and C2 be two cyclic codes with defining set Z(C1) and Z(C2), respectively. Then Z(C1 ∩ C2) = Z(C1) ∪ Z(C2).

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The General Cyclic Case

Lemma

Let C1 and C2 be two cyclic codes with defining set Z(C1) and Z(C2), respectively. Then Z(C1 ∩ C2) = Z(C1) ∪ Z(C2).

Theorem

Let C1 and C2 be two cyclic codes with parameters [n, k1, d1]q and [n, k2, d2]q, respectively. Then there is an QUENTA code with pa- rameters [[n, k1 −|Z(C⊥

1 )∩Z(C2)|, min{d1, d2}; n−k2 −|Z(C⊥ 1 )∩

Z(C2)|]]q.

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The General Cyclic Case

Corollary

Let C be a LCD cyclic code with parameters [n, k, d]q. Then there is a maximal entanglement QUENTA code with parameters [[n, k, d; n− k]]q. In particular, if C is MDS, so it is the QUENTA code derived from it.

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Quantum Reed-Solomon Entanglement-Assisted Codes

Theorem

Let C1 = RSk1(n, b1) and C2 = RSk2(n, b2) be two Reed-Solomon codes over Fq with 0 ≤ b1 ≤ k1, b2 ≥ 0, and b1 +b2 ≤ k1 +1. Then we have two possible cases:

• 1. For k1 − b1 ≥ b2, there is an QUENTA code with parameters

[[n, b1 + b2 − 1, n − min{k1, k2} + 1; n + b1 + b2 − k1 − k2 − 1]]q;

• 2. For k1 − b1 < b2, there is an QUENTA code with parameters

[[n, k1, n − min{k1, k2} + 1; n − k2]]q.

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Quantum Reed-Solomon Entanglement-Assisted Codes

Corollary

Let C = RSk(n, b) be a Reed-Solomon codes over Fq with 0 ≤ b ≤ (k + 1)/2. Then there is an MDS QUENTA code with parameters [[n, 2b−1, n−k+1; n+2b−2k−1]]q. In particular, for b = (k+1)/2, there is a maximal entanglement MDS QUENTA code.

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Quantum BCH Entanglement-Assisted Codes

Theorem

Let C1 and C2 be two BCH codes over Fq with Z(Ci) = {bi, . . . , bi+ δ − 2}, for i = 1, 2, with b1 + b2 = n + 2 − δ. If the parameters are given by [n, k1, δ]q and [n, k2, δ]q for C1 and C2, respectively, then there is an QUENTA code with parameters [[n, k1+k2−n+c, δ; c]]q, with c = n − max{k1, k2}.

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Thanks for your attention!

Any questions?