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 1/27
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 2/27
Classical and Quantum Information ◮ Classical information often represented by a finite alphabet, e.g., bits 0 and 1 ◮ Quantum-bit (qubit) basis states: � 1 � � 0 � ∈ C 2 ∈ C 2 | 0 � = | 1 � = 0 1 general pure state where α, β ∈ C , | α | 2 + | β | 2 = 1 | ψ � = α | 0 � + β | 1 � measurement (read-out): “0” with probability | α | 2 “1” with probability | β | 2 3/27
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: | b 1 � ⊗ · · · ⊗ | b n � = | b 1 . . . b n � = | b � where b i ∈ { 0 , 1 } general pure state: | c x | 2 = 1 � � | ψ � = c x | x � where x ∈{ 0 , 1 } n x ∈{ 0 , 1 } n 4/27
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: | b 1 � ⊗ · · · ⊗ | b n � = | b 1 . . . b n � = | b � where b i ∈ { 0 , 1 } general pure state: | c x | 2 = 1 � � | ψ � = c x | x � where x ∈{ 0 , 1 } n x ∈{ 0 , 1 } n 1 For example, | Φ � = 2 ( | 00 � + | 11 � ) √ 4/27
Error Basis ◮ Pauli Matrices � 0 � � 0 � 1 − i X = , Y = , 1 0 i 0 � 1 � � 1 � 0 0 Z = , I = 0 − 1 0 1 vector space basis of all 2 × 2 matrices 5/27
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 6/27
Super Dense Coding How does it work? b 1 b 0 Z � sender id receiver | Φ � • ˜ H b 0 � ˜ b 1 7/27
Super Dense Coding How does it work? b 1 b 0 Z � sender id receiver | Φ � • ˜ H b 0 � ˜ b 1 ◮ Two classical bits with one use of a quantum channel ◮ Proposed by Bennett and Weisner in 1992 7/27
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 8/27
Attempt of Repetition Code ◮ Good candidate for Bit-flip channels | ψ � = α | 0 � + β | 1 � → | ψ enc � = α | 000 � + β | 111 � 9/27
Attempt of Repetition Code ◮ Good candidate for Bit-flip channels | ψ � = α | 0 � + β | 1 � → | ψ enc � = α | 000 � + β | 111 � ◮ Possibles 1-qubit error in our setting | 000 � | 000 � I ⊗ I ⊗ I − → | 111 � | 111 � 9/27
Attempt of Repetition Code ◮ Good candidate for Bit-flip channels | ψ � = α | 0 � + β | 1 � → | ψ enc � = α | 000 � + β | 111 � ◮ Possibles 1-qubit error in our setting | 000 � | 000 � I ⊗ I ⊗ I − → | 111 � | 111 � | 000 � | 100 � X ⊗ I ⊗ I − → | 111 � | 011 � 9/27
Attempt of Repetition Code ◮ Good candidate for Bit-flip channels | ψ � = α | 0 � + β | 1 � → | ψ enc � = α | 000 � + β | 111 � ◮ Possibles 1-qubit error in our setting | 000 � | 000 � I ⊗ I ⊗ I − → | 111 � | 111 � | 000 � | 100 � X ⊗ I ⊗ I − → | 111 � | 011 � | 000 � | 010 � I ⊗ X ⊗ I − → | 111 � | 101 � 9/27
Attempt of Repetition Code ◮ Good candidate for Bit-flip channels | ψ � = α | 0 � + β | 1 � → | ψ enc � = α | 000 � + β | 111 � ◮ Possibles 1-qubit error in our setting | 000 � | 000 � I ⊗ I ⊗ I − → | 111 � | 111 � | 000 � | 100 � X ⊗ I ⊗ I − → | 111 � | 011 � | 000 � | 010 � I ⊗ X ⊗ I − → | 111 � | 101 � | 000 � | 001 � I ⊗ I ⊗ X − → | 111 � | 110 � 9/27
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 10/27
Entanglement-Assisted Quantum Error Correcting Code ◮ The first QUENTA code was proposed by Bowen 1 ◮ The stabilizer formalism for qubits QUENTA code was done by Brun et al. 2 ◮ This class of codes can violate the quantum Hamming bound 3 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) 11/27
Entanglement-Assisted Quantum Error Correcting Code ◮ The first QUENTA code was proposed by Bowen 1 ◮ The stabilizer formalism for qubits QUENTA code was done by Brun et al. 2 ◮ This class of codes can violate the quantum Hamming bound 3 Super Dense Coding Quantum Error Correction Entanglement-Assisted Quantum Error Correcting Code 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) 11/27
QUENTA Code Scheme n − k − c qudits | 0 n − k − c � n qudits E k qudits | ψ � c qudits sender N receiver | Φ ⊗ c � k qudits | ψ ′ � D c qudits id ⊗ c 12/27
Euclidean Construction Method Proposition Let C 1 and C 2 be two linear codes with parameters [ n, k 1 , d 1 ] q and [ n, k 2 , d 2 ] q and parity check matrices H 1 and H 2 , respectively. Then there is an QUENTA code with parameters [[ n, k 1 + k 2 − n + c, d ≥ min { d 1 , d 2 } ; c ]] q that requires c = rank ( H 1 H T 2 ) 13/27
Euclidean Construction Method Proposition Let C 1 and C 2 be two linear codes with parameters [ n, k 1 , d 1 ] q and [ n, k 2 , d 2 ] q and parity check matrices H 1 and H 2 , respectively. Then there is an QUENTA code with parameters [[ n, k 1 + k 2 − n + c, d ≥ min { d 1 , d 2 } ; c ]] q that requires c = rank ( H 1 H T 2 ) = dim( C ⊥ 1 ) − dim( C ⊥ 1 ∩ C 2 ) maximally entangled states. 13/27
Euclidean Construction Method Proposition Let C 1 and C 2 be two linear codes with parameters [ n, k 1 , d 1 ] q and [ n, k 2 , d 2 ] q and parity check matrices H 1 and H 2 , respectively. Then there is an QUENTA code with parameters [[ n, k 1 + k 2 − n + c, d ≥ min { d 1 , d 2 } ; c ]] q that requires c = rank ( H 1 H T 2 ) = dim( C ⊥ 1 ) − dim( C ⊥ 1 ∩ C 2 ) maximally entangled states. A entanglement-assisted quantum code is ◮ MDS if d = ( n − k + c ) / 2 + 1 ◮ Maximal entanglement if c = n − k 13/27
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 14/27
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 15/27
Notation ◮ Let q � = 2 be a prime power and F q be a finite field ◮ n denotes the code length, with gcd ( n, q ) = 1 , and m = ord n ( q ) ◮ Z ( C ) denotes the defining set of a cyclic code C ◮ Lastly, g ( x ) is the generator polynomial of C 16/27
BCH Codes Definition Let b ≥ 0 , δ ≥ 1 , and α ∈ F q m . A cyclic code C of length n over F q is a BCH code with designed distance δ if g ( x ) = lcm { m b ( x ) , m b +1 ( x ) , . . . , m b + δ − 2 ( x ) } where m i ( x ) is the minimal polynomial of α i over F q . In particular, Z ( C ) = { b, b + 1 , . . . , b + δ − 2 } . If n = q m − 1 then the BCH code is called primitive , and if b = 1 it is called narrow sense . 17/27
BCH Codes Definition Let b ≥ 0 , δ ≥ 1 , and α ∈ F q m . A cyclic code C of length n over F q is a BCH code with designed distance δ if g ( x ) = lcm { m b ( x ) , m b +1 ( x ) , . . . , m b + δ − 2 ( x ) } where m i ( x ) is the minimal polynomial of α i over F q . In particular, Z ( C ) = { b, b + 1 , . . . , b + δ − 2 } . If n = q m − 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 17/27
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 ⊥ ) = Z n \ {− i | i ∈ Z ( C ) } and the generator polynomial is given by the lcm between the min- imal polynomials over F q of the elements α j such that j ∈ Z ( C ⊥ ) . 18/27
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