Quantum Error-Correcting Codes: Discrete Math meets Physics Markus - PowerPoint PPT Presentation

Quantum Error-Correcting Codes LAWCI Latin American Week on Coding and Information UniCamp – Campinas, Brazil 2018, July 20–27 Quantum Error-Correcting Codes: Discrete Math meets Physics Markus Grassl Markus.Grassl@mpl.mpg.de www.codetables.de Markus Grassl – 1– 24.07.2018

Quantum Error-Correcting Codes LAWCI Simple Quantum Error-Correcting Code Repetition code: | 0 � �→ | 000 � , | 1 � �→ | 111 � Encoding of one qubit: α | 0 � + β | 1 � �→ α | 000 � + β | 111 � . This defines a two-dimensional subspace H C ≤ H 2 ⊗ H 2 ⊗ H 2 bit-flip quantum state subspace no error α | 000 � + β | 111 � ( 1 ⊗ 1 ⊗ 1 ) H C 1 st position α | 100 � + β | 011 � ( X ⊗ 1 ⊗ 1 ) H C 2 nd position α | 010 � + β | 101 � ( 1 ⊗ X ⊗ 1 ) H C 3 rd position α | 001 � + β | 110 � ( 1 ⊗ 1 ⊗ X ) H C Hence we have an orthogonal decomposition of H 2 ⊗ H 2 ⊗ H 2 Markus Grassl – 27– 24.07.2018

Quantum Error-Correcting Codes LAWCI Simple Quantum Error-Correcting Code s s s | ψ � | c 1 � ❣ s | 0 � | c 2 � ❣ s s | 0 � | c 3 � ☛✟ ✂ ✂ ✍ s 1 | 0 � ❣ ❣ q ☛✟ ❆ ❑ s 2 ❣ ❣ | 0 � q encoding syndrome computation measurement Error X ⊗ I ⊗ I gives syndrome s 1 s 2 = 10 Error I ⊗ X ⊗ I gives syndrome s 1 s 2 = 01 Error I ⊗ I ⊗ X gives syndrome s 1 s 2 = 11 Markus Grassl – 30– 24.07.2018

Quantum Error-Correcting Codes LAWCI Linearity of Syndrome Computation Different Errors: Error X ⊗ I ⊗ I syndrome 10 Error I ⊗ X ⊗ I syndrome 01 Suppose the (non-unitary) error is of the form E = α X ⊗ I ⊗ I + β I ⊗ X ⊗ I. Then syndrome computation yields α ( X ⊗ I ⊗ I | ψ enc � ⊗ | 10 � ) + β ( I ⊗ X ⊗ I | ψ enc � ⊗ | 01 � ) . (error correction) �→ | ψ enc � ( α | 10 � + β | 01 � ) Markus Grassl – 33– 24.07.2018

Quantum Error-Correcting Codes LAWCI Shor’s Nine-Qubit Code Hadamard basis: 1 1 | + � = 2 ( | 0 � + | 1 � ) , |−� = 2 ( | 0 � − | 1 � ) √ √ Bit-flip code: | 0 � �→ | 000 � , | 1 � �→ | 111 � . Phase-flip code: | 0 � �→ | + ++ � , | 1 � �→ | − −−� . Concatenation with bit-flip code gives: 1 | 0 � �→ 2 3 ( | 000 � + | 111 � )( | 000 � + | 111 � )( | 000 � + | 111 � ) √ 1 | 1 � �→ 2 3 ( | 000 � − | 111 � )( | 000 � − | 111 � )( | 000 � − | 111 � ) √ Claim: This code can correct one error, i. e., it is an [ [ n, k, d ] ] 2 = [ [9 , 1 , 3] ] 2 . Markus Grassl – 34– 24.07.2018

Quantum Error-Correcting Codes LAWCI Shor’s Nine-Qubit Code Bit-flip code: | 0 � �→ | 000 � , | 1 � �→ | 111 � Effect of single-qubit errors: • X -errors change the basis states, but can be corrected • Z -errors at any of the three positions:   Z | 000 � = | 000 �  “encoded” Z -operator Z | 111 � = −| 111 � = ⇒ can be corrected by the second level of encoding Markus Grassl – 35– 24.07.2018

Quantum Error-Correcting Codes LAWCI Knill-Laflamme Conditions Revisited • code with ONB {| c i �} , channel with Kraus operators { E k } • conditions: � c i | E † (i) k E ℓ | c j � = 0 for i � = j � c i | E † k E ℓ | c i � = � c j | E † (ii) k E ℓ | c j � = α kl • interpretation: (i) orthogonal states remain orthogonal under errors (ii) errors “rotate” all basis states the same way ✻ ✻ E ℓ | c j � ✗ ✄ ✄ errors | c j � ✄ = ⇒ E k | c j � ❇ ▼ α ❇ ✄ ✏ ✶ ✏✏ E k | c i � ✲ ❇ ✄ ✲ ❳❳❳❳ α ❳ ③ | c i � E ℓ | c i � Markus Grassl – 36– 24.07.2018

Quantum Error-Correcting Codes LAWCI General Decoding Algorithm E 1 C E 2 C · · · E k C · · · V 0 E 1 | c 0 � E 2 | c 0 � · · · E k | c 0 � · · · V 1 E 1 | c 1 � E 2 | c 1 � · · · E k | c 1 � · · · . . . . ... . . . . . . . . V i E 1 | c i � E 2 | c i � · · · E k | c i � · · · . . . . ... . . . . . . . . � c i | E † k E l | c j � = δ i,j α k,l (1) Markus Grassl – 37– 24.07.2018

Quantum Error-Correcting Codes LAWCI General Decoding Algorithm E 1 C E 2 C · · · E k C · · · V 0 E 1 | c 0 � E 2 | c 0 � · · · E k | c 0 � · · · V 1 E 1 | c 1 � E 2 | c 1 � · · · E k | c 1 � · · · rows are orthogonal as . . . . ... . . . . . . . . � c i | E † k E l | c j � = 0 for V i E 1 | c i � E 2 | c i � · · · E k | c i � · · · i � = j . . . . ... . . . . . . . . � c i | E † k E l | c j � = δ i,j α k,l (1) Markus Grassl – 38– 24.07.2018

Quantum Error-Correcting Codes LAWCI General Decoding Algorithm E 1 C E 2 C · · · E k C · · · V 0 E 1 | c 0 � E 2 | c 0 � · · · E k | c 0 � · · · V 1 E 1 | c 1 � E 2 | c 1 � · · · E k | c 1 � · · · rows are orthogonal as . . . . ... . . . . . . . . � c i | E † k E l | c j � = 0 for V i E 1 | c i � E 2 | c i � · · · E k | c i � · · · i � = j . . . . ... . . . . . . . . inner product between columns is constant as � c i | E † k E l | c i � = α k,l � c i | E † k E l | c j � = δ i,j α k,l (1) Markus Grassl – 39– 24.07.2018

Quantum Error-Correcting Codes LAWCI General Decoding Algorithm E 1 C E 2 C · · · E k C · · · V 0 E 1 | c 0 � E 2 | c 0 � · · · E k | c 0 � · · · V 1 E 1 | c 1 � E 2 | c 1 � · · · E k | c 1 � · · · rows are orthogonal as . . . . ... . . . . . . . . � c i | E † k E l | c j � = 0 for V i E 1 | c i � E 2 | c i � · · · E k | c i � · · · i � = j . . . . ... . . . . . . . . inner product between columns is constant as � c i | E † k E l | c i � = α k,l = ⇒ simultaneous Gram-Schmidt orthogonalization within the spaces V i Markus Grassl – 40– 24.07.2018

Quantum Error-Correcting Codes LAWCI Orthogonal Decomposition E ′ E ′ E ′ 1 C 2 C · · · k C · · · E ′ E ′ E ′ V 0 1 | c 0 � 2 | c 0 � · · · k | c 0 � · · · E ′ E ′ E ′ V 1 1 | c 1 � 2 | c 1 � · · · k | c 1 � · · · rows are mutually . . . . ... . . . . . . . . orthogonal E ′ E ′ E ′ V i 1 | c i � 2 | c i � · · · k | c i � · · · . . . . ... . . . . . . . . columns are mutually orthogonal • new error operators E ′ k are linear combinations of the E l • projection onto E ′ k C determines the error • exponentially many orthogonal spaces E ′ k C Markus Grassl – 41– 24.07.2018

Quantum Error-Correcting Codes LAWCI CSS Prelims: Linear Binary Codes Generator matrix: C = [ n, k, d ] 2 is a k -dim. subspace of F n 2 ⇒ basis with k (row) vectors, G ∈ F k × n = 2 C = { i G : i ∈ F k 2 } Parity check matrix: C = [ n, k, d ] 2 is a k -dim. subspace of F n 2 ⇒ kernel of n − k homogeneous linear equations, H ∈ F ( n − k ) × n = 2 2 | v H t = 0 } C = { v : v ∈ F n Error syndrome: erroneous codeword v = c + e ⇒ error syndrome v H t = c H t + e H t = e H t = Markus Grassl – 42– 24.07.2018

Quantum Error-Correcting Codes LAWCI CSS Prelims: Dual of a Classical Code Let C ≤ F n 2 be a linear code. Then � � n � C ⊥ := y ∈ F n 2 : x i y i = 0 for all x ∈ C i =1 We denote the scalar product by x · y := � n i =1 x i y i . Lemma: Let C be a linear binary code. Then   � for x ∈ C ⊥ , | C | ( − 1) x · c =  ∈ C ⊥ . 0 for x / c ∈ C Markus Grassl – 43– 24.07.2018

Quantum Error-Correcting Codes LAWCI Bit-flips and Phase-flips Let C ≤ F n 2 be a linear code. Then the image of the state � 1 � | c � | C | c ∈ C under a bit-flip x ∈ F n 2 and a phase-flip z ∈ F n 2 is given by � 1 ( − 1) z · c | c + x � . � | C | c ∈ C Hadamard transform H ⊗ . . . ⊗ H maps this to � ( − 1) xz ( − 1) x · b | b + z � � | C ⊥ | b ∈ C ⊥ Markus Grassl – 44– 24.07.2018

Quantum Error-Correcting Codes LAWCI Proof: � 1 Hadamard transform H ⊗ n = ( − 1) a · b | b �� a | √ 2 n a , b ∈ F n 2 � � � � � 1 1 H ⊗ n ( − 1) z · c | c + x � ( − 1) z · c + a · b | b �� a | c + x � � = � | C | 2 n | C | a , b ∈ F n c ∈ C c ∈ C 2 � � 1 ( − 1) z · c +( c + x ) · b | b � = � | C | 2 n b ∈ F n c ∈ C 2 � ( − 1) x · b � 1 ( − 1) c · ( b + z ) | b � = � | C | 2 n b ∈ F n c ∈ C 2 � ( − 1) x · ( b + z ) � 1 ( − 1) c · b | b + z � = � | C | 2 n b ∈ F n c ∈ C 2 � = ( − 1) x · z ( − 1) x · b | b + z � � | C ⊥ | b ∈ C ⊥ Markus Grassl – 45– 24.07.2018

Quantum Error-Correcting Codes LAWCI CSS Codes Introduced by R. Calderbank, P. Shor, and A. Steane [Calderbank & Shor PRA, 54 , 1098–1105, 1996] [Steane, PRL 77 , 793–797, 1996] Construction: Let C 1 = [ n, k 1 , d 1 ] q and C 2 = [ n, k 2 , d 1 ] q be classical linear codes with C ⊥ 2 ≤ C 1 . Let { x 1 , . . . , x K } be representatives for the cosets C 1 /C ⊥ 2 . Define quantum states � 1 | x i + C ⊥ 2 � := � | x i + y � | C ⊥ 2 | y ∈ C ⊥ 2 Theorem: Then the vector space C spanned by these states is a quantum code with parameters [ [ n, k 1 + k 2 − n, d ] ] q where d ≥ min( d 1 , d 2 ) . Markus Grassl – 46– 24.07.2018