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– 23.07.2018

Quantum Error-Correcting Codes LAWCI Classical & 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” ˆ = | 0 � = “1” ˆ = | 1 � = 0 1 general pure state: where α, β ∈ C , | α | 2 + | β | 2 = 1 | ψ � = α | 0 � + β | 1 � measurement (read-out): result “0” with probability | α | 2 result “1” with probability | β | 2 Markus Grassl – 2– 23.07.2018

Quantum Error-Correcting Codes LAWCI Classical & 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: � � x ∈{ 0 , 1 } n | c x | 2 = 1 | ψ � = c x | x � where x ∈{ 0 , 1 } n → normalised vector in ( C 2 ) ⊗ n ∼ = C 2 n − Markus Grassl – 3– 23.07.2018

Quantum Error-Correcting Codes LAWCI Classical & Quantum Information Larger alphabet messages represented as vectors over a finite field, i. e., x ∈ F n q Qudit register basis states: | b 1 � ⊗ . . . ⊗ | b n � =: | b 1 . . . b n � = | b � where b i ∈ F q general pure state: � � | c x | 2 = 1 | ψ � = c x | x � where x ∈ F n x ∈ F n q q = C q n ∼ → normalised vector in ( C q ) ⊗ n ∼ = C [ F n − q ] (isomorphic as vector spaces) Markus Grassl – 4– 23.07.2018

Quantum Error-Correcting Codes LAWCI Bra-Ket Notation • column vectors denoted by | x � “ket x ” • elements of the dual vector space (row vectors) denoted by � y | “bra y ” q � | x � = α i | i � i =1 q � � x | = α i � i | i =1 • inner product � x | y � “bra-c-ket” • linear operators M = � i,j m i,j | i �� j | in particular: rank-one orthogonal projections P | ψ � = | ψ �� ψ | Markus Grassl – 5– 23.07.2018

Quantum Error-Correcting Codes LAWCI Quantum Operations Continuous time: Schr¨ odinger equation i � ∂ ∂t | ψ ( t ) � = H ( t ) | ψ ( t ) � time-independent Hamiltonian H : | ψ ( t ) � = e iH ( t − t 0 ) | ψ ( t 0 ) � Discrete time: Unitary operations | ψ ( t 1 ) � = U ( t 1 , t 0 ) | ψ ( t 0 ) � • composite systems, independent operation: U = U 1 ⊗ U 2 • single-qudit operations U (1) = U ⊗ I ⊗ I ⊗ . . . U (2) = I ⊗ U ⊗ I ⊗ . . . Markus Grassl – 6– 23.07.2018

Quantum Error-Correcting Codes LAWCI Quantum Operations CNOT on basis states: | x �| y � �→ | x �| x + y � Toffoli gate on basis states: | x �| y �| z � �→ | x �| y �| z + xy � • Every reversible classical Boolean function can be decomposed into Toffoli gates (plus constants). • Every unitary operation U ∈ U (2 n ) can be decomposed into single qubit gates and CNOT. • • U ✐ ✐ • V T ✐ • W Markus Grassl – 7– 23.07.2018

Quantum Error-Correcting Codes LAWCI Quantum Measurement Physics textbook: • observable A given by self-adjoint operator A = A ∗ = A † • real eigenvalues λ i correspond to physical quantity • expectation value � A � = � ψ | A | ψ � from many repetitions with identically prepared quantum states | ψ � Quantum Information Processing: • spectral decomposition A = � i λ i P i , with P i orthogonal projection onto eigenspace i • single-shot experiment, one random outcome i (“click”) • probability p i = � ψ | P i | ψ � 1 • post-measurement state | ψ ′ � = √ p i P i | ψ � Markus Grassl – 8– 23.07.2018

Quantum Error-Correcting Codes LAWCI Quantum Measurement: Examples Single Qubit: • observable σ z = diag(1 , − 1) = (+1) | 0 �� 0 | + ( − 1) | 1 �� 1 | • qubit state | ψ � = α | 0 � + β | 1 � where α, β ∈ C ⇒ outcome “0” or “1” with probability | α | 2 or | β | 2 , resp. = Two Qubits: • observable σ (1) = diag(1 , − 1) ⊗ I = | 0 �� 0 | ⊗ I − | 1 �� 1 | ⊗ I z • two-qubit state | ψ � = α | 00 � + β | 01 � + γ | 10 � + δ | 11 � ⇒ outcome “0” with probability | α | 2 + | β | 2 = � � 1 post-measurement state � α | 00 � + β | 01 � | α | 2 + | β | 2 Markus Grassl – 9– 23.07.2018

Quantum Error-Correcting Codes LAWCI Quantum Entanglement 1 1 • two-qubit state | Ψ � = 2 | 00 � + 2 | 11 � √ √ • measuring σ (1) z , i. e., the first qubit = ⇒ outcome “0” or “1” with probability 1 / 2 • post-measurement state first outcome “0”: | 00 � first outcome “1”: | 11 � • measuring σ (2) z , i. e., the second qubit = ⇒ second outcome is identical to the first outome Spooky action at a distance. Markus Grassl – 10– 23.07.2018

Quantum Error-Correcting Codes LAWCI Quantum Error Correction General scheme s action environment inter- no access s system | φ � entanglement ⇒ decoherence Markus Grassl – 11– 23.07.2018

Quantum Error-Correcting Codes LAWCI Quantum Error Correction General scheme s interaction environment no access s s encoding system | φ � s s encoding ancilla | 0 � three-party entanglement Markus Grassl – 12– 23.07.2018

Quantum Error-Correcting Codes LAWCI Quantum Error Correction General scheme s s interaction environment no access s s s encoding system | φ � correction code entanglement s s s encoding ancilla | 0 � error s syndrome ancilla | 0 � environment/ancilla Markus Grassl – 13– 23.07.2018

Quantum Error-Correcting Codes LAWCI Quantum Error Correction General scheme s s s interaction environment no access s s s encoding decoding system | φ � | φ � correction s s s encoding ancilla | 0 � | 0 � error s s syndrome ancilla | 0 � Markus Grassl – 14– 23.07.2018

Quantum Error-Correcting Codes LAWCI Quantum Error Correction General scheme s s s interaction environment no access s s s encoding decoding system | φ � | φ � correction s s s encoding ancilla | 0 � | 0 � error s s syndrome ancilla | 0 � Basic requirement some knowledge about the interaction between system and environment Common assumptions • no initial entanglement between system and environment • local/uncorrelated errors, i. e., only a few qubits are disturbed or interaction with symmetry ( = ⇒ decoherence free subspaces) Markus Grassl – 15– 23.07.2018

Quantum Error-Correcting Codes LAWCI Interaction System/Environment “Closed” System � environment | ε � action � � inter- = U env/sys | ε �| φ � system | φ � “Channel” � E i ρ in E † Q : ρ in := | φ �� φ | �− → ρ out := Q ( | φ �� φ | ) := i i with Kraus operators (error operators) E i Local/low correlated errors • product channel Q ⊗ n where Q is “close” to identity • Q can be expressed (approximated) with error operators ˜ E i such that each ˜ E i acts on few subsystems, e. g. quantum gates Markus Grassl – 16– 23.07.2018

Quantum Error-Correcting Codes LAWCI Quantum Channel: Product Channel Assumption: only local interactions , i. e., each subsystem interacts with a separate environment.   ✲ ✲    | ˜  | ψ 1 � ψ 1 �  subsystem 1     ✸ ✑ ✑ ✸  ✑ ✑ ◗ ◗ s ◗ ◗ s       ✲ ✲   | ǫ 1 � | ˜ ǫ 1 �   environment 1               ✲ ✲   | ˜  | ψ 2 �  subsystem 2 ψ 2 �     ✑ ✸ ✸ ✑ ✑ ✑  ◗ ◗  ◗ s ◗ s   � ✲ ✲ | ˜ | ǫ 2 � | ˜ ǫ 2 � | ψ � ⊗ | ǫ � environment 2 ψ � ⊗ | ˜ ǫ �           .   .   .           ✲ ✲   | ˜ | ψ n �   ψ n � subsystem n     ✑ ✸ ✸ ✑ ✑ ✑  ◗ ◗  s ◗ ◗ s     ✲ ✲     | ǫ n � | ˜ ǫ n � environment n     Markus Grassl – 17– 23.07.2018

Quantum Error-Correcting Codes LAWCI Computer Science Approach: Discretize QECC Characterization [Knill & Laflamme, PRA 55 , 900–911 (1997)] A subspace C of H with orthonormal basis {| c 1 � , . . . , | c K �} is an error-correcting code for the error operators E = { E 1 , E 2 , . . . } , if there exists constants α k,l ∈ C such that for all | c i � , | c j � and for all E k , E l ∈ E : � c i | E † k E l | c j � = δ i,j α k,l . (1) It is sufficient that (1) holds for a vector space basis of E . Markus Grassl – 18– 23.07.2018