LOCAL APPROXIMATE ERROR CORRECTION CODES Michael J. Kastoryano w/ Steve Flammia, Jeongwan Haah and Isaac Kim Quantum 1, 4 (2017) JHEP 04 (2017) 40 September 14 2017, QEC U. Maryland
MOTIVATION Intriguing example Crépeau et. al. (2005), quant-ph/0503139 Consequence of no-cloning theorem No quantum code can correct more than n/ 4 arbitrary errors Classical codes (Ex: repetition code) can correct up to arbitrary classical errors b n/ 2 c Crépeau et. al. (2005) construct an approximate quantum code that can correct up to arbitrary quantum errors! b n/ 2 c Indication that approximate codes can outperform exact codes!
MOTIVATION What about topological codes? Codes often characterised by three numbers: length ; distance ; encoded (qu-)bits k d n Tradeoff bounds kd 2 ≤ cn Commuting projector codes Bravyi, Poulin, Terhal kd ≤ cn Subsystem codes Bravyi kd 1 / 2 ≤ cn Classical lattice systems Bravyi, Poulin, Terhal; Yoshida Where do approximate quantum codes sit?
Lattice commuting projector codes C S j = S 2 [ S j , S k ] = 0 { S j } j A Y Π = S j C = {| ψ i , Π | ψ i = | ψ i } B Λ j C is the codespace Erasure errors
Lattice commuting projector codes C S j = S 2 [ S j , S k ] = 0 { S j } j A Y Π = S j C = {| ψ i , Π | ψ i = | ψ i } B Λ j C is the codespace Erasure errors C C C C C
C C C C C (i) Topological order C A B Λ
C C C C C (i) Topological order C (ii) Decoupling I ρ ( A : CR ) = S ( A ) + S ( AB ) − S ( B ) A B Λ
C C C C C (i) Topological order C (ii) Decoupling A (iii) Error correction B Λ
C C C C C (i) Topological order C (ii) Decoupling A (iii) Error correction (iv) Disentangling unitary B Λ
C C C C C (i) Topological order C (ii) Decoupling A (iii) Error correction (iv) Disentangling unitary B Λ (v) Cleaning
CLEANABILITY
CLEANABILITY
C C C C C Which properties can be extended to approximate codes? Focus on topological codes; tradeoff bounds
BPT BOUND? Tradeoff bound kd 2 ≤ cn Subspace or commuting projector codes Bravyi, Poulin, Terhal Toric code saturates the bound in 2D Proof: Expansion bound Union bound Counting degrees of freedom
BPT BOUND? kd 2 ≤ cn Expansion Lemma: If is correctable and is correctible, then is correctable. A B A ∪ B Proof: ρ ACD = ω A ⊗ ρ CD (iv) correctable A ⇒ A B C D R ABC AC ( ρ ACD ) = ρ ABCD (iii) correctable ⇒ B Λ Define a map AC ( ω A ⊗ ρ CD ) F ABC ( ρ CD ) = R ABC C AC ( ω A ⊗ ρ CD ) = R ABC Show (iii) F ABC ( ρ CD ) = R ABC AC ( ρ ACD ) = ρ ABCD C
BPT BOUND? kd 2 ≤ cn Union Lemma: If is correctable and is correctible, then is correctable. A B A ∪ B Proof: ∂ A ∂ B (iv) R B ∂ B ∂ B ( ρ Λ \ B ) = ρ Λ correctable A ⇒ B A (iii) correctable ⇒ R B ∂ B ∂ B ( ρ Λ \ A ) = ρ Λ B C Λ R AB ∂ B ∂ AB ( ρ Λ \ AB ) = ρ Λ Clearly,
kd 2 ≤ cn BPT bound: Proof: B 1 B 2 Construct the largest square correctible region by adding ‘onion’ rings. A Largest square region d 2 Λ Decompose the lattice as in Fig 2. and are correctable X Y X Y I ( X : R ) = S ( X ) + S ( R ) − S ( XR ) = 0 Z S ( Y ) + S ( R ) − S ( Y R ) = 0 Sum the two and use subadditivity to get Fig 2 S ( R ) ≤ S ( Z ) Take identity state on code space kd 2 ≤ cn and S ( R ) = k log(2) S ( Z ) ≤ cn/d 2 ⇒
C C C C C Which properties can be extended to approximate codes? Focus on topological codes; tradeoff bounds Take as our basic definition
AQEC? Definition (approximate correctability): There exists a recovery map such that for any code state R AB B ρ ABR ∈ C the following holds: B ( ρ ABR , R AB B ( ρ BR )) ≤ δ B ( ρ , σ ) 2 = 1 − F ( ρ , σ ) Bures distance q √ σρ √ σ ] R F ( ρ , σ ) = tr[ A B Stabilised distance; is a copy of the logical R space.
AQEC? Definition (local approximate correctability): There exists a recovery map such that for any code state R AB B ρ ABCR ∈ C the following holds: B ( ρ ABCR , R AB B ( ρ BCR )) ≤ δ C state can be recovered without modifying C R ` A B
EQUIVALENT FORMULATIONS Definition (information-disturbance tradeoff): ρ ABCR B ( ω A ⊗ ρ CR , ρ ACR ) = inf ρ ABCR B ( R AB B ( ρ BCR , ρ ABCR ) inf sup sup ω A R AB B ⇢ ABCR B ( ω A ⊗ ρ CR , ρ ACR ) δ ` ( A ) := inf sup ! A ρ ABCR C is in the code space R ω A is some fixed state on A ` A B ρ ABCR is in the code space
C C C C C Which properties can be extended to approximate codes? (iii) <=> (iv)
EQUIVALENT FORMULATIONS Definition (information-disturbance tradeoff): ρ ABCR B ( ω A ⊗ ρ CR , ρ ACR ) = inf ρ ABCR B ( R AB B ( ρ BCR ) , ρ ABCR ) inf sup sup ω A R AB B C ` R A ⇢ ABCR B ( ω A ⊗ ρ CR , ρ ACR ) δ ` ( A ) := inf sup ! A B Definition (decoupling): 1 9 δ ` ( A ) 2 ≤ ⇢ ABCR B ( ρ ACR , ρ A ⊗ ρ CR ) ≤ 2 δ ` ( A ) sup
C C C C C Which properties can be extended to approximate codes? (iii) <=> (iv) (iii) <=> (ii) but with different error order
CLEANABILITY Error correction cleanability: ⇒ If is locally correctable: B ( R AB B ( ρ BCR ) , ρ ABCR ) ≤ δ A V BC = ( R AB Then for any logical unitary , the pull-back U ABC B ) ∗ ( U ABC ) satisfies √ || ( U ABC − V BC ) Π || ≤ 4 δ Error correction cleanability: ⇐ || ( U ABC − V BC ) Π || ≤ δ If for any there exists a on s.t. || V B || ≤ 1 U AB B || ρ AB − ω A ⊗ ρ R || 1 ≤ 5 δ ω A Then there exists s.t. C R A R A B B ⇒ ⇐
C C C C C Which properties can be extended to approximate codes? (iii) <=> (iv) Topological quantum (iii) <=> (ii) but with different error order order seems to be (iii) <=> (v) but with different error order different! and different locality constraints
APPROXIMATE BPT Tradeoff bound (1 − cn � d log d kd 2 ≤ cn n � ) kd 2 ≤ c 0 n ` 4 becomes Proof: D Approximate expansion bound A B C Need (iv) and (iii) Approximate union bound A B Need locality of recovery C
(1 − cn � d log d n � ) kd 2 ≤ c 0 n ` 4 BPT bound: Proof: B 1 B 2 Construct the largest square correctible region by adding ‘onion’ rings. A Largest square region d 2 Λ Decompose the lattice as in Fig 2. and are correctable X Y X Y I ( X : R ) = S ( X ) + S ( R ) − S ( XR ) = 0 Z S ( Y ) + S ( R ) − S ( Y R ) = 0 Sum the two and use subadditivity to get Fig 2 S ( R ) ≤ S ( Z ) Take identity state on code space kd 2 ≤ cn and S ( R ) = k log(2) S ( Z ) ≤ cn/d 2 ⇒
(1 − cn � d log d n � ) kd 2 ≤ c 0 n ` 4 BPT bound: Proof: B 1 B 2 Construct the largest square correctible region by adding ‘onion’ rings. A Need (iii)=(iv) Largest square region d 2 Λ Decompose the lattice as in Fig 2. and are correctable X Y X Y I ( X : R ) = S ( X ) + S ( R ) − S ( XR ) = 0 Z S ( Y ) + S ( R ) − S ( Y R ) = 0 Continuity of mutual information Sum the two and use subadditivity to get Fig 2 S ( R ) ≤ S ( Z ) Take identity state on code space kd 2 ≤ cn and S ( R ) = k log(2) S ( Z ) ≤ cn/d 2 ⇒
EXAMPLES Perturbations of commuting projector codes ( i ) Follows from the stability of topological order and Lieb-Robinson bounds
EXAMPLES Perturbations of commuting projector codes ( i ) Follows from the stability of topological order and Lieb-Robinson bounds MERA codes ( ii )
MERA CODES “Disentangling” unitary = Isometry Logical space = Physical space
MERA MODEL | ρ s i = W 1 W 2 · · · W s | φ ( s ) i | φ ( s ) i 2 H s “Disentangling” unitary = Isometry The MERA circuit encodes the subspace into as H s H 0 Logical space = C s ⊂ H s Physical space
MERA MODEL Local operators get mapped to local operators!
MERA MODEL h ρ s | O s | σ s i = h ρ s +1 | Φ s +1 ( O s ) | σ s +1 i s is a quantum channel in the Heisenberg picture Φ ( O ) Exponentially fast in n. Φ n ( O ) ≈ 1tr[ ρ O ] O Local operators get mapped to local operators!
AQEC? Definition (information-disturbance tradeoff): ρ ABCR B ( ω A ⊗ ρ CR , ρ ACR ) = inf ρ ABCR B ( R AB B ( ρ BCR ) , ρ ABCR ) inf sup sup ω A R AB B 1 9 δ ` ( A ) 2 ≤ ⇢ ABCR B ( ρ ACR , ρ A ⊗ ρ CR ) ≤ 2 δ ` ( A ) sup ⇢ ABCR B ( ω A ⊗ ρ CR , ρ ACR ) δ ` ( A ) := inf sup ! A √ More familiar distance measure 2 B 2 ( ρ , σ ) ≤ || ρ − σ || 1 ≤ 2 2 B ( ρ , σ ) To show the existence of a good local recovery map, we need to bound: Proof is very similar to showing is small || ρ A ⊗ ρ CR − ρ ACR || 1 decay of correlations
RESULT ✓ | A | ◆ ν / 2 “Disentangling” unitary ||R AB B ( ρ BCR ) − ρ ABCR || 1 ≤ c | AB | Isometry Proof is similar to that for decay of Logical space = correlations in MERA Physical space
PROOF SKETCH || ρ A ⊗ ρ CR − ρ ACR || 1 = sup tr[ O ACR ( ρ A ⊗ ρ CR − ρ ACR )] “Disentangling” unitary O ACR X tr[ O ACR ρ ] = tr[ Φ s ( O ACR ) ρ ( s )] = tr[ Φ s ( O Aj ) ⊗ Φ s ( O CRj ) ρ ( s )] Isometry j Logical space = X tr[1 ⊗ Φ s ( O CRj ) ρ ( s )]tr[ O Aj σ ] ≈ Physical space j
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