Approximate Operator Quantum Error Correction Prabha Mandayam - - PowerPoint PPT Presentation

▶
approximate operator quantum error correction
SMART_READER_LITE
LIVE PREVIEW

Approximate Operator Quantum Error Correction Prabha Mandayam - - PowerPoint PPT Presentation

Sep 28, 2022 249 likes •752 views

Approximate Operator Quantum Error Correction Prabha Mandayam Institute of Mathematical Sciences, Chennai Joint work with Hui Khoon Ng (CQT, Singapore) Reference: Phys.Rev.A , 81 , 062342 (2010) P. Mandayam and H.K. Ng (in prep.) Prabha

slide-1
SLIDE 1

Approximate Operator Quantum Error Correction

Prabha Mandayam Institute of Mathematical Sciences, Chennai Joint work with Hui Khoon Ng (CQT, Singapore)

Reference: Phys.Rev.A , 81, 062342 (2010)

  • P. Mandayam and H.K. Ng (in prep.)

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 1 / 14

slide-2
SLIDE 2

Talk Outline

The Transpose Channel and its role in perfect QEC

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 2 / 14

slide-3
SLIDE 3

Talk Outline

The Transpose Channel and its role in perfect QEC A simple, analytical approach to approximate (subspace) QEC

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 2 / 14

slide-4
SLIDE 4

Talk Outline

The Transpose Channel and its role in perfect QEC A simple, analytical approach to approximate (subspace) QEC Finding good approximate codes using the transpose channel

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 2 / 14

slide-5
SLIDE 5

Talk Outline

The Transpose Channel and its role in perfect QEC A simple, analytical approach to approximate (subspace) QEC Finding good approximate codes using the transpose channel From subspace to subsystem codes: Approximate Operator Quantum Error Correction

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 2 / 14

slide-6
SLIDE 6

Talk Outline

The Transpose Channel and its role in perfect QEC A simple, analytical approach to approximate (subspace) QEC Finding good approximate codes using the transpose channel From subspace to subsystem codes: Approximate Operator Quantum Error Correction Conclusion: A unified framework for approximate QEC via the Transpose Channel

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 2 / 14

slide-7
SLIDE 7

The Transpose Channel

Definition For a given noise channel E ∼ {Ei}, and a code C (with projector P), Transpose Channel : RT ∼ {Ri}N

i=1 , Ri ≡ PE† i E(P)−1/2

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 3 / 14

slide-8
SLIDE 8

The Transpose Channel

Definition For a given noise channel E ∼ {Ei}, and a code C (with projector P), Transpose Channel : RT ∼ {Ri}N

i=1 , Ri ≡ PE† i E(P)−1/2

Composed of three CP maps: RT = P ◦ E† ◦ N –

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 3 / 14

slide-9
SLIDE 9

The Transpose Channel

Definition For a given noise channel E ∼ {Ei}, and a code C (with projector P), Transpose Channel : RT ∼ {Ri}N

i=1 , Ri ≡ PE† i E(P)−1/2

Composed of three CP maps: RT = P ◦ E† ◦ N –

E † is the adjoint channel P is the projection onto C N is the normalization map N(·) = E(P)−1/2(·)E(P)−1/2

⇒ RT Independent of the Kraus representation.

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 3 / 14

slide-10
SLIDE 10

The Transpose Channel

Definition For a given noise channel E ∼ {Ei}, and a code C (with projector P), Transpose Channel : RT ∼ {Ri}N

i=1 , Ri ≡ PE† i E(P)−1/2

Composed of three CP maps: RT = P ◦ E† ◦ N –

E † is the adjoint channel P is the projection onto C N is the normalization map N(·) = E(P)−1/2(·)E(P)−1/2

⇒ RT Independent of the Kraus representation. RT is trace-preserving on the support of E(P).

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 3 / 14

slide-11
SLIDE 11

The Transpose Channel

Definition For a given noise channel E ∼ {Ei}, and a code C (with projector P), Transpose Channel : RT ∼ {Ri}N

i=1 , Ri ≡ PE† i E(P)−1/2

Composed of three CP maps: RT = P ◦ E† ◦ N –

E † is the adjoint channel P is the projection onto C N is the normalization map N(·) = E(P)−1/2(·)E(P)−1/2

⇒ RT Independent of the Kraus representation. RT is trace-preserving on the support of E(P). If E is perfectly correctable on C, RT is the recovery map that recovers the information encoded in C.

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 3 / 14

slide-12
SLIDE 12

Role of RT in perfect QEC

Lemma (Alternative form of perfect QEC conditions1) A channel E ∼ {Ei}N

i=1 satisfies the Knill-Laflamme conditions for a code C iff

PE†

i E(P)−1/2EjP = βijP, ∀i, j = 1, ..., N

for some positive matrix β.

1H.K. Ng and P.Mandayam, PRA, 81 (2010).

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 4 / 14

slide-13
SLIDE 13

Role of RT in perfect QEC

Lemma (Alternative form of perfect QEC conditions1) A channel E ∼ {Ei}N

i=1 satisfies the Knill-Laflamme conditions for a code C iff

PE†

i E(P)−1/2EjP = βijP, ∀i, j = 1, ..., N

for some positive matrix β. The LHS consists of the Kraus operators of RT ◦ E.

1H.K. Ng and P.Mandayam, PRA, 81 (2010).

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 4 / 14

slide-14
SLIDE 14

Role of RT in perfect QEC

Lemma (Alternative form of perfect QEC conditions1) A channel E ∼ {Ei}N

i=1 satisfies the Knill-Laflamme conditions for a code C iff

PE†

i E(P)−1/2EjP = βijP, ∀i, j = 1, ..., N

for some positive matrix β. The LHS consists of the Kraus operators of RT ◦ E. E is perfectly correctable on a code space C iff the action of RT ◦ E is a simple projection onto C. The recovery operation is manifestly clear!

1H.K. Ng and P.Mandayam, PRA, 81 (2010).

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 4 / 14

slide-15
SLIDE 15

Role of RT in perfect QEC

Lemma (Alternative form of perfect QEC conditions1) A channel E ∼ {Ei}N

i=1 satisfies the Knill-Laflamme conditions for a code C iff

PE†

i E(P)−1/2EjP = βijP, ∀i, j = 1, ..., N

for some positive matrix β. The LHS consists of the Kraus operators of RT ◦ E. E is perfectly correctable on a code space C iff the action of RT ◦ E is a simple projection onto C. The recovery operation is manifestly clear! Can be perturbed to obtain sufficient conditions for approximate QEC. Size of the perturbation is directly related to the fidelity due to RT.

1H.K. Ng and P.Mandayam, PRA, 81 (2010).

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 4 / 14

slide-16
SLIDE 16

Approximate Quantum Error Correction

Worst-case fidelity: For a codespace C, under the action of the noise channel E and recovery R, F 2

min[C, R ◦ E] = min |ψ ∈C F 2[|ψ, R ◦ E(|ψψ|)] ,

  • F 2[|ψ, σ] ≡ ψ|σ|ψ
  • .

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 5 / 14

slide-17
SLIDE 17

Approximate Quantum Error Correction

Worst-case fidelity: For a codespace C, under the action of the noise channel E and recovery R, F 2

min[C, R ◦ E] = min |ψ ∈C F 2[|ψ, R ◦ E(|ψψ|)] ,

  • F 2[|ψ, σ] ≡ ψ|σ|ψ
  • .

Fidelity-loss : ηR = 1 − min|ψ∈C F 2[|ψ, (R ◦ E)(|ψψ|)].

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 5 / 14

slide-18
SLIDE 18

Approximate Quantum Error Correction

Worst-case fidelity: For a codespace C, under the action of the noise channel E and recovery R, F 2

min[C, R ◦ E] = min |ψ ∈C F 2[|ψ, R ◦ E(|ψψ|)] ,

  • F 2[|ψ, σ] ≡ ψ|σ|ψ
  • .

Fidelity-loss : ηR = 1 − min|ψ∈C F 2[|ψ, (R ◦ E)(|ψψ|)]. Channel E is approximately correctable on code space C if ∃ a physical (CPTP) map R such that F 2

min[C, R ◦ E] ≈ 1.

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 5 / 14

slide-19
SLIDE 19

Approximate Quantum Error Correction

Worst-case fidelity: For a codespace C, under the action of the noise channel E and recovery R, F 2

min[C, R ◦ E] = min |ψ ∈C F 2[|ψ, R ◦ E(|ψψ|)] ,

  • F 2[|ψ, σ] ≡ ψ|σ|ψ
  • .

Fidelity-loss : ηR = 1 − min|ψ∈C F 2[|ψ, (R ◦ E)(|ψψ|)]. Channel E is approximately correctable on code space C if ∃ a physical (CPTP) map R such that F 2

min[C, R ◦ E] ≈ 1.

Finding the optimal recovery for worst-case fidelity is not a convex-optimization problem!

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 5 / 14

slide-20
SLIDE 20

Approximate Quantum Error Correction

Worst-case fidelity: For a codespace C, under the action of the noise channel E and recovery R, F 2

min[C, R ◦ E] = min |ψ ∈C F 2[|ψ, R ◦ E(|ψψ|)] ,

  • F 2[|ψ, σ] ≡ ψ|σ|ψ
  • .

Fidelity-loss : ηR = 1 − min|ψ∈C F 2[|ψ, (R ◦ E)(|ψψ|)]. Channel E is approximately correctable on code space C if ∃ a physical (CPTP) map R such that F 2

min[C, R ◦ E] ≈ 1.

Finding the optimal recovery for worst-case fidelity is not a convex-optimization problem!

Optimizing for entanglement fidelity is tractable via SDP, convex-optimization2 . Channel similar to RT is close to optimal for entanglement fidelity3. Analytically, close-to-optimal recovery maps have been constructed for worst-case entanglement fidelity4.

2Fletcher et al. PRA, 75, 021338 (2007), Kosut et al. PRL, 100, 020502 (2008) 3Barnum and Knill, JMP, 43, 2097 (2002) 4Beny and Oreshkov, PRL, 104, 120501 (2010)

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 5 / 14

slide-21
SLIDE 21

A simple, analytical approach to approximate QEC5

Theorem (Near-optimality of the transpose channel) Given a code space C of dimension d and optimal recovery map Rop with optimal fidelity loss ηop, such that F 2[|ψ, (Rop ◦ E)(|ψψ|)] ≥ 1 − ηop, then, F 2[|ψ, (RT ◦ E)(|ψψ|)] ≥ 1 − (d + 1)ηop for any |ψ ∈ C.

5H.K. Ng and P. Mandayam, PRA, 81, 062342 (2010)

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 6 / 14

slide-22
SLIDE 22

A simple, analytical approach to approximate QEC5

Theorem (Near-optimality of the transpose channel) Given a code space C of dimension d and optimal recovery map Rop with optimal fidelity loss ηop, such that F 2[|ψ, (Rop ◦ E)(|ψψ|)] ≥ 1 − ηop, then, F 2[|ψ, (RT ◦ E)(|ψψ|)] ≥ 1 − (d + 1)ηop for any |ψ ∈ C. Corollary: ηop ≤ ηT ≤ ηop[(d + 1) + O(ηop)].

5H.K. Ng and P. Mandayam, PRA, 81, 062342 (2010)

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 6 / 14

slide-23
SLIDE 23

A simple, analytical approach to approximate QEC5

Theorem (Near-optimality of the transpose channel) Given a code space C of dimension d and optimal recovery map Rop with optimal fidelity loss ηop, such that F 2[|ψ, (Rop ◦ E)(|ψψ|)] ≥ 1 − ηop, then, F 2[|ψ, (RT ◦ E)(|ψψ|)] ≥ 1 − (d + 1)ηop for any |ψ ∈ C. Corollary: ηop ≤ ηT ≤ ηop[(d + 1) + O(ηop)]. For any noise-channel E, RT does not perform much worse than Rop - at most adds a factor of (d + 1) to the fidelity-loss.

5H.K. Ng and P. Mandayam, PRA, 81, 062342 (2010)

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 6 / 14

slide-24
SLIDE 24

A simple, analytical approach to approximate QEC5

Theorem (Near-optimality of the transpose channel) Given a code space C of dimension d and optimal recovery map Rop with optimal fidelity loss ηop, such that F 2[|ψ, (Rop ◦ E)(|ψψ|)] ≥ 1 − ηop, then, F 2[|ψ, (RT ◦ E)(|ψψ|)] ≥ 1 − (d + 1)ηop for any |ψ ∈ C. Corollary: ηop ≤ ηT ≤ ηop[(d + 1) + O(ηop)]. For any noise-channel E, RT does not perform much worse than Rop - at most adds a factor of (d + 1) to the fidelity-loss. When ηop = 0, ηT = ηop implying that RT is indeed the optimal recovery map for perfect QEC!

5H.K. Ng and P. Mandayam, PRA, 81, 062342 (2010)

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 6 / 14

slide-25
SLIDE 25

A simple, analytical approach to approximate QEC5

Theorem (Near-optimality of the transpose channel) Given a code space C of dimension d and optimal recovery map Rop with optimal fidelity loss ηop, such that F 2[|ψ, (Rop ◦ E)(|ψψ|)] ≥ 1 − ηop, then, F 2[|ψ, (RT ◦ E)(|ψψ|)] ≥ 1 − (d + 1)ηop for any |ψ ∈ C. Corollary: ηop ≤ ηT ≤ ηop[(d + 1) + O(ηop)]. For any noise-channel E, RT does not perform much worse than Rop - at most adds a factor of (d + 1) to the fidelity-loss. When ηop = 0, ηT = ηop implying that RT is indeed the optimal recovery map for perfect QEC! Sufficient and necessary conditions for approximate subspace QEC .

5H.K. Ng and P. Mandayam, PRA, 81, 062342 (2010)

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 6 / 14

slide-26
SLIDE 26

Examples: Codes for the Amplitude Damping channel

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.7 0.75 0.8 0.85 0.9 0.95 1 Damping parameter (γ) Worst−case fidelity(F2) No error−correction [4,1] code, Leung recovery [4,1] code, RT recovery [4,1] code, SDP recovery [[5,1,3]] code

[4, 1] code: Leung et al. PRA 56, 2567 (1997)

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 7 / 14

slide-27
SLIDE 27

Examples

Random 2-,3-,4-qubit AQEC codes for the Amplitude Damping channel

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.75 0.8 0.85 0.9 0.95 1 Damping parameter (γ) Worst−case fidelity (F2) [[5,1,3]] code random 4−qubit code, RT recovery random 3−qubit code, RT recovery random 2−qubit code, RT recovery random 4−qubit code, Id reocvery Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 8 / 14

slide-28
SLIDE 28

Beyond Subspace Codes

Operator Quantum Error Correction6 A fixed partition of the physical (system) Hilbert space: H = HA ⊗ HB ⊕ K. Information is encoded only in subsystem HA.

6Kribs et al., PRL, 94, 180501, (2005)

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 9 / 14

slide-29
SLIDE 29

Beyond Subspace Codes

Operator Quantum Error Correction6 A fixed partition of the physical (system) Hilbert space: H = HA ⊗ HB ⊕ K. Information is encoded only in subsystem HA. Noise channel E is perfectly correctable on subsystem HA, if there exists a physical map R, such that (R ◦ E)(ρA ⊗ ρB) = ρA ⊗ σB, ∀ ρA ∈ HA, ρB ∈ HB.

6Kribs et al., PRL, 94, 180501, (2005)

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 9 / 14

slide-30
SLIDE 30

Beyond Subspace Codes

Operator Quantum Error Correction6 A fixed partition of the physical (system) Hilbert space: H = HA ⊗ HB ⊕ K. Information is encoded only in subsystem HA. Noise channel E is perfectly correctable on subsystem HA, if there exists a physical map R, such that (R ◦ E)(ρA ⊗ ρB) = ρA ⊗ σB, ∀ ρA ∈ HA, ρB ∈ HB. Theorem (Alternate conditions for perfect OQEC7) A code C ≡ HA ⊗ HB with projector P ≡ PA ⊗ PB is perfectly correctable on subsystem HA under E ∼ {Ei} iff ∃ operators B′

ij on HB such that,

PE†

i E(P)−1/2EjP = PA ⊗ B′ ij,

∀i, j,

6Kribs et al., PRL, 94, 180501, (2005)

  • 7P. Mandayam and H.K. Ng, in preparation.

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 9 / 14

slide-31
SLIDE 31

Approximate Operator Quantum Error Correction

RT is the appropriate recovery map: Subsystem HA is perfectly correctable under E iff RT ◦ E is the identity operation on HA.

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 10 / 14

slide-32
SLIDE 32

Approximate Operator Quantum Error Correction

RT is the appropriate recovery map: Subsystem HA is perfectly correctable under E iff RT ◦ E is the identity operation on HA. This gives a direct route to approximate OQEC.

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 10 / 14

slide-33
SLIDE 33

Approximate Operator Quantum Error Correction

RT is the appropriate recovery map: Subsystem HA is perfectly correctable under E iff RT ◦ E is the identity operation on HA. This gives a direct route to approximate OQEC. Theorem (Sufficient Conditions for Approximate OQEC) A subsystem code C is ǫ-correctable on HA under the action of E if ∃ operators B′

ij on B such that

PE†

i E(P)−1/2EjP = PA ⊗ B′ ij + ∆ij,

∀i, j, where ∆ij is an operator on HA ⊗ HB such that

i,j ∆ijtr ≤ ǫ.

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 10 / 14

slide-34
SLIDE 34

Approximate Operator Quantum Error Correction

RT is the appropriate recovery map: Subsystem HA is perfectly correctable under E iff RT ◦ E is the identity operation on HA. This gives a direct route to approximate OQEC. Theorem (Sufficient Conditions for Approximate OQEC) A subsystem code C is ǫ-correctable on HA under the action of E if ∃ operators B′

ij on B such that

PE†

i E(P)−1/2EjP = PA ⊗ B′ ij + ∆ij,

∀i, j, where ∆ij is an operator on HA ⊗ HB such that

i,j ∆ijtr ≤ ǫ.

  • i,j ∆ijtr is directly related to the worst-case fidelity using the transpose

channel : min|ψA∈HA F 2[|ψA, trB{(RT ◦ E)(|ψAψ| ⊗ ρB)}] .

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 10 / 14

slide-35
SLIDE 35

Transpose channel recovery for approximate OQEC?

We prove RT is near-optimal when E = EA ⊗ EB, and HA is approximately correctable under EA

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 11 / 14

slide-36
SLIDE 36

Transpose channel recovery for approximate OQEC?

We prove RT is near-optimal when E = EA ⊗ EB, and HA is approximately correctable under EA System HB starts out in the maximally mixed state.

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 11 / 14

slide-37
SLIDE 37

Transpose channel recovery for approximate OQEC?

We prove RT is near-optimal when E = EA ⊗ EB, and HA is approximately correctable under EA System HB starts out in the maximally mixed state. Theorem Let σmax

B

be the maximally-mixed state on the noisy subsystem HB. Then, F2 [|ψA, trB {(RT ◦ E)(|ψAψ| ⊗ σmax

B

)}] ≥ 1 − (dA + 1)ηop, for all |ψA ∈ HA.

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 11 / 14

slide-38
SLIDE 38

Transpose channel recovery for approximate OQEC?

We prove RT is near-optimal when E = EA ⊗ EB, and HA is approximately correctable under EA System HB starts out in the maximally mixed state. Theorem Let σmax

B

be the maximally-mixed state on the noisy subsystem HB. Then, F2 [|ψA, trB {(RT ◦ E)(|ψAψ| ⊗ σmax

B

)}] ≥ 1 − (dA + 1)ηop, for all |ψA ∈ HA. Special case : For qubit codes in an independent error model, where individual error operators on HB are scaled Paulis, F2 [|ψA, trB {(RT ◦ E)(|ψAψ| ⊗ ρB)}] is independent of ρB.

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 11 / 14

slide-39
SLIDE 39

Transpose channel recovery for approximate OQEC

RT is also provably near-optimal when E destroys most of the information in subsystem HB.

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 12 / 14

slide-40
SLIDE 40

Transpose channel recovery for approximate OQEC

RT is also provably near-optimal when E destroys most of the information in subsystem HB. Theorem For any pure state |ψA ∈ HA and any state ρB ∈ HB F2 [|ψA, trB {(RT ◦ E)(|ψAψ| ⊗ ρB)}] ≥ 1 − (dA + 1)ηop − 3δ, where 0 ≤ δ < 1 is the smallest possible value such that E [|ψAψ| ⊗ (ρB − σmax

B

)]tr ≤ δρB − σmax

B

tr

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 12 / 14

slide-41
SLIDE 41

Transpose channel recovery for approximate OQEC

RT is also provably near-optimal when E destroys most of the information in subsystem HB. Theorem For any pure state |ψA ∈ HA and any state ρB ∈ HB F2 [|ψA, trB {(RT ◦ E)(|ψAψ| ⊗ ρB)}] ≥ 1 − (dA + 1)ηop − 3δ, where 0 ≤ δ < 1 is the smallest possible value such that E [|ψAψ| ⊗ (ρB − σmax

B

)]tr ≤ δρB − σmax

B

tr For example, if E is strictly-contractive on HB with δ << 1, the transpose channel can recover with high fidelity for any state on HB. If E nearly destroys all information in HB, it is close to being completely disentangling on HA and HB.

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 12 / 14

slide-42
SLIDE 42

Summary

We have outlined a simple unifying approach to approximate error correction, based on the transpose channel, RT.

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 13 / 14

slide-43
SLIDE 43

Summary

We have outlined a simple unifying approach to approximate error correction, based on the transpose channel, RT. Understanding the crucial role played by RT in perfect OQEC, leads to a set

  • f sufficient conditions for approx. OQEC.

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 13 / 14

slide-44
SLIDE 44

Summary

We have outlined a simple unifying approach to approximate error correction, based on the transpose channel, RT. Understanding the crucial role played by RT in perfect OQEC, leads to a set

  • f sufficient conditions for approx. OQEC.

Provides a simple algorithm to check if a given code is approximately correctable – enables us to construct good AQEC codes of shorter lengths. Compares favorably with previous approaches to AQEC which involve numerical optimization.

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 13 / 14

slide-45
SLIDE 45

Summary

We have outlined a simple unifying approach to approximate error correction, based on the transpose channel, RT. Understanding the crucial role played by RT in perfect OQEC, leads to a set

  • f sufficient conditions for approx. OQEC.

Provides a simple algorithm to check if a given code is approximately correctable – enables us to construct good AQEC codes of shorter lengths. Compares favorably with previous approaches to AQEC which involve numerical optimization. Near-optimality of RT: Established for subspace codes.

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 13 / 14

slide-46
SLIDE 46

Summary

We have outlined a simple unifying approach to approximate error correction, based on the transpose channel, RT. Understanding the crucial role played by RT in perfect OQEC, leads to a set

  • f sufficient conditions for approx. OQEC.

Provides a simple algorithm to check if a given code is approximately correctable – enables us to construct good AQEC codes of shorter lengths. Compares favorably with previous approaches to AQEC which involve numerical optimization. Near-optimality of RT: Established for subspace codes. Established for subsystem codes, when:

Noisy subsystem HB starts in a maximally mixed state; Qubit codes, independent error model with scaled Pauli error operators on HB; Noise channel nearly destroys all information in the noisy subsystem HB.

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 13 / 14

slide-47
SLIDE 47

Some Open Questions...

Closing the gap between our sufficient and necessary conditions - is it possible to avoid the factor of codespace dimension (d or dA) in F 2

min?

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 14 / 14

slide-48
SLIDE 48

Some Open Questions...

Closing the gap between our sufficient and necessary conditions - is it possible to avoid the factor of codespace dimension (d or dA) in F 2

min?

Implementation of RT efficiently using POVMs and gates. RT ∼ {Ri ≡ PE†

i E(P)−1/2} → Ri = Ui

  • R†

i Ri Mi

(Generalized measurement followed by a conditional unitary.)

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 14 / 14

slide-49
SLIDE 49

Some Open Questions...

Closing the gap between our sufficient and necessary conditions - is it possible to avoid the factor of codespace dimension (d or dA) in F 2

min?

Implementation of RT efficiently using POVMs and gates. RT ∼ {Ri ≡ PE†

i E(P)−1/2} → Ri = Ui

  • R†

i Ri Mi

(Generalized measurement followed by a conditional unitary.) Approximate error correction as the first step of encoding in a fault-tolerant architecture?

Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 14 / 14