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
Talk Outline The Transpose Channel and its role in perfect QEC Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 2 / 14
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
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
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
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
The Transpose Channel Definition For a given noise channel E ∼ { E i } , and a code C (with projector P ), i =1 , R i ≡ PE † Transpose Channel : R T ∼ { R i } N i E ( P ) − 1 / 2 Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 3 / 14
The Transpose Channel Definition For a given noise channel E ∼ { E i } , and a code C (with projector P ), i =1 , R i ≡ PE † Transpose Channel : R T ∼ { R i } N i E ( P ) − 1 / 2 Composed of three CP maps: R T = P ◦ E † ◦ N – Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 3 / 14
The Transpose Channel Definition For a given noise channel E ∼ { E i } , and a code C (with projector P ), i =1 , R i ≡ PE † Transpose Channel : R T ∼ { R i } N i E ( P ) − 1 / 2 Composed of three CP maps: R T = 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 ⇒ R T Independent of the Kraus representation. Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 3 / 14
The Transpose Channel Definition For a given noise channel E ∼ { E i } , and a code C (with projector P ), i =1 , R i ≡ PE † Transpose Channel : R T ∼ { R i } N i E ( P ) − 1 / 2 Composed of three CP maps: R T = 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 ⇒ R T Independent of the Kraus representation. R T is trace-preserving on the support of E ( P ) . Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 3 / 14
The Transpose Channel Definition For a given noise channel E ∼ { E i } , and a code C (with projector P ), i =1 , R i ≡ PE † Transpose Channel : R T ∼ { R i } N i E ( P ) − 1 / 2 Composed of three CP maps: R T = 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 ⇒ R T Independent of the Kraus representation. R T is trace-preserving on the support of E ( P ) . If E is perfectly correctable on C , R T is the recovery map that recovers the information encoded in C . Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 3 / 14
Role of R T in perfect QEC Lemma (Alternative form of perfect QEC conditions 1 ) A channel E ∼ { E i } N i =1 satisfies the Knill-Laflamme conditions for a code C iff PE † i E ( P ) − 1 / 2 E j P = β ij P, ∀ i, j = 1 , ..., N for some positive matrix β . 1 H.K. Ng and P.Mandayam, PRA, 81 (2010). Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 4 / 14
Role of R T in perfect QEC Lemma (Alternative form of perfect QEC conditions 1 ) A channel E ∼ { E i } N i =1 satisfies the Knill-Laflamme conditions for a code C iff PE † i E ( P ) − 1 / 2 E j P = β ij P, ∀ i, j = 1 , ..., N for some positive matrix β . The LHS consists of the Kraus operators of R T ◦ E . 1 H.K. Ng and P.Mandayam, PRA, 81 (2010). Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 4 / 14
Role of R T in perfect QEC Lemma (Alternative form of perfect QEC conditions 1 ) A channel E ∼ { E i } N i =1 satisfies the Knill-Laflamme conditions for a code C iff PE † i E ( P ) − 1 / 2 E j P = β ij P, ∀ i, j = 1 , ..., N for some positive matrix β . The LHS consists of the Kraus operators of R T ◦ E . E is perfectly correctable on a code space C iff the action of R T ◦ E is a simple projection onto C . The recovery operation is manifestly clear! 1 H.K. Ng and P.Mandayam, PRA, 81 (2010). Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 4 / 14
Role of R T in perfect QEC Lemma (Alternative form of perfect QEC conditions 1 ) A channel E ∼ { E i } N i =1 satisfies the Knill-Laflamme conditions for a code C iff PE † i E ( P ) − 1 / 2 E j P = β ij P, ∀ i, j = 1 , ..., N for some positive matrix β . The LHS consists of the Kraus operators of R T ◦ E . E is perfectly correctable on a code space C iff the action of R T ◦ 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 R T . 1 H.K. Ng and P.Mandayam, PRA, 81 (2010). Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 4 / 14
Approximate Quantum Error Correction Worst-case fidelity: For a codespace C , under the action of the noise channel E and recovery R , � � F 2 | ψ � ∈C F 2 [ | ψ � , R ◦ E ( | ψ �� ψ | )] , F 2 [ | ψ � , σ ] ≡ � ψ | σ | ψ � min [ C , R ◦ E ] = min . Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 5 / 14
Approximate Quantum Error Correction Worst-case fidelity: For a codespace C , under the action of the noise channel E and recovery R , � � F 2 | ψ � ∈C F 2 [ | ψ � , R ◦ E ( | ψ �� ψ | )] , F 2 [ | ψ � , σ ] ≡ � ψ | σ | ψ � min [ C , R ◦ E ] = min . Fidelity-loss : η R = 1 − min | ψ �∈C F 2 [ | ψ � , ( R ◦ E )( | ψ �� ψ | )] . Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 5 / 14
Approximate Quantum Error Correction Worst-case fidelity: For a codespace C , under the action of the noise channel E and recovery R , � � F 2 | ψ � ∈C F 2 [ | ψ � , R ◦ E ( | ψ �� ψ | )] , F 2 [ | ψ � , σ ] ≡ � ψ | σ | ψ � min [ C , R ◦ E ] = min . Fidelity-loss : η R = 1 − min | ψ �∈C F 2 [ | ψ � , ( R ◦ E )( | ψ �� ψ | )] . 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
Approximate Quantum Error Correction Worst-case fidelity: For a codespace C , under the action of the noise channel E and recovery R , � � F 2 | ψ � ∈C F 2 [ | ψ � , R ◦ E ( | ψ �� ψ | )] , F 2 [ | ψ � , σ ] ≡ � ψ | σ | ψ � min [ C , R ◦ E ] = min . Fidelity-loss : η R = 1 − min | ψ �∈C F 2 [ | ψ � , ( R ◦ E )( | ψ �� ψ | )] . 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
Approximate Quantum Error Correction Worst-case fidelity: For a codespace C , under the action of the noise channel E and recovery R , � � F 2 | ψ � ∈C F 2 [ | ψ � , R ◦ E ( | ψ �� ψ | )] , F 2 [ | ψ � , σ ] ≡ � ψ | σ | ψ � min [ C , R ◦ E ] = min . Fidelity-loss : η R = 1 − min | ψ �∈C F 2 [ | ψ � , ( R ◦ E )( | ψ �� ψ | )] . 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-optimization 2 . Channel similar to R T is close to optimal for entanglement fidelity 3 . Analytically, close-to-optimal recovery maps have been constructed for worst-case entanglement fidelity 4 . 2 Fletcher et al. PRA, 75 , 021338 (2007), Kosut et al. PRL, 100 , 020502 (2008) 3 Barnum and Knill, JMP, 43 , 2097 (2002) 4 Beny and Oreshkov, PRL, 104 , 120501 (2010) Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 5 / 14
A simple, analytical approach to approximate QEC 5 Theorem (Near-optimality of the transpose channel) Given a code space C of dimension d and optimal recovery map R op with optimal fidelity loss η op , such that F 2 [ | ψ � , ( R op ◦ E )( | ψ �� ψ | )] ≥ 1 − η op , then, F 2 [ | ψ � , ( R T ◦ E )( | ψ �� ψ | )] ≥ 1 − ( d + 1) η op for any | ψ � ∈ C . 5 H.K. Ng and P. Mandayam, PRA, 81 , 062342 (2010) Prabha Mandayam (IMSc) QEC’11 7 Dec 2011 6 / 14
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