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Prabha Mandayam Institute for Quantum Information California Institute of Technology (Work done jointly with David Poulin) Quantum Error Correction 2007 12/21/2007 Background : Standard approach to quantum error correction: ideal notion of correctability - Codespace C is correctable under the action of a (trace-preserving)quantum channel A iff ∃ a trace-preserving quantum channel R such that ∀ ρ ∈ C, R◦ A(ρ) = ρ Condition for perfect error correction (Knill, Laflamme ’97) - Channel A with Kraus operators {Ai } , is correctable on a codespace C with projector P iff ∃ scalars {λij } such that PAi† AjP = λijP - Independent of the Kraus representation; easily checkable once P is given. 12/21/2007 Quantum Error Correction 2007 2 Formulating the problem:• Define a notion of approximate correctability ∃ a TP channel R : F[R◦A(ρ), ρ] ≥ 1 - δ ∀ ρ ∈ C (F: fidelity measure) • Questions: - Characterizing approximate correctability : Can we find an easily checkable condition for approximate error correction? - Robustness of perfect error correction: What happens when the KnillLaflamme condition is only approximately satisfied? • Choice of the fidelity measure F : - Entanglement fidelity - Worst case fidelity: Every state in the codespace is recovered with high fidelity. 12/21/2007 Quantum Error Correction 2007 3 BK condition for approximate correction with high entanglement fidelity : To correct the action of channel A = {Ai } on a codespace C (dimension d ) with projector P, construct the recovery map RBK with Kraus operators : RiBK = { (P/d)1/2 Ai† A(P/d)-1/2 } Barnum, Knill (J Math Phys, 43 2097 [2002 ]) : For any TP recovery channel R, Fent[P/d, RBK ◦ A] ≥ ( Fent [P/d, R ◦ A] )2 ⇒ If Fent [P/d, R ◦ A] ≥ 1 - δ , then Fent[P/d, RBK ◦ A] ≥ 1 – 2δ Easily checkable condition for approximate correctability: A is correctable on codespace C with high entanglement fidelity iff Fent[P/d, RBK ◦ A] ≥ 1 - δ - Recovery is optimal to within a factor of 2. Alternate approaches to find the optimal recovery map : A.Fletcher, P.W.Shor, MZ.Win quant-ph/0606035 (2006) using Semi-Definite Programming. R.L.Kosut, D.A.Lidar quant-ph/0606078 (2006) using Convex Optimization 12/21/2007 Quantum Error Correction 2007 4 Approximate Correction with high worst case fidelity: Attempt to construct approximate codes by relaxing the perfect error correction condition. Recall that for a perfectly correctable channel A on codespace C, there exists a representation in which the Kraus operators map C to mutually orthogonal subspaces in a unitary fashion: ∃ {Ai} such that PAi† AjP = λi δij P H H A1 C A2 A3 A4 A5 A : B ( C) → B (H) 12/21/2007 Quantum Error Correction 2007 5 The Leung et al. 4-qubit code: Example of a code that corrects with high worst case fidelity: Leung et al. 4-qubit code (1997) for the Amplitude Damping Channel . - satisfies the Knill-Laflamme condition approximately. Kraus operators of the channel map C to mutually orthogonal subspaces which are not unitary transforms of the codespace. H H A1 A2 C A3 A4 A5 A : B ( C) → B (H) 12/21/2007 Quantum Error Correction 2007 6 Algebraic condition motivated by the 4-qubit code: If the action of channel A = { Ai , i = 1,2…N } on codespace C is such that ∃ scalars { λi > 0 } and positive operators {Bi } such that the Kraus operators in the canonical representation (Tr[Ai† AjP] = 0 ∀ i ≠ j) satisfy with then, there exists a TP recovery R such that The map R is constructed with Kraus operators { Ri = PUi† } where [ Ui : unitary that approximates the action of Ai ] 12/21/2007 Quantum Error Correction 2007 7 Approximate correctability of a general channel: Algebraic condition stated above characterizes approximately correcting codes for a restricted class of channels. For an arbitrary channel : Codespace gets distorted under the action of different Kraus operators and is mapped to subspaces that are not necessarily mutually orthogonal. ( PAi† AjP ≠ 0 ∀ i ≠ j ) H H A1 C A5 A2 A3 A4 A : B ( C) → B (H) 12/21/2007 Quantum Error Correction 2007 8 A general condition for approximate correctability: If A = { Ai , i = 1,2…N } acts on C such that (i) ∃ real numbers λi > 0 , and positive operators {Bi } satisfying with (ii) For i ≠ j ( M = Spectral norm = Max. singular value of M) then, ∃ TP R such that 12/21/2007 Quantum Error Correction 2007 9 Constructing the optimal recovery channel: Construct partial isometries {Wi } by a Gram-Schmidt like procedure: where - (PWk† WKP)-1 exist on the codespace since Orthogonality: PWi† Wj P = 0 ∀ i ≠ j 12/21/2007 Quantum Error Correction 2007 10 Properties of the recovery map: By constructing {Wi }, we have generated a set of mutually orthogonal subspaces that approximate the action of the channel with high enough fidelity. {Mi = WiPWi† } : positive operators that have disjoint support {Supp(Mi)} : a set of mutually orthogonal subspaces of H Recovery map with Kraus operators Ri = PWi† recovers with high worst case fidelity. 12/21/2007 Quantum Error Correction 2007 11 To conclude... We have outlined a sufficient condition for approximate correctability that characterizes a general class of approximately correcting codes. In the process, we have shown that the perfect error correction condition is robust. Can we use this condition to obtain trade-offs between code lengths and fidelity for non-Pauli error channels? Is this condition also necessary for approximate correctability? This would imply that every approximately correcting code can be visualized as a perturbation of the Knill-Laflamme condition! Is the Barnum-Knill recovery close to optimal for worst case fidelity as well? This would lead to a characterization of approximate codes, independent of the KnillLaflamme condition. 12/21/2007 Quantum Error Correction 2007 12