Download ppt

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