1. Linear Quantum Error Correction
Alireza Shabani and Daniel Lidar
arXiv/
Department of Electrical Engineering,
University of Southern California
December 2007
QEC07

2. Linear Quantum Error Correction
A map-based formulation to fault-tolerant quantum
Computation in presence of non-Markovian noise
arXiv/

3. - Any noise process can be modeled as a linear map
Outline:
Why linear quantum error correction?
A linear map representation for dynamics of an open quantum system.
- Any noise process can be modeled as a linear map
Quantum error correcting codes for linear quantum maps.
- Recovery: CP (completely positive) and non-CP
- How to implement a non-CP map?
Implications of these results for fault-tolerant
quantum computation.

4. Why Linear Quantum Error Correction?
Standard QECC theory explicitly uses the setting of CP maps:
- This is how noise is modeled.
- This is how recovery is constructed.
noise
recovery
Current versions of fault-tolerant quantum computation (FTQC) theory are founded
mainly based on the standard quantum coding. Two main features:
1. Computation, noise and error correction processes are all discrete in time.
No fault-tolerant continuous quantum error correction!
2. The assumption that system and bath are in a product state:
These conditions are well satisfied in Markovian regime of decoherence,
but what if noise is not in that regime?

5. Fault-Tolerant Quantum Computation
Map-based formulation:
[Aharanov 96, 99], [Gottesman 98], [Knill, 98, 01, 05], [Preskill 98], …
Markovian noise
Probabilistic error model ?
Non-Markovian noise
Hamiltonian-based formulation:
[Terhal 05], [Aliferis 06], [Aharonov 06], [Novais 07]
Non-Markovian noise is studied in system-bath Hamiltonian level:
short time approximation
Errors are over-counted.

6. “Reduced” Quantum Dynamics
Bath
System
The standard view
Completely Positive (CP) Maps:

7. Linear Map Representation for Open Quantum Systems
Initial system-bath state: or
System state:
Singular value decomposition of
Linear Map:

8. Linear Map Representation for Open Quantum Systems
Arbitrary initial system-bath state:
The second term does not contribute to the state of the system:
Constant
Jordan’s theorem: Any affine map can be equivalently represented by a linear map.
[PRA 71, ]
Linear quantum map: ,

9. Examples of non–CP maps
Periodic evolution:
Inverse of a CP map is almost never CP
(unless it’s unitary)
Inverse of a phase-flip map:
How would we error-correct this map?
opposite sign  non-CP map

10. Quantum Error Correction for Linear Maps
A code space is a subspace of a system + ancilla Hilbert space which its erroneous
information can be recovered by applying a recovery quantum map.
Recovery can be a CP or non-CP map.
CP Recovery:
Theorem 1:
Consider a linear noise map , and associate to it an
“extended” CP map Then any QEC
code and corresponding CP recovery map for are also a QEC code and CP
recovery map for
Good News!
We can use standard quantum codes for correcting non-CP noise map.

11. Quantum Error Correction for Linear Maps
What about a non-CP recovery?
A linear recovery map corrects a linear noise map
over a code space.
Theorem 2:
Sufficient conditions for correctability of a linear map by another map :
i) , ii)

12. How to Implement a Linear Map?
CP map:
Noise
Non-CP map:
Noise
Code qubits and recovery ancilla qubits are initially
correlated, or entangled.

13. Example of non-CP Recovery
No encoding (single qubit):
Data
Ancilla
If , then is recovered with perfect fidelity.
Linear non-CP recovery:

14. Entanglement -Assisted QEC
[4,1,3;1] code:
CP noise
Condition for CP recovery:
code space projectors
but
Non-CP Linear Recovery

15. Implications for Fault-Tolerant Quantum Computation
The assumption is not required.
Computation, noise and error correction processes are all discrete in time.
Unlike the existing Hamiltonian-based formulation of FTQC, there is no need
to consider a first order approximation in time, in order to obtain a discrete
model of the dynamics.

16. Conclusion by a linear map.
Reduced dynamics of an open quantum system can be represented
by a linear map.
Remarkably, every linear noise map can be fixed using quantum
error correcting codes with CP recovery operations.
Non-CP recovery maps can be implemented by creating initial
correlation between the encoding and recovery ancilla qubits.
LQEC equips us with a tool required for a map-based formulation of
fault-tolerant quantum computation in presence of non-Markovian
noise process.
arXiv/