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Linear Quantum Error Correction
Alireza Shabani and Daniel Lidar
arXiv/0708.1953
Department of Electrical Engineering, University
of Southern California
December 2007
QEC07
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Linear Quantum Error Correction
A map-based formulation to fault-tolerant quantum
Computation in presence of non-Markovian noise
arXiv/0708.1953
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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.

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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.

recovery
noise

• 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

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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.
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Reduced Quantum Dynamics
The standard view Completely Positive (CP) Maps
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Linear Map Representation for Open Quantum
Systems
Initial system-bath state
or
System state
Singular value decomposition of
Linear Map
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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
Jordans theorem Any affine map can be
equivalently represented by a linear map.
PRA 71, 034101
Linear quantum map
,
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Examples of nonCP maps
Periodic evolution
Inverse of a CP map is almost never CP (unless
its unitary)
Inverse of a phase-flip map
How would we error-correct this map?
opposite sign ? non-CP map
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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.
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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)
.
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How to Implement a Linear Map?
Noise
CP map
Non-CP map
Noise
Code qubits and recovery ancilla qubits are
initially correlated, or entangled.
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Example of non-CP Recovery
No encoding (single qubit)
Data
Ancilla
If , then
is recovered with perfect fidelity.
Linear non-CP recovery
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Entanglement -Assisted QEC
4,1,31 code
CP noise
Condition for CP recovery
code space projectors
but
Non-CP Linear Recovery
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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.

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Conclusion

• 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/0708.1953