Linear Quantum Error Correction Alireza Shabani and Daniel Lidar arXiv/0708.1953 Department of Electrical Engineering, University of Southern California December 2007 QEC07

Linear Quantum Error Correction A map-based formulation to fault-tolerant quantum Computation in presence of non-Markovian noise arXiv/0708.1953

- 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.

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?

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.

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

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

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, 034101] Linear quantum map: ,

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

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.

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) .

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

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

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

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.

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/0708.1953