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Channel-adaptedQuantum Error Recovery (QER)

Encoder

Channel

Recovery

- QEC scheme specifies Encoder and Recovery
- Generic methods do this independently of channel
- Channel-adapted QER selects given encoder and channel, seeking to maximize entanglement fidelity
- Essentially, maximize probability of correct transmission
- Optimization problem can be solved exactly using a semidefinite program (SDP)

Quantum Operations

- What are valid choices for the recovery ?
- Quantum operations are completely positive and trace preserving (CPTP)
- Standard expression for operation uses Kraus form
- Many sets of operators correspond to the same mapping
- Less common Choi matrix is more convenient
- Use Jamiolkowski isomorphism
- Operation is uniquely specified by a positive operator
- Constraints:
- Entanglement fidelity has simple form with Choi matrix

Optimum Channel-adapted QER

- The optimal recovery operation has a nice form
- Linear objective function:
- Linear equality constraint:
- Semidefinite matrix constraint:
- Optimization problem is a semidefinite program
- is the Choi matrix for the channel and input
- Convex optimization problem with efficient solution

Channel-adapted recoveries yielded better entanglement fidelity in both examples

Improved performance even for lower noise channels

Channel-adaptation extended the region where error correction was effective

Doubled for amplitude damping

QER Example: [5,1] Code,Amplitude Damping Channel[4,1] Channel-adapted Code ofLeung et. al.

- Channel-adaptation can make more efficient codes
- The 4 qubit code and recovery presented by Leung et.al. matches the 5 qubit QEC
- Known as approximate error correction as the recovery permits small distortion
- By channel-adapting the recovery operations, the 4 and 5 qubit codes continue their equivalence

Outline

- Numerical Tools for Channel-Adaptation
- Optimum Channel-adapted Quantum Error Recovery (QER)
- Structured Near-optimal Channel-adapted QER
- Channel-Adaptation for the Amplitude Damping Channel
- Conclusions and Open Questions

Motivation for Near-optimal Channel-adapted QER

Three drawbacks of optimal QER

- SDP for n-length code requires 4n+1 optimization variables
- Difficult to compute for codes beyond 5 qubits
- Optimal recovery may be difficult to implement
- Constrained to be valid quantum operation, but circuit complexity is not considered
- Optimal recovery operation provides little insight into channel-adapted mechanism
- Numerical result is hard to analyze for intuition

Optimum QER demonstrates inefficiency of generic QEC and provides a performance bound, but is only a first step toward channel-adapted error correction.

Projective Syndrome Measurement

Determine Projective

Measurement

Operator P.

Given Outcome P

Determine

Correction Term.

- Recovery operations can always be interpreted as a syndrome measurement followed by a syndrome correction
- Projective measurements are intuitively and physically simpler to understand than the general measurement
- Examination of optimal recovery examples suggest that projective syndromes approximate optimality
- By selecting a projective measurement, we partition the recovery problem into a set of smaller problems
- Challenge is to select a near-optimal projective measurement

Connection to Eigen-analysis

- Consider constraining recoveries to projective measurements followed by unitary operations
- Done in CSS codes, stabilizer codes
- One consequence:
- Write the entanglement fidelity problem in terms of the eigenvectors (Kraus operators) of the Choi matrix
- If were the only constraint, the solution would be the eigen-decomposition of
- CPTP constraint is not the same, but they are similar
- From this observation, we construct a near-optimal algorithm dubbed ‘EigQER’

EigQER Algorithm

- Initialize .

For the kth iteration:

- Determine the eigenvector associated with the largest eigenvalue of .
- Determine recovery operator as the closest isometry to using the singular value decomposition.
- Update by projecting out the space spanned by :

Iterate until the recovery operation is complete:

EigQER Example5 Qubit Amplitude Damping Channel

- For small g, EigQER and Optimal QER are nearly indistinguishable
- Performances diverge somewhat as noise level increases
- Asymptotic behavior approaching g=0 are identical

EigQER ExampleAmplitude Damping for Long Codes

- QEC performance worse for longer codes
- Generic recovery only corrects single qubit errors
- Strong performance of Channel-adapted Shor code (9 qubits)
- 8 redundant qubits aids adaptability
- Steane code (7 qubits) performance surprising
- Not well adapted to amplitude damping errors

From optimization theory: Every problem has an associated dual

Dual feasible point is an upper bound for performance

If Dual=Primal then we know it is optimal

Numerical algorithm: construct a dual feasible point given a projective recovery

Near-optimality Claim:Lagrange Dual Upper BoundOutline

- Numerical Tools for Channel-Adaptation
- Optimum Channel-adapted Quantum Error Recovery (QER)
- Structured Near-optimal Channel-adapted QER
- Channel-Adaptation for the Amplitude Damping Channel
- Conclusions and Open Questions

The 5 qubit code has 24=16 syndrome measurements

QEC uses

10 syndromes to correct single X or Y errors

5 syndromes to correct single Z errors

1 syndromes to “correct” Identity (No Error)

Channel-adapted QER uses

1 Syndrome to “correct” Identity Error

5 Syndromes to correct X+iY Errors (approximately)

Remaining 10 syndromes to correct higher order errors

(i.e.Z errors and 2 qubit dampings)

Channel-adaptation more efficiently utilizes the redundancy of the error correcting code

Degrees of freedom targeted to expected errors

Amplitude Damping Error Syndromes|a|2

E=aI+bX+gY

|g|2

|b|2

Code Subspace

g/4

YError

g/4

New Syndrome

Subspace

XError

[4,1] Code – A Second Look

Amplitude damping error on an arbitrary encoded state:

These are clearly orthogonal subspaces correctable errors

Some subspaces only reached by multiple damped qubits; each correspond to |0Li:

Optimal recovery has three components:

- Standard `perfect’ recovery from damping errors
- Partial correction for some multi-qubit damping errors
- Approximate correction of `no damping’ case (optional for small g)

Amplitude Damping Errors in the Stabilizer Formalism

- Amplitude damping errors have the form
- How does this act on a state stabilized by g?
- Three cases of interest
- g has an I on the ith qubit
- g has a Z on the ith qubit
- g has an X (or Y) on the ith qubit
- We also know that Ziis a generator
- We can thus determine stabilizers for the damped subspaces.

[4,1] Stabilizer Illustration

Code Subspace:

Damped Subspaces:

- We can clearly see each damped subspace is orthogonal to the code subspace
- Mutual orthogonality easier to see by rewriting the generators of the 2nd and 4th
- While not shown, stabilizer analysis allows easy understanding of multiple dampings

[2(M+1),M] Amplitude Damping Codes

- [4,1] code generalizes directly to higher rate codes
- Paired qubit structure makes guarantees orthogonality of damped subspaces
- Perfectly corrects first order damping errors
- Partially corrects multiple qubit dampings
- Straightforward quantum circuit implementation for encoding and recovery

[7,3] Hamming Amplitude Damping Code

- First 3 generators are the classical Hamming code
- [7,4] code that corrects a single bit error
- Fourth generator distinguishes between X and Y errors
- Dedicate 2 syndrome measurements for every damping error
- Perfectly corrects single qubit dampings
- No corrections for multiple qubit dampings
- All X generator generalizes other classical linear codes
- Must be even-parity, single error correcting
- Amplitude damping “redemption” of the Steane code

Summary

- Optimal QER is a semidefinite program
- Phys. Rev. A 75(1):021338, 2007 (quant-ph/0606035)
- Optimality conditions
- Robustness analysis
- Analytically constructed optimal QER for stabilizer code, Pauli errors, and completely mixed input
- Structured near-optimal QER operations
- Phys. Rev. A (to appear) (quant-ph/0708.3658)
- EigQER, OrderQER, Block SDP QER
- More computationally scaleable, more physically realizable
- Performance upper bounds via Lagrange duality
- Gershgorin upper bound
- Iterated dual bound
- Amplitude damping channel-adapted codes
- Analysis of optimal QER for [4,1] code of Leung et. al.
- [2(M+1),M] stabilizer codes
- Even parity classical linear codes (1-error correcting)
- Both classes have Clifford group recovery operations
- quant-ph/0710.1052

Open Question: Channel-adaptedFault Tolerant Quantum Computing

- QEC is the foundation for research in fault tolerant quantum computing (FTQC)
- QEC models noisy channel between two perfect quantum computers
- FTQC explores computing with faulty quantum gates
- Channel-adapted theory will have practical value when extended to channel-adapted FTQC
- Must demonstrate universal set of fault tolerant gates
- Must show that errors do not propagate
- Requires determining a physical noise model and designing a channel-adapted scheme
- Principles and tools of QEC will be the launching point for this analysis

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