Channel-Adapted Quantum Error Correction

1. Channel-Adapted Quantum Error Correction Andrew Fletcher QEC ‘07 21 December 2007

2. 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)

3. 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

4. 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

5. 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

6. [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

7. 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

8. 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.

9. 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

10. 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’

11. 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:

12. 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

13. 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

14. 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 Bound

15. 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

16. 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

17. [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)

18. 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.

19. [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

20. [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

21. [6,2] vs. [4,1]2

22. [2(M+1),M] Normalized Comparison

23. [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

24. [7,3] vs. [8,3]

25. 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

26. 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