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Channel-Adapted Quantum Error Correction. Andrew Fletcher QEC ‘07 21 December 2007. Channel-adapted Quantum Error Recovery (QER). Encoder. Channel. Recovery. QEC scheme specifies Encoder and Recovery Generic methods do this independently of channel

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channel adapted quantum error correction

Channel-Adapted Quantum Error Correction

Andrew Fletcher

QEC ‘07

21 December 2007

channel adapted quantum error recovery qer
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
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
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
qer example 5 1 code amplitude damping channel
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 of leung et al
[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
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
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
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
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
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 example 5 qubit amplitude damping channel
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 example amplitude damping for long codes
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
near optimality claim lagrange dual upper bound
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
outline15
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
amplitude damping error syndromes
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
[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 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
[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
[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
[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
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 adapted fault tolerant quantum computing
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