Channel adapted quantum error correction
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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 l.jpg

Channel-Adapted Quantum Error Correction

Andrew Fletcher

QEC ‘07

21 December 2007

Channel adapted quantum error recovery qer l.jpg
Channel-adaptedQuantum Error Recovery (QER)




  • 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 l.jpg
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 l.jpg
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 l.jpg

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 l.jpg
[4,1] Channel-adapted Code of fidelity in both examplesLeung et. al.

  • Channel-adaptation can make more efficient codes

  • The 4 qubit code and recovery presented by Leung 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 l.jpg
Outline fidelity in both examples

  • 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 l.jpg
Motivation for Near-optimal fidelity in both examplesChannel-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 l.jpg
Projective Syndrome Measurement fidelity in both examples

Determine Projective


Operator P.

Given Outcome P


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 l.jpg
Connection to Eigen-analysis fidelity in both examples

  • 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 l.jpg
EigQER Algorithm fidelity in both examples

  • 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 l.jpg
EigQER Example fidelity in both examples5 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 l.jpg
EigQER Example fidelity in both examplesAmplitude 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 l.jpg

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 l.jpg
Outline dual

  • 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 l.jpg

The 5 qubit code has 2 dual4=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





Code Subspace




New Syndrome



4 1 code a second look l.jpg
[4,1] Code – A Second Look dual

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 l.jpg
Amplitude Damping Errors in the Stabilizer Formalism dual

  • 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 l.jpg
[4,1] Stabilizer Illustration dual

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 l.jpg
[2(M+1),M] Amplitude Damping Codes dual

  • [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 l.jpg
[7,3] Hamming Amplitude Damping Code dual

  • 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 l.jpg
Summary dual

  • 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 l.jpg
Open Question: Channel-adapted dualFault 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