qec’07-1 asf 6/13/2015 mit lincoln laboratory channel-adapted quantum error correction andrew...

QEC’07-1ASF 04/18/23

MIT Lincoln Laboratory

Channel-Adapted Quantum Error Correction

Andrew Fletcher

QEC ‘07

21 December 2007

MIT Lincoln LaboratoryQEC’07-2

ASF 04/18/23

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)

ChannelEncoder Recovery


ASF 04/18/23

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


ASF 04/18/23

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

trX R C½;E

maxX RtrX R C½;E,

such that X R ¸ 0, troutX R = I

troutX R = I

X R ¸ 0


ASF 04/18/23

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


ASF 04/18/23

[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


ASF 04/18/23

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


ASF 04/18/23

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.


ASF 04/18/23

Projective Syndrome Measurement

• 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

Determine Projective MeasurementOperator P.

Given Outcome P Determine

Correction Term.


ASF 04/18/23

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’


ASF 04/18/23

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:


ASF 04/18/23

EigQER Example5 Qubit Amplitude Damping Channel

• For small , EigQER and Optimal QER are nearly indistinguishable

– Performances diverge somewhat as noise level increases

• Asymptotic behavior approaching =0 are identical


ASF 04/18/23

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


ASF 04/18/23

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


ASF 04/18/23

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


ASF 04/18/23

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

||2

||2

||2

=I+X+Y

X Error

Code Subspace

Y Error

New SyndromeSubspace

/4

/4

E 1 E 0 E 0 ¢¢¢¼

E 1 I I ¢¢¢


ASF 04/18/23

[4,1] Code – A Second Look

j0L i = j0000i + j1111i , j1L i = j0011i + j1100i

®j0111i + ¯ j0100i®j1011i + ¯ j1000i®j1101i + ¯ j0001i®j1110i + ¯ j0010i

®j0L i + ¯ j1L i

Amplitude damping error on an arbitrary encoded state:

These are clearly orthogonal subspaces correctable errors

j1010i j0110i j0101i j1001iSome subspaces only reached by multiple damped qubits; each correspond to |0Li:

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

Optimal recovery has three components:


ASF 04/18/23

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 Zi is a generator

• We can thus determine stabilizers for the damped subspaces.

X j + iYj

(X i + iYi) jÃi = (X i + iYi)gjÃi = g(X i + iYi) jÃi

(X i + iYi) jÃi = (X i + iYi)gjÃi = ¡ g(X i + iYi) jÃi

(X i + iYi) jÃi = (X i + iYi)gjÃi = g(X i ¡ iYi) jÃi

Zi(X i + iYi) jÃi = (iYi ¡ i2X i) jÃi = (X i + iYi) jÃi


ASF 04/18/23

[4,1] Stabilizer Illustration

X 1 + iY1:-ZZ I II I ZZZ I I I

X 2 + iY2:-ZZ I II I ZZI Z I I

X 3 + iY3:ZZ I I

- I I ZZI I Z I

X 4 + iY4:ZZ I I

- I I ZZI I I Z

X X X XZ Z I II I Z Z

Damped Subspaces:

Code Subspace:

•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

j0L i = j0000i + j1111ij1L i = j0011i + j1100i

X 2 + iY2:-ZZ I II I ZZ

-Z I I I

X 4 + iY4:ZZ I I

- I I ZZ- I I Z I


ASF 04/18/23

[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

[6;2] codeX X X X X XZ Z I I I II I Z Z I II I I I Z Z

[8;3] codeX X X X X X X XZ Z I I I I I II I Z Z I I I II I I I Z Z I II I I I I I Z Z


ASF 04/18/23

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


ASF 04/18/23

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


ASF 04/18/23

[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

[7;3] linear codeI I I Z Z Z ZI Z Z I I Z ZZ I Z I Z I ZX X X X X X X


ASF 04/18/23

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


ASF 04/18/23

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


ASF 04/18/23

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

qec’07-1 asf 6/13/2015 mit lincoln laboratory channel-adapted quantum error correction andrew...

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