Turning error-reducing quantum turbo codes into error-correcting - - PowerPoint PPT Presentation

Turning error-reducing quantum turbo codes into error-correcting codes

Mamdouh Abbara (MEc), Iryna Andriyanova (ENSEA), Jean-Pierre Tillich (INRIA) QEC14 December the 17th, 2014

introduction

The 5−qubit code

Probability of error after decoding the 5-qubit code 1/34

introduction

An alternative strategy for concatenation

0>

C C C C C C C

inner convolutional code of rate 1 interleaver 0> 0> 0> 0> 0> 0> 0> 0> 0> 0> 0> 0> 0> 0> 0> 0> 0> 0> 0> 0> 0> 0> 0> 0> 0> 0> 0>

2/34

introduction

The message of this talk

◮ It is possible to concatenate with a rate 1 code (so no protection against errors at all...) and still achieve something nontrivial when the rate 1 code is a convolutional code. 3/34

introduction

Improving the 5−qubit code

Figure 1: Probability of error after decoding – complexity of encoding ≈ complexity of encoding a 5-qubit code – Rate 1

5 → 1 8

– same complexity of decoding as the 5-qubit code – modified quantum turbo-code construction 4/34

Introduction

serial quantum turbo-codes

◮ as for quantum LDPC codes it is possible to build such codes and decode them with iterative decoding algorithms. ◮ freedom to introduce randomness in the construction what we do not have for quantum LDPC codes. ◮ much simpler to construct. ◮ but there are also some problems related to encoding issues... 5/34

serial concatenation

• 2. Concatenation of codes

– Pn Pauli group over n quits – Clifford transformation U : U †PnU ∈ Pn

k

0> φ> 0>

U

0>

P

1

P P P

n 2 3

U

−1 ... ... ...

k n−k

... ... ...

L L S S

n−k

L1

2

S1

2

6/34

serial concatenation

◮ Physical error P = P1P2 . . . Pn ◮ Logical error,syndrome LS = L1L2 . . . Lk

• logical error

S1 . . . Sn−k

• syndrome

= U †PU

n

S

n−k k

L P U

7/34

serial concatenation

Stabilizer, Normalizer

◮ Stabilizer set S corresponds to L = I . . . I, S ∈ {I, Z}n−k : S =

• U(I . . . I

k

S)U†, S ∈ {I, Z}n−k

• ◮ Normalizer set N corresponds to S ∈ {I, Z}n−k.

N =

• U(L, S)U†, S ∈ {I, Z}n−k

◮ Quantum minimum distance dquantum = min{|P| ∈ N \ S} dclassical = min {|P| ∈ N \ {I . . . I}} 8/34

introduction

Serial concatenation of codes

symbols L

• ut

U Π Uinner

E E E L2

1 1 2

S S

N 1 n

L’ L’

1 N−n

S2

n−k

S S1

k

L

... ... ... ... ... ... ... ... ...

information syndrome code

• f the inner

syndrome code

• f the outer

9/34

minimum distance

• 3. Minimum Distance Properties

When the inner code is a juxtaposition of small codes

U

.....

L S L S L S P

P

PN

N N 1 1 1

U

U

U Clifford transformation on n qubits, Din ≤ n, Dcont ≤ Doutn. 10/34

minimum distance

• ut

U Dout

• ut

L Sout Sin

Dcon ≤ Doutn. 11/34

minimum distance

The problem (I)

L1...Lk1 S1...Sr1

Sout

S′

1...S′ r2 Sin Uout

→ L′

1, ..., L′ k1+r1 S′ 1, ..., S′ r2

• Sin

Π

→ L′

π(1), ..., L′ π(k1+r1), S′ 1, ..., S′ r2 Uin

→ P1, ..., Pk1+r1+r2 12/34

minimum distance

The problem (II)

Assume that there exists for the inner code a bound D such that for each i ∈ {1, . . . , k1 + r1} and every P ∈ {X, Y, Z} there exists a choice for the S′

j’s in {I, Z} such that

• Uin(

i−1 times

I . . . I P

k1+r1−i times

I . . . I S′

1, . . . S′ r2)U † in

• ≤ D

then if the minimum distance of the outer code is Dout the minimum distance of the concatenated code is upper bounded by DoutD 13/34

stabilizer

The problem(III)

L1...Lk1 S1...Sr1

Sout

S′

1...S′ r2 Sin Uout

→ L′

1, ..., L′ k1+r1 S′ 1, ..., S′ r2

• Sin

with

• L′

1, ..., L′ k1+r1

• =

Dout

Π

→ L′

π(1), ..., L′ π(k1+r1), S′ 1, ..., S′ r2

for each of the L′

π(i) = I

consider the corresponding S′i

1 . . . S′i r2

and mutiply them to obtain S′

1, . . . , S′ r2 Uin

→ P1, ..., Pk1+r1+r2 with |P1 . . . Pk1+r1+r2| ≤ DoutD 14/34

minimum distance

When the inner encoder is convolutional

k n−k n

U

Li Mi Mi

− 1

Si Pi

m m

S0 L1 S2 S1 L2 LN SN SN+1 SN+t PN+t PN+1 PN P2 P1 PN+t−1 SN+t−1{

{ {

}

. . . . . . U U U U U U

Din = O(1) Dcon ≤ ? 15/34

minimum distance

• ut

U L S

16/34

minimum distance

Classical setting

◮ Choose Uout and Uin as (classical) convolutional encoders. ◮ [Kahale-Urbanke-ISIT 1998] In the classical case, by an averaging argument, if the free distance of Cout is dout and if Uin is a non- catastrophic and recursive encoder, then the minimum distance of the resulting code is typically of order Θ

• N

dout−2 dout

• .

◮ Generalizes easily to the quantum setting ? 17/34

minimum distance

A first problem

Theorem 1. [Poulin-Tillich-Ollivier-ISIT 2008] There are no quantum convolutional encoders which are at the same time non- catastrophic and recursive. 18/34

minimum distance

Catastrophic/recursive

(S0, L1, S1, . . . , Li, Si, . . . )

• conv. encoder

− → P = (P1, P2, . . . , ) with S0 ∈ {I, Z}m, Si ∈ {I, Z}n−k for i ≥ 1 L

def

= L1, L2, . . . , ◮ Non-catastrophic encoder : supp(P) finite ⇒ supp(L) finite. ◮ Recursive encoder : |L| = 1 ⇒ supp(P) infinite. 19/34

minimum distance

A crucial argument used in the classical setting

Consider convolutional encoders for which |L| ≤ |P| 20/34

minimum distance

A quantum convolutional encoder that does the job

21/34

minimum distance

A theorem

Theorem 2. [Abbara-Tillich - ITW 2011] If the inner code is the aforementioned convolutional code of rate 1 and the outer code is a juxtaposition of copies of a quantum code of classical minimum distance dclassical and quantum minimum distance dquantum, then with probability → 1 as the length N of the inner code → ∞ the minimum distance Dcon of the concatenated scheme satisfies – Dcon = Ω

• N

dclassical−2 dclassical

• if dclassical > 2

– Dcon = Ω

• log N

log log N

• if dclassical = 2 and dquantum ≥ 3.

22/34

construction

The construction

A first attempt

interleaver

C

0> 0> 0>

C

0> 0> 0>

C

0> 0> 0>

C

0> 0> 0>

C

0> 0> 0>

C

0> 0> 0>

C

0> 0> 0> inner convolutional code of rate 1

23/34

construction

Decoding

n’

C

0> 0> 0>

C

0> 0> 0>

C

0> 0> 0>

C

0> 0> 0>

C

0> 0> 0>

C

0> 0> 0>

C

0> 0> 0> inner convolutional code of rate 1 interleaver

P P P

1 2 n

L 1 L n L 1 L n

decoding the inner code deinterleaving decoding the outer code

S L’

1 1

L’

n

S

24/34

construction

The problem

• uter code

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.6 1.8 2.0 1.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.6 1.8 2.0

entropy of the output of the inner code entropy of the input of the inner code

1.4

starting point = endpoint entropy of the output of the outer code entropy of the input of the inner code inner code

25/34

construction

The modified construction

0>

C C C C

0> 0> 0> 0> 0> 0> 0> 0> 0> 0> 0> 0> inner convolutional code of rate 1 interleaver 0> 0> 0> 0> 0> 0> 0>

26/34

Results

QuBit-error probability after decoding

1e-05 0.0001 0.001 0.01 0.1 1 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 Bitwise error probability Depolarizing channel strength K = 17143 K = 1143 K = 143

27/34

Results

Probability of error per block

0.01 0.1 1 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 Decoding error probability Depolarizing channel strength N = 60000 N = 4000 N = 500

28/34

Results

Entropy evolution during decoding

(P1) + 1st position of the outer code sent directly to the channel decoding trajectory entropy of the output of the inner code starting point endpoint 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 entropy of the input of the outer code entropy of the output of the outer code entropy of the input of the inner code inner code

• uter code

29/34

further

• 5. Going further : a multilevel construction

C C C C

0> 0> 0> 0> 0>

C

0> 0> 0>

C

0> 0> 0> 0> 0> 0> 0>

C

0> 0> 0> 0> 0>

C

0> 0> 0> 0> 0> inner convolutional code of rate 1 interleaver code of rate 1 interleaver 0> 0> 0> 0> 0> 0> 0> 0> 0> 0> 0> 0> 0> 0> 0> inner convolutional

channel channel

30/34

further

Analysis on the erasure channel

Theorem 3. Let t be the number of stages of the concatenated construction where we assume that the underlying block code C is

• f minimum distance 3. Then the probability pt that a logical qubit

stays erased after transmission of the encoded words over an erasure channel of erasure probability p is given by pt = O

• p3t+1+3t−3

t 1 2 3 pt O(p9) O(p33) O(p105) 31/34

further

Results

Figure 2: Probability of error after decoding/comparison with the 5-qubit code 32/34

Summary

Theorem 4. [Poulin-Tillich-Ollivier-08] There are no quantum convolutional encoders which are at the same time non-catastrophic and recursive. 33/34

Summary/Conclusion

◮ Non-catastrophic and non-recursive encoders[Poulin-Tillich- Ollivier-09] : – ⇒ Constant minimum distance... – Might be interesting up to moderate blocklength. ◮ catastrophic and recursive encoders – iterative decoding does not converge (the scheme has to be modified). – the minimum distance might be unbounded. The work presented here : exploring the option catastrophic and recursive encoder. 34/34