Quantum Codes Suitable for Iterative Decoding

Quantum Codes Suitable for Iterative
Decoding
Jean-Pierre Tillich (INRIA, Projet SECRET)
May 28th, 2009

1/26
introduction
1. Quantum codes suitable for iterative decoding
◮ Powerful alternatives to concatenated schemes currently under
consideration for fault tolerant quantum computing architectures.
◮ Better lower bounds on the capacity of quantum channels (basically
unknown for most channels).
◮ Quantum LDPC codes: suggested in 2003 by McKay&al.
◮ Quantum (serial) turbo-codes:
2005.
suggested by Ollivier-Tillich in

Introduction
Quantum LDPC codes
◮ Would have the same advantages as in the classical setting : easy
to decode, possible to operate with them successfully at rates near
capacity, better than concatenated coding schemes.
◮ Have apparently drawbacks that classical LDPC codes do not have
• Problem 1: Almost all constructions of families of quantum
LDPC codes either have either vanishing rate or bounded
minimum distance, exception: [Freedman-Meyer-Luo2002] with
d = Ω(log n).
• Problem 2: commutation constraints → many 4-cycles in the
Tanner graph: problems with iterative decoding and proving
that it converges.
• Problem 3: oscillatory behavior of iterative decoding.
• Problem 4: Random constructions?
2/26

Why does randomness help?
Introduction
◮ In general, very difficult to prove that a specific code has good
error correction capacity.
◮ Often it is easier to prove that among a large family of codes
F, most codes have good error correction capacity, by using the
probabilistic method:
1. Choosing a probability distribution P on F,
2. Picking codes from F according to P,
3. Computing the expectation E(T ) of quantities T related to the
correction capacity,
4. concluding that most codes of F have good error correction
capacity.
3/26

4/26
Serial quantum turbo-codes
Introduction
◮ 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...
Theorem 1. [Poulin-Tillich-Ollivier-07] There are no encoders
which are at the same time non-catastrophic and recursive.

5/26
Entanglement assisted coding schemes
[Brun-Devetak-Hsieh2006]
introduction
◮ enable to define quantum codes without having to satisfy the
commutation constraints on the stabilizer matrix by relying on
shared entanglement between encoder/decoder, classical coding
schemes can then be used to attain the hashing bound.
◮ amount of entanglement which is a priori linear in the code length
can be arbitrarily reduced with the help of a concatenated code
scheme.

6/26
Another approach
introduction
◮ Solving the 4-cycle problem with a variant of LDPC codes:
Tornado codes.
◮ Obtaining a quantum code construction with provably good
performance under iterative decoding.
◮ polynomial minimal distance.
◮ attaining the capacity of the quantum erasure channel with linear
decoding complexity .

7/26
2. Error Model
Quantum
Much richer error model than in the classical setting
qubit flip(X) phase flip (Z) both ! (Y )
|0〉 → |1〉
|1〉 → |0〉
|0〉 → |0〉
|1〉 → −|1〉
|0〉 → −i|1〉
|1〉 → i|0〉
X =
( ) 0 1
1 0
Z =
( ) 1 0
0 −1
Y =
( ) 0 −i
i 0

8/26
Quantum
The Pauli group over n qubits G n
G n = {E 1 ⊗ E 2 ⊗ · · · ⊗ E n |E i ∈ G 1 }
∼= {I, X, Y, Z} n × {±1, ±i}
X 2 = Y 2 = Z 2 = I,
XY = −Y X = iZ,
XZ = −ZX = −iY

9/26
The depolarizing channel
Quantum
The depolarizing channel of probability of error p : E = E 1 ⊗
E 2 ⊗ · · · ⊗ E n ∈ G n such that the E i ’s are independent and
Prob(E i = I) = 1 − p
Prob(E i = X) = Prob(E i = Y ) = Prob(E i = Z) = p 3 .

10/26
Quantum code and encoding
Quantum
◮ A quantum code C protecting k qubits by embedding them in an
n-qubit system : subspace of dimension 2 k of C 2n .
◮ An encoding for such a code : unitary transform U : C 2n → C 2n
such that
(
)
C = U C 2k ⊗ |0 n−k 〉 .

11/26
Definition of a stabilizer code
Quantum
◮ Let S be an Abelian subgroup of G n where all its elements are
of order 2 and such that −1 /∈ S. Such a subgroup is called an
admissible group.
◮ A stabilizer code C associated to an admissible group S is the
subspace of C 2n defined by
C = {|ψ〉 ∈ C 2n |∀M ∈ S, M|ψ〉 = |ψ〉}
S is called the stabilizer group of C. If S is generated by n − k
independent generators then dim C = 2 k .

12/26
◮ For E, F ∈ G n :
The syndrome
Quantum
E ⋆ F = 0 if E and F commute
E ⋆ F = 1 otherwise.
◮ For any choice of generators M 1 , . . . , M n−k of generators of
the stabilizer group, there exists a quantum measurement which
reveals the syndrome of the error
s(E) def
= (E ⋆ M i ) 1≤i≤n−k .
◮ The matrix H of size (n−k)×n whose rows are M 1 , M 2 , . . . , M n−k
is called a parity-check matrix of the stabilizer code.

13/26
Harmful/less errors-Minimum Distance
Quantum
◮ Harmless errors: errors ∈ the stabilizer group.
◮ Harmful errors: errors with zero syndrome /∈ the stabilizer group.
◮ Minimum distance: minimum Hamming weight of a harmful error.

14/26
3.Quantum LDPC code
Q-LDPC
◮ Stabilizer code for which there exist a sparse set of generators for
the stabilizer group.
◮ These codes can be iteratively decoded with algorithms similar to
algorithms used for classical LDPC codes.

15/26
Q-LDPC
The problem : constructing Q-LDPC codes with a
good minimum distance
Fact 1. If there is a column of the parity-check matrix which
contains at most one type of element different from I, then there is
an error with zero-syndrome of weight 1.
X X X X
Ζ Ζ I I
I I X X
X
Ζ
Here, I ⊗ I ⊗ X ⊗ I is such an error.
4-cycles are unavoidable.
Minimum distance > 1 ⇒

16/26
Iterative decoding
Q-LDPC
(0.7,0.1,0.1,0.1)
(0.7,0.1,0.1,0.1)
(0.7,0.1,0.1,0.1) (0.7,0.1,0.1,0.1)

17/26
Iterative decoding (II)
Q-LDPC
(0.7,0.1,0.1,0.1) (0.4,0.1,0.1,0.4)
(0.7,0.1,0.1,0.1)
(0.304,0.304,0.196,0.196)
(0.7,0.1,0.1,0.1) (0.7,0.1,0.1,0.1)

18/26
A construction
Q-LDPC
(v ,v )
1 2
v
1
V 1 ... C 1 ...
c 1
V
2
v
2
C
... 2 ...
c
2
V x V
1 2
(c ,c )
1 2
(v ,c )
1 2
V x C
1
(c ,v )
1 2
2
C x C
1 2
X
Ζ
C x V
1
2

Q-LDPC
The toric code=an instance of this construction
=
X
n = 2t 2 , k = 2, R = 1 t 2. 19/26
X
X
X
Z
Z
Z
Z

Q-LDPC
X
n = 2t 2 , k = 2, R = 1 t 2, d = t 20/26
X
X
X
Z
Z
Z
Z

21/26
Properties
Q-LDPC
◮ Complete freedom on the constituent graphs ⇒ optimization on
their degree distribution ?
◮ Theorem 2. [Tillich-Zémor09] classical LDPC code of
parameters [n, k, d] ⇒ quantum LDPC code of parameters
[
}
n 2 + (n
{{
− k)
}
2 , }{{} k 2 , }{{} d ].
size of the enc. state # prot. qubits min. distance
◮ Corollary: Families of quantum codes and minimum distance
Ω ( n 1/2) for any rate 0 < R < 1 (previously [Freedman-Meyer-
Luo02] Ω (log n)).

22/26
Decoding the construction
Q-LDPC
◮ No 4-cycles in the Tanner graph used for decoding if the graph is
split into two graphs blue edges/red edges.
◮ Errors = row of the parity-check matrix, are undetected ⇒
oscillating behavior of the iterative decoder. Change the decoding
algorithm / use parity-checks of fairly large support?
◮ Iterative decoding seems quite difficult to analyze rigorously even
on the erasure channel.

Quantum Tornado codes
◮ Split the (n − k) × n parity-check matrix H in two halves.
k n−k
Tornado
H =
L R n−k.
◮ the first part L of size (n − k) × k is chosen by a certain degree
distribution and without 4-cycles.
◮ The second part R of size (n − k) × (n − k) is chosen in such a
way that the rows of H commute. If L is sparse then R can be
chosen to be sparse ⇒ quantum LDPC code.
◮ Iterative decoding can be analyzed rigorously if the error is known
on the last n − k positions.
23/26

24/26
Tornado
◮ Re-encode the last n − k positions by a quantum LDPC code of
the same type.
◮ Iterate the process.
k = k 0
|0>
...
U k 0
...
k = r(k )
1 0
|0>
... ...
k = r(k )
2 1
|0>
|0>
...
U k
1
...
...
k = r(k )
t t−1
|0>
|0>
...
...
U k
s
t−1
|0>
|0>
V
...

25/26
Q-Tornado codes
quantum codes
Theorem 3. [Luby-Mitzenmacher-Shokrollahi-Spielman-1997]
Classical Tornado codes attain the capacity of the erasure channel
with linear encoding/decoding complexity.
Theorem 4. [Tillich09] With a last level of size O(n 1 3) chosen
at random and rate < 1 − 2p, quantum Tornado codes attain the
capacity of the erasure channel of erasure probability p with linear
decoding complexity.
Drawback ( : size of the last level ⇒ probability that the decoding
fails Ω e −αn1/3) .

Open problems
quantum codes
◮ Increase the size of the last level in the quantum Tornado code
construction.
◮ Obtaining Q-LDPC codes with linear minimum distance?
◮ Using Q-LDPC codes to improve the lower bounds known on the
quantum channel capacity.
◮ Analyze rigorously iterative decoding of Q-LDPC codes.
◮ Find a modification of quantum serial turbo-code construction
with unbounded minimum distance and where iterative decoding
converges to the right solution for good enough channel conditions.
◮ Generalize other iterative decoding schemes.
◮ Use such codes to improve existing fault tolerant architectures.
26/26