General Entanglement-Assisted Quantum Error-Correcting Codes

Todd A. Brun, Igor Devetak and Min-Hsiu HsiehCommunication Sciences Institute

QEC07

[[n,k]] quantum error correcting codemeasure + correct

Pauli unitaries

Z X

Pauli groupDiscretization of errors

Shor ’95; Steane ‘96; Gottesman ’96; Calderbank, Rains, Shor, Sloane ‘96

• An [[n,k]] quantum error correcting code is described by a (n-k) £ 2n parity check matrix H. Its rowspace B(H) is an isotropic subspace of

commutingstabilizer generators

dual containing code

and commute

n=5, k=1

• The symplectic product is defined by

• and commute (anti-commute) iff

Classical symplectic codes

• The correctable error set E is defined by:

degenerate code

• The code space is defined as the simultaneous +1 eigenspace of the stabilizer operators

• Correction involves measuring the “error syndrome” (i.e. the simultaneous eigenvector of the stabilizer generators) ,

distinct error syndromes

Y error on 4th q-bit

Quantum stabilizer codes

If E1 and E2 are in E, then at least one of the two conditions hold:

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1) E2tE1 ∉ Z(S)

2) E2tE1 ∈ S

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≡S

Properties of Stabilizer Codes

We can see that stabilizer codes have the following properties:

1. The code corresponds to an isotropic (that is, dual-containing) classical code over a symplectic space.

2. The error correcting conditions are almost the same as classical (except for the existence of degenerate quantum codes, in which distinct errors share the same error syndrome).

3. Correction consists of measuring an error syndrome and performing an appropriate correcting action (a unitary).

Entanglement-assisted error correction

[[n,k;c]] EA quantum error correcting code

Alice

Bob

e-bit

c e-bits

Bowen ‘03; Brun, Devetak, Hsieh, Science 2006; quant-ph/0608027

Entanglement-assisted stabilizer formalism

It turns out that we can establish a simple extension of the usual stabilizer formalism to describe entanglement-assisted codes. We again establish a “stabilizer” which is a subgroup of the Pauli group on n q-bits; but we no longer require this subgroup to be Abelian. For such a subgroup, we can find a set of generators which fall into two groups:

Isotropic generators, which commute with all other generators; and

Symplectic generators, which come in anticommuting pairs; each pair commutes with all other generators.

• An [[n,k;c]] EA quantum error correcting code is described by a (n-k) £ 2n parity check matrix H. B = rowspace(H). Again,

• Take a general symplectic matrix H. Its rowspace B can be written as

• Canonical example

symplectic pairs

Entanglement-assisted stabilizer formalism

The isotropic generators generate SI and the symplectic generators generate SE.

stabilized by Measure in the simultaneous eigenbasis of

n = 3, k = 1, c = 2

[[3, 1; 2]] code

• The correctable error set E is defined by:

degenerate code

• The code space is defined as the simultaneous +1 eigenspace of the stabilizer generators

• Decoding involves measuring the “error syndrome” (i.e. the simultaneous eigenvector of the stabilizer generators) ,

nn nc cc

If E1 and E2 are in E, then at least one of the two conditions hold:

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1) E2tE1 ∉ Z( SI ,SE )

2) E2tE1 ∈ SI

Properties of the entanglement-assisted stabilizer formalism

We can now compare the properties of an EAQECC to those of an ordinary QECC:

1. The code corresponds to a classical code over a symplectic space. (No longer needs to be dual-containing!)

2. The error correcting conditions are almost the same as classical (except for the existence of degenerate quantum codes, in which distinct errors share the same error syndrome).

3. Correction consists of measuring an error syndrome and performing an appropriate correcting action (a unitary).

The GF(4) Construction

• Natural isometry between GF(4) and (Z2)2

• Any dual containing classical [n,k,d]4 code can be made into a [[n,2k–n,d]] QECC

• Now: Any classical [n,k,d]4 code can be made into a [[n,2k-n+c,d;c]] catalytic QECC for some c

• When the classical code attains the Singleton bound n-k ¸ d-1 the quantum code attains the quantum Singleton bound n-k ¸ 2(d-1)

• When the classical code attains the Shannon limit 2 – H4(1 – 3p, p,p,p) on a quaternary symmetric channel, the quantum code attains the Hashing limit 1-H2(1-3p, p,p,p).

• Modern classical codes (LDPC, turbo) can now be made quantum without having to be dual-containing.

Operator QECCs

Poulin; Bacon; Aly, Klappenecker and Sarvepalli

The basic idea of operator QECCs is that part of the system (the noisy or gauge part) contains no information about either the quantum information to be transmitted or the errors which occur. We allow arbitrary noise to affect this gauge subsystem.

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Ψe Ψe = ψ ψ ⊗ 0n−k−r 0n−k−r ⊗ ρG

Because the gauge subsystem can be in any arbitrary state, OQECCs are not subspaces. (That is, the superposition of two valid states in the OQECC is not a valid state unless they have the same state for the gauge subsystem.)

An [[n,k;r]] OQECC: r = number of gauge q-bits

• Once again, write rowspace B of symplectic matrix H as

OQECC Stabilizer Formalism

• Now, symplectic pairs to correspond to operations on the gauge qubits

• The correctable error set E is defined by:

If E1 and E2 are in E, then at least one of the two conditions hold:

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1) E2tE1 ∉ Z(SI )

2) E2tE1 ∈ SI ,SG

Entanglement-assisted operator quantum error-correcting codes

Both EAQECCs and OQECCs extend the stabilizer formalism by allowing a noncommuting “stabilizer” group. EAQECCs add an extra e-bit for each symplectic pair in the set of generators. OQECCs drop these noncommuting pairs of generators, and identify states which differ by the action of gauge operators.

One can combine these approaches: use entanglement to retain some of the symplectic group in the stabilizer, while gauging out the rest. The same classical symplectic code can thus produce a range of quantum codes, combining different degrees of entanglement-assistance and passive correction. This is the most general extension of stabilizer codes so far constructed.

Hsieh, Devetak, Brun, ISIT June 2007; quant-ph/0708.2142 (to appear PRA); Gilbert et al. quant-ph/0709.0128

• Once again, write rowspace B of symplectic matrix H as

EAOQECC Stabilizer Formalism

• Some symplectic pairs to correspond to operations on the gauge qubits, and others to pre-existing entanglement

• The correctable error set E is defined by:If E1 and E2 are in E, then at least one of the two conditions hold:

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1) E2tE1 ∉ Z( SI ,SE )

2) E2tE1 ∈ SI ,SG

generate SI

generate SE

generate SG

[[n,k,d;r;c]] code

Examples

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

The [8,1,3;1] EAQECC can be made an [8,1,3;2;1] EAOQECC by dropping two generators from SI and adding them and their symplectic partners to SG.

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⇒

(This example is based on the 9-bit Shor code.)

We can construct another example from classical BCH codes.

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H =

1 α α 2 L

1 α 3 α 6 L

1 α 5 α 10 L

1 α 7 α 14 L

α n−1

α 3(n−1)

α 5(n−1)

α 7(n−1)

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

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α ∈ GF(26)

α 6 +α +1= 0

n = 63

Replacing α by its binary representation produces a [63,39,9] classical BCH code, which gives a [[63,21,9;6]] EAQECC. Making some symplectic pairs into gauge operators yields:

n k d r c

63 21 9 0 6

63 21 7 1 5

63 21 7 2 4

63 21 7 3 3

63 21 7 4 2

63 21 7 5 1

63 21 7 6 0

Conclusions

• Almost all known quantum error-correcting codes are stabilizer codes. These have a connection to classical symplectic codes, but only for dual-containing codes.

• Two extensions beyond this scheme which do not require this constraint are entanglement-assisted QECCs and operator QECCs. Both yield non-commuting “stabilizers” which are dealt with in different ways.

• These two extensions can usefully be combined into general EAOQECCs, which gives considerable flexibility in constructing QECCs from classical codes.

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