[PPT] - Local stabilizer codes in 3D without string logical operators PowerPoint Presentation

SLIDE 1

Local stabilizer codes in 3D without string logical operators

arXiv:1101.1962 Jeongwan Haah IQIM, Caltech 6 Dec 2011 Quantum Error Correction 2011

SLIDE 2

Problem

Theme

◮ Find a noise-free subspace/subsystem from a physical system,

SLIDE 3

Problem

Theme

◮ Find a noise-free subspace/subsystem from a physical system, ◮ on which manipulations are still possible.

SLIDE 4

Problem

Theme

◮ Find a noise-free subspace/subsystem from a physical system, ◮ on which manipulations are still possible.

Many-body system

SLIDE 5

Problem

Theme

◮ Find a noise-free subspace/subsystem from a physical system, ◮ on which manipulations are still possible.

Many-body system

◮ Local indistinguishability – Topological order

SLIDE 6

Problem

Theme

◮ Find a noise-free subspace/subsystem from a physical system, ◮ on which manipulations are still possible.

Many-body system

◮ Local indistinguishability – Topological order ◮ Stable at zero temperature

SLIDE 7

Problem

Theme

◮ Find a noise-free subspace/subsystem from a physical system, ◮ on which manipulations are still possible.

Many-body system

◮ Local indistinguishability – Topological order ◮ Stable at zero temperature

Topological QC ?

◮ TQC with anyons ∼ Excited states ◮ (Self-correcting) Memory ∼ Ground state

SLIDE 8

Why hard?

Kitaev 2D toric code

H = −

s X X X X Z Z

−

p Z Z ◮ Locally indistinguishable 4-fold degenerate ground state.

SLIDE 9

Why hard?

Kitaev 2D toric code

H = −

s X X X X Z Z

−

p Z Z ◮ Locally indistinguishable 4-fold degenerate ground state. ◮ But, |ψ0 ⇐

⇒ |ψ1 by dragging a quasi-particle across.

Z

Z · · · Z •

SLIDE 10

Why hard?

Kitaev 2D toric code

H = −

s X X X X Z Z

−

p Z Z ◮ Locally indistinguishable 4-fold degenerate ground state. ◮ But, |ψ0 ⇐

⇒ |ψ1 by dragging a quasi-particle across.

Z

Z · · · Z •

Bad at T > 0.

SLIDE 11

Why hard?

Kitaev 2D toric code

H = −

s X X X X Z Z

−

p Z Z ◮ Locally indistinguishable 4-fold degenerate ground state. ◮ But, |ψ0 ⇐

⇒ |ψ1 by dragging a quasi-particle across.

Z

Z · · · Z •

Bad at T > 0. The same had happened for all low dimensional models. Self-correction is possible in 4D or higher.

SLIDE 12

3D Cubic Code

IZ

ZI
ZI

ZZ II IZ IZ ZI

IX
XI
XI

II XX IX IX XI

SLIDE 13

3D Cubic Code

IZ

ZI
ZI

ZZ II IZ IZ ZI

IX
XI
XI

II XX IX IX XI

◮ Two spins per site on a simple cubic lattice.

◮ Local interaction. All terms commuting. Frustration-free.

SLIDE 14

3D Cubic Code

IZ

ZI
ZI

ZZ II IZ IZ ZI

IX
XI
XI

II XX IX IX XI

◮ Two spins per site on a simple cubic lattice.

◮ Local interaction. All terms commuting. Frustration-free. ◮ Rigorous definition for string in the discrete lattice. ◮ The model has only short string segments.

SLIDE 15

3D Cubic Code

IZ

ZI
ZI

ZZ II IZ IZ ZI

IX
XI
XI

II XX IX IX XI

◮ Two spins per site on a simple cubic lattice.

◮ Local interaction. All terms commuting. Frustration-free. ◮ Rigorous definition for string in the discrete lattice. ◮ The model has only short string segments. ◮ Exotic degeneracy: k(L = 2p) = 4L − 2, k(L = 2p + 1) = 2.

SLIDE 16

Stabilizer/Additive codes

SLIDE 17

Stabilizer/Additive codes

Abelianized Pauli group

◮ σX · σX = I ⇐

⇒ (10) + (10) = (00)

◮ σX · σZ = −iσY ⇐

⇒ (10) + (01) = (11)

SLIDE 18

Stabilizer/Additive codes

Abelianized Pauli group

◮ σX · σX = I ⇐

⇒ (10) + (10) = (00)

◮ σX · σZ = −iσY ⇐

⇒ (10) + (01) = (11)

◮ [σX, σZ] = 0 ⇐

⇒ (10)λ1(01)T = 1

◮ [σX ⊗ σX, σZ ⊗ σZ] = 0 ⇐

⇒ (1010)λ2(0101)T = 0

SLIDE 19

Stabilizer/Additive codes

Abelianized Pauli group

◮ σX · σX = I ⇐

⇒ (10) + (10) = (00)

◮ σX · σZ = −iσY ⇐

⇒ (10) + (01) = (11)

◮ [σX, σZ] = 0 ⇐

⇒ (10)λ1(01)T = 1

◮ [σX ⊗ σX, σZ ⊗ σZ] = 0 ⇐

⇒ (1010)λ2(0101)T = 0

◮ Abelianized Pauli group is a symplectic vector space over Fp with

λn = In −In

.

SLIDE 20

Stabilizer/Additive codes

Abelianized Pauli group

◮ σX · σX = I ⇐

⇒ (10) + (10) = (00)

◮ σX · σZ = −iσY ⇐

⇒ (10) + (01) = (11)

◮ [σX, σZ] = 0 ⇐

⇒ (10)λ1(01)T = 1

◮ [σX ⊗ σX, σZ ⊗ σZ] = 0 ⇐

⇒ (1010)λ2(0101)T = 0

◮ Abelianized Pauli group is a symplectic vector space over Fp with

λn = In −In

.

Codes

SLIDE 21

Stabilizer/Additive codes

Abelianized Pauli group

◮ σX · σX = I ⇐

⇒ (10) + (10) = (00)

◮ σX · σZ = −iσY ⇐

⇒ (10) + (01) = (11)

◮ [σX, σZ] = 0 ⇐

⇒ (10)λ1(01)T = 1

◮ [σX ⊗ σX, σZ ⊗ σZ] = 0 ⇐

⇒ (1010)λ2(0101)T = 0

◮ Abelianized Pauli group is a symplectic vector space over Fp with

λn = In −In

.

Codes

◮ Commuting set of Pauli operators = Null subspace S

= stabilizer group

◮ Symmetry group = Hyperbolic subspace in S⊥

= group of logical operators.

SLIDE 22

Cubic codes

Generally

◮ Simple cubic lattice with m qubits per site. ◮ t types of cubic interactions with code space dimension > 1.

SLIDE 23

Cubic codes

Generally

◮ Simple cubic lattice with m qubits per site. ◮ t types of cubic interactions with code space dimension > 1. ◮ No single site logical operator. ◮ No obvious string operator.

SLIDE 24

Cubic codes

Generally

◮ Simple cubic lattice with m qubits per site. ◮ t types of cubic interactions with code space dimension > 1. ◮ No single site logical operator. ◮ No obvious string operator.

Simply

◮ t = 2 : one for X, one for Z type. ◮ Product of all terms in the Hamiltonian be Id. ◮ Logical operator on a site be Id. ◮ Logical operator on a straight line be Id.

SLIDE 25

Cubic codes

Generally

◮ Simple cubic lattice with m qubits per site. ◮ t types of cubic interactions with code space dimension > 1. ◮ No single site logical operator. ◮ No obvious string operator.

Simply

◮ t = 2 : one for X, one for Z type. ◮ Product of all terms in the Hamiltonian be Id. ◮ Logical operator on a site be Id. ◮ Logical operator on a straight line be Id.

Finally

◮ Translate the conditions into linear algebra equations on Pm ◮ → Linear equations, rank constraints.

SLIDE 26

17 / 237

D′

Z

C ′

Z

AZ

BZ B′

Z

A′

Z

CZ DZ

D′

X

C ′

X

AX

BX B′

X

A′

X

CX DX

SLIDE 27

17 / 237

D′

Z

C ′

Z

AZ

BZ B′

Z

A′

Z

CZ DZ

D′

X

C ′

X

AX

BX B′

X

A′

X

CX DX

◮ 64 commutation relations determine everything.

SLIDE 28

17 / 237

D′

Z

C ′

Z

AZ

BZ B′

Z

A′

Z

CZ DZ

D′

X

C ′

X

AX

BX B′

X

A′

X

CX DX

◮ 64 commutation relations determine everything.

◮ 27 linear equations ensuring commuting Hamiltonian.

SLIDE 29

17 / 237

D′

Z

C ′

Z

AZ

BZ B′

Z

A′

Z

CZ DZ

D′

X

C ′

X

AX

BX B′

X

A′

X

CX DX

◮ 64 commutation relations determine everything.

◮ 27 linear equations ensuring commuting Hamiltonian. ◮ Find by brute-force the solutions of remaining 30 equations.

SLIDE 30

17 / 237

D′

Z

C ′

Z

AZ

BZ B′

Z

A′

Z

CZ DZ

D′

X

C ′

X

AX

BX B′

X

A′

X

CX DX

◮ 64 commutation relations determine everything.

◮ 27 linear equations ensuring commuting Hamiltonian. ◮ Find by brute-force the solutions of remaining 30 equations. ◮ 17 different solutions up to symmetry group of cube !

SLIDE 31

Code 1 / 17

IZ

ZI
ZI

ZZ II IZ IZ ZI

IX
XI
XI

II XX IX IX XI

Duality QZ ↔ QX.

SLIDE 32

Code 1 / 17

IZ

ZI
ZI

ZZ II IZ IZ ZI

IX
XI
XI

II XX IX IX XI

Duality QZ ↔ QX.

This model has no string...

◮ Wait, is it topologically ordered? ◮ What do you mean by string? What if there is a thick string?

SLIDE 33

No local observables distinguish ground states

If O has small support, then ΠGSOΠGS = c(O)ΠGS.

SLIDE 34

No local observables distinguish ground states

If O has small support, then ΠGSOΠGS = c(O)ΠGS.

◮ Claim: Any Pauli operator of bounded support commuting with all

stabilizers is a stabilizer.

SLIDE 35

No local observables distinguish ground states

If O has small support, then ΠGSOΠGS = c(O)ΠGS.

◮ Claim: Any Pauli operator of bounded support commuting with all

stabilizers is a stabilizer.

◮ Any operator is a linear combination of Pauli operators.

SLIDE 36

No local observables distinguish ground states

If O has small support, then ΠGSOΠGS = c(O)ΠGS.

◮ Claim: Any Pauli operator of bounded support commuting with all

stabilizers is a stabilizer.

◮ Any operator is a linear combination of Pauli operators. ◮ Because QX ↔ QZ, it suffices to consider X-type Pauli operator.

SLIDE 37

Tool – Eraser

Exercise

If ? is one of II, XI, IX, XX;

◮ [ZZ, ?] = 0

SLIDE 38

Tool – Eraser

Exercise

If ? is one of II, XI, IX, XX;

◮ [ZZ, ?] = 0

= ⇒ ? = II, XX

SLIDE 39

Tool – Eraser

Exercise

If ? is one of II, XI, IX, XX;

◮ [ZZ, ?] = 0

= ⇒ ? = II, XX

◮

[IZ, ?] = 0

[ZI, ?] = 0

SLIDE 40

Tool – Eraser

Exercise

If ? is one of II, XI, IX, XX;

◮ [ZZ, ?] = 0

= ⇒ ? = II, XX

◮

[IZ, ?] = 0

[ZI, ?] = 0 = ⇒ ? = II

SLIDE 41

Tool – Eraser

Exercise

If ? is one of II, XI, IX, XX;

◮ [ZZ, ?] = 0

= ⇒ ? = II, XX

◮

[IZ, ?] = 0

[ZI, ?] = 0 = ⇒ ? = II

◮

[IZ, ?] = 0

[ZZ, ?] = 0

SLIDE 42

Tool – Eraser

Exercise

If ? is one of II, XI, IX, XX;

◮ [ZZ, ?] = 0

= ⇒ ? = II, XX

◮

[IZ, ?] = 0

[ZI, ?] = 0 = ⇒ ? = II

◮

[IZ, ?] = 0

[ZZ, ?] = 0 = ⇒ ? = II

SLIDE 43

No local observables distinguish ground states

Proof

?

?
?

? ? ? ? XI

IZ
ZI
ZI

ZZ II IZ IZ ZI

IX
XI
XI

II XX IX IX XI

SLIDE 44

No local observables distinguish ground states

Proof

?

?
?

? ? ? ? XI

IZ
ZI
ZI

ZZ II IZ IZ ZI

IX
XI
XI

II XX IX IX XI

?
?
?

? ? ? II

SLIDE 45

No local observables distinguish ground states

Proof

?

?
?

? ? ? ? XI

IZ
ZI
ZI

ZZ II IZ IZ ZI

IX
XI
XI

II XX IX IX XI

?
?
?

? ? ? II ?

?
?

? ? II

SLIDE 46

No local observables distinguish ground states

Proof

?

?
?

? ? ? ? XI

IZ
ZI
ZI

ZZ II IZ IZ ZI

IX
XI
XI

II XX IX IX XI

?
?
?

? ? ? II ?

?
?

? ? II ?

II
?

?

SLIDE 47

No local observables distinguish ground states

Proof

?

?
?

? ? ? ? XI

IZ
ZI
ZI

ZZ II IZ IZ ZI

IX
XI
XI

II XX IX IX XI

?
?
?

? ? ? II ?

?
?

? ? II ?

II
?

? Q.E.D.

SLIDE 48

Elementary Excitations (Syndromes)

Can you make a pair of defects out of these ?

SLIDE 49

String?

SLIDE 50

String?

SLIDE 51

String?

SLIDE 52

String?

SLIDE 53

String?

SLIDE 54

String?

SLIDE 55

String?

SLIDE 56

String?

SLIDE 57

String?

SLIDE 58

String?

Surface?

SLIDE 59

String segment

String?

◮ 1D object..... – Not well-defined; only for special models.

SLIDE 60

String segment

String?

◮ 1D object..... – Not well-defined; only for special models. ◮ Need to deal with a family of Hamiltonians.....

SLIDE 61

String segment

String?

◮ 1D object..... – Not well-defined; only for special models. ◮ Need to deal with a family of Hamiltonians.....

Definition

A finite Pauli operator that creates excitations at most two locations.

SLIDE 62

String segment

String?

◮ 1D object..... – Not well-defined; only for special models. ◮ Need to deal with a family of Hamiltonians.....

Definition

A finite Pauli operator that creates excitations at most two locations.

◮ Anchors envelop excitations. ◮ width = the size of the anchors. ◮ length = the distance between the anchors.

SLIDE 63

String segment

String?

◮ 1D object..... – Not well-defined; only for special models. ◮ Need to deal with a family of Hamiltonians.....

Definition

A finite Pauli operator that creates excitations at most two locations.

◮ Anchors envelop excitations. ◮ width = the size of the anchors. ◮ length = the distance between the anchors. ◮ No geometric restriction.

SLIDE 64

Trivial string segments

X X X X X X X Z Z Z Z Z Z Z Z Z Z X X Z Z Z

SLIDE 65

Trivial string segments

X X X X X X X Z Z Z Z Z Z Z Z Z Z X X Z Z Z

◮ Trivial, if there is a equivalent P s.t. supp(P) = C1 ∪ C2 and

dist(C1, C2) > 1.

SLIDE 66

Trivial string segments

X X X X X X X Z Z Z Z Z Z Z Z Z Z X X Z Z Z

◮ Trivial, if there is a equivalent P s.t. supp(P) = C1 ∪ C2 and

dist(C1, C2) > 1.

◮ A trivial string segment only creates trivial charges.

examples...

SLIDE 67

String segment: Example

In 2D toric code:

Z

Z Z Z

SLIDE 68

String segment: Example

In 2D toric code:

Z

Z Z Z

◮ Dots are enclosed by two anchors.

◮ Pauli operator commutes with local terms in H except at the

anchors.

◮ Non-trivial string segment. ◮ Width = 1, length = ∞

SLIDE 69

String segment: Example

In 2D Ising model: H = −

ij ZiZj

SLIDE 70

String segment: Example

In 2D Ising model: H = −

ij ZiZj ◮ Non-trivial X-type string segment. ◮ Two anchors must be adjacent. ◮ length = 0

SLIDE 71

No-strings rule

IZ

ZI
ZI

ZZ II IZ IZ ZI

IX
XI
XI

II XX IX IX XI

A string segment of width w and length ≥ 15w is trivial.

SLIDE 72

No-strings rule

X X X X X X X Z Z Z Z Z Z Z Z Z Z X X Z Z Z

Proof

◮ Any long string segment is equivalent to a product of “flat” ones. ◮ Any long flat segment is trivial.

SLIDE 73

No-strings rule : Proof

Recall the Eraser: IZ

ZI
ZI

ZZ II IZ IZ ZI

IX
XI
XI

II XX IX IX XI

?
?
?

? ? ? ? XI

?
?
?

? ? ? II ?

?
?

? ? ?

◮ IZ–ZI has two independent ends.

SLIDE 74

No-strings rule : Proof

IZ ZI ZI ZZ II IZ IZ ZI

SLIDE 75

No-strings rule : Proof

IZ ZI ZI ZZ II IZ IZ ZI Reduced to flat segments.

SLIDE 76

No-strings rule : Proof

IZ

ZI
ZI

ZZ II IZ IZ ZI

SLIDE 77

No-strings rule : Proof

IZ

ZI
ZI

ZZ II IZ IZ ZI

[O1 − O2, IZ − ZI] = 0

[O1 − O2, II − IZ] = 0

SLIDE 78

No-strings rule : Proof

IZ

ZI
ZI

ZZ II IZ IZ ZI

[O1 − O2, IZ − ZI] = 0

[O1 − O2, II − IZ] = 0 ⇐ ⇒ O1 − O2 =          II − II XI − II IX − XI XX − XI

SLIDE 79

No-strings rule : Proof

IZ

ZI
ZI

ZZ II IZ IZ ZI

[O1 − O2, IZ − ZI] = 0

[O1 − O2, II − IZ] = 0 ⇐ ⇒ O1 − O2 =          II − II XI − II IX − XI XX − XI          II − II − II − · · · XX − XI − II − · · · IX − XI − II − · · · XI − II − II − · · ·

SLIDE 80

No-strings rule : Proof

IZ

ZI
ZI

ZZ II IZ IZ ZI

[O1 − O2, IZ − ZI] = 0

[O1 − O2, II − IZ] = 0 ⇐ ⇒ O1 − O2 =          II − II XI − II IX − XI XX − XI          II − II − II − · · · XX − XI − II − · · · IX − XI − II − · · · XI − II − II − · · · Q.E.D.

SLIDE 81

No-strings rule

◮ You can’t drag the defect. ◮ Annihilate it, and then create it.

SLIDE 82

Code space dimension under periodic boundary conditions

L k L k 2 6 3 2 4 14 5 2 6 6 7 2 8 30 9 2 10 6 11 2 12 14 13 2 14 6 15 50 16 62 17 2

SLIDE 83

Code space dimension under periodic boundary conditions

L k L k 2 6 3 2 4 14 5 2 6 6 7 2 8 30 9 2 10 6 11 2 12 14 13 2 14 6 15 50 16 62 17 2

Formula(?) for odd L

k + 2 4 = degx gcd

1 + (1 + x)L, 1 + (1 + tx)L, 1 + (1 + t2x)L

F4

where t2 + t + 1 = 0.

SLIDE 84

Summary

IZ

ZI
ZI

ZZ II IZ IZ ZI

IX
XI
XI

II XX IX IX XI

◮ Strictly local commuting Hamiltonian, frustration-free.

◮ Defined and proved no-strings rule, with aspect ratio 15. ◮ Exotic degeneracy: k(L = 2p) = 4L − 2, k(L = 2p + 1) = 2. ◮ Logarithmic energy barrier leads to very long memory time