XS-Stabilizer Formalism Xiaotong Ni (MPQ) joint work with - - PowerPoint PPT Presentation

XS-Stabilizer Formalism

Xiaotong Ni (MPQ) joint work with Buerschaper, Van den Nest

http://arxiv.org/abs/1404.5327 QIP 15’

Outline

• Motivation
• Example: double semion model
• Summary of properties

• Pauli-S group:
• Given

We call a state XS-stabilizer state if (uniquely) When not unique, we call it XS-stabilizer code

Definition

S = diag(1, i)

Ps

n = hα, X, Si⊗n

α = √ i

G = hg1, . . . , gmi ⇢ Ps

n

|ψi

gj|ψi = |ψi

S−1XS = −iXZ X ⊗ S ⊗ Z

Motivation

Pauli stabilizer formalism

• (Innocently looking) tensor product operators
• Most properties from commutation relation and

linear algebra

• Numerous applications: Fault tolerance,

measurement based computation, etc

XS stabilizer

• (Still innocently looking) tensor product operators
• Many properties from commutation relation and

linear algebra

• Simple to learn

Toric (surface) code

Toric (surface) code

• Practical way to build an active quantum memory

Toric (surface) code

• Practical way to build an active quantum memory
• Great example to understand basic properties of

systems with topological order

• Exactly solvable and simple
• Contains features like anyons, string operators,

boundary, twist, etc.

Bravyi Kitaev 98’ Bombin 10’

XS-stabilizer: double semion and more

S S S S S S X X X X X X Z Z Z

Other motivations

Other motivations

• (Efficient) representation of a larger class of

quantum states

Other motivations

• (Efficient) representation of a larger class of

quantum states

• A class of commuting projector problems that are

in NP (P)

An introduction to the Double semion model

Double semion model

S S S S S S X X X X X X Z Z Z

X

x is close loops

(1)number of loops|xi gp gv gp|ψi = gv|ψi = |ψi |0i |1i Levin, Wen 05’

Z-type operator

S S S S S S X X X X X X Z Z Z

|0i |1i

Gauge invariant subspace

Plaquette operator

X S Z

Plaquette operator

X S Z

2 =

Plaquette operator

• The square is equal to 1 inside gauge invariant subspace

X S Z

2 =

Plaquette operator

• The square is equal to 1 inside gauge invariant subspace

X S Z

2 =

Plaquette operator

• The square is equal to 1 inside gauge invariant subspace
• Eigenvalue of original operator is ±1 inside the subspace

X S Z

2 =

Commutator

X S Z XS XS3 [X,S]=XSX-1S-1=iZ

Commutator

X S Z XS XS3 [X,S]=XSX-1S-1=iZ

=

Commutator

X S Z XS XS3 [X,S]=XSX-1S-1=iZ

= = =

Commutator

X S Z XS XS3 [X,S]=XSX-1S-1=iZ

= = = Different non-black color: Z

Commutator

=

X S Z

Different non-black color: Z

Commutator

=

X S Z

Different non-black color: Z

+1 inside the subspace

Commuting Hamiltonians

• The Plaquette operators are hermitian and

commuting in the gauge invariant subspace

• The gauge invariant subspace = locally project into

the +1 eigenspace of Z-type operators

X S Z

Commuting Hamiltonians

• The Plaquette operators are hermitian and

commuting in the gauge invariant subspace

• The gauge invariant subspace = locally project into

the +1 eigenspace of Z-type operators

X S Z

This is a general procedure!

String operators

X S Z XS XS3

String operators

X S Z XS XS3

String operators

X S Z XS XS3

Commutator

Different non-black color: Z

X S Z XS XS3

=

Twisted quantum double

• Closed loops on each layer, with a phase add to

each configuration

• (A subclass) can be described by XS stabilizer.

Some of them support non-abelian anyons.

Hu, Wan, Wu 2012

Summary of properties

Computational complexity

• Given , is there a state stabilized

by it?

G = hg1, . . . , gmi ⇢ Ps

n

Computational complexity

• Given , is there a state stabilized

by it?

G = hg1, . . . , gmi ⇢ Ps

n

i3Sj ⊗ Sk ⊗ Sl . . .

NP-complete 1 in 3 SAT

Computational complexity

• Given , is there a state stabilized

by it?

G = hg1, . . . , gmi ⇢ Ps

n

Diagonal stabilizers have no S

Efficient Degeneracy 2k

i3Sj ⊗ Sk ⊗ Sl . . .

NP-complete 1 in 3 SAT

Computational complexity

• Given , is there a state stabilized

by it?

G = hg1, . . . , gmi ⇢ Ps

n

Diagonal stabilizers have no S

Efficient Degeneracy 2k

Double semion i3Sj ⊗ Sk ⊗ Sl . . .

NP-complete 1 in 3 SAT

Form of the state

• We can construct a specific basis for the code

space.

• For each , we can efficiently find a circuit of (first)

Clifford and (then) which generate the state

• can be computed for Pauli operator

efficiently.

{ψj}

ψj {T, CS, CCZ}

hψj|P|ψki

P

Entanglement property

Entanglement property

• For a given XS-stabilizer state and a bipartition

(A, B), we can efficiently find a Pauli state and
 such that .

|ψi

|ϕABi

UA ⌦ UB|ψi = |ϕABi

UA ⊗ UB

Entanglement property

• For a given XS-stabilizer state and a bipartition

(A, B), we can efficiently find a Pauli state and
 such that .

• Indirect way to compute entropy for XS states.

|ψi

|ϕABi

UA ⌦ UB|ψi = |ϕABi

UA ⊗ UB

Entanglement property

• For a given XS-stabilizer state and a bipartition

(A, B), we can efficiently find a Pauli state and
 such that .

• Indirect way to compute entropy for XS states.
• Reflects the fact that toric code and double semion

have very similar entanglement properties.

|ψi

|ϕABi

UA ⌦ UB|ψi = |ϕABi

UA ⊗ UB

Flammia et al. 09

Outlooks

Outlooks

• Better codes?

Outlooks

• Better codes?
• A more generalized (and interesting) stabilizer

formalism?

Outlooks

• Better codes?
• A more generalized (and interesting) stabilizer

formalism?

• A larger class of commuting projector problems that

in NP

Outlooks

• Better codes?
• A more generalized (and interesting) stabilizer

formalism?

• A larger class of commuting projector problems that

in NP

• Understanding entanglement properties better

Thanks

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