Is entanglement “robust”? - PowerPoint Presentation

Slide1

Is entanglement “robust”?

Lior

Eldar

MIT

/

CTP

Joint work with Aram HarrowSlide2

Highlights

We prove a conjecture of Freedman and Hastings [‘12] that there exist quantum systems of “robust entanglement”- called NLTS.

Tightly connected to the quantum PCP conjecture:

one of the major open problems in quantum complexity theory.

Connected to the conjecture on quantum locally-testable codes

(

qLTC

).

Implies that the folklore that quantum entanglement is “fragile” is an artifact of considering spatially-localized systems.Slide3

Quantum PCP

Complexity perspectiveSlide4

Local Hamiltonians

n

qubits

Sum of few-body terms, WLOG

projectors.

Fractional

energy of a Hamiltonian:Slide5

In complexity theory

Local Hamiltonian problem (LHP):

Decide: is

min

Ψ

E

(

Ψ)=0 or minΨE(Ψ)>1/poly.

QMA - the quantum analog of MA: quantum prover, quantum verifier, bounded error

.

Theorem [

Kitaev

‘98]: LHP is QMA-complete.Slide6

Classical constraint satisfaction problems (CSP)

CSP = classical version of LH. All local terms are diagonal in standard basis.

Like 3-SAT formulas.

PCP theorem [

AS,ALMSS,Dinur

]: NP-hard even to distinguish minimal energy 0 from minimal energy > 0.1Slide7

Quantum PCP: hardness of LHP

Approximate-LHP: Given LH, decide whether there exists a state t such that E(t) = 0 or whether E(t)>1/3 for all states t.

Approximate LHP is NP-hard, and in QMA

.

Quantum

PCP conjecture [AALV ‘09] : Approximate-LHP is QMA-hard

.Slide8

NLTS

Entanglement perspectiveSlide9

Quantify entanglement

U

U

U

U

U

U

U

Complexity of a state = minimal depth circuit

|0>

|0>

|0>

|0>

|0>Slide10

“Robustly-quantum” systems 2:

NLTS

[Freedman, Hastings ‘13]: No low-energy trivial states.

Definition: “trivial states” -

depth = O(1).

NLTS Conjecture: there

is a

constant c>0, and an infinite family of LH’s in which all “trivial states” have fractional energy at least c.Slide11

qPCP NLTS

If NLTS is false

 NP to every Hamiltonian

Suppose

Ѱ

= U |000..0>.

qPCP

NLTS

U

+

U

K-Local

measurement on

Ѱ

,<

Ѱ

|H|

Ѱ

>

k2

O

(d

)

-

local measurement on |00..0>

HSlide12

Main Result

Theorem [E., Harrow ‘15]:

There exists an explicit infinite family of 7-local commuting Pauli Hamiltonians, and a constant c such that:

If a circuit U generates a quantum state whose energy is at most c



U has depth at least

Ω

(log(#

qubits

)).

Proof techniques: products of hyper-graphs, locally-testable codes, Hamiltonian (graph)-powering, degree-reduction, uncertainty principle Slide13

Perspective on NLTSSlide14

Example of highly-entangled

states

: quantum code-states

Fact:

|Ψ

1

>,

|Ψ2> are quantum code-

states, M local measurementsthen <Ψ1

|

M

|Ψ

1

>

=

<Ψ

2

|

M

|Ψ

2

>

Corollary

: quantum code-states require large circuit depth.

Proof

:

Let U be a depth-d circuit : |

Ψ⟩ = U|00..0⟩Define LH : H = Σ

iU|0⟩⟨0|iU+H distinguishes

Ψ from any orthogonal code-state but is 2d-local contradiction.

 no codestate can be locally generated

 Ω(log n) circuit lower-bound.Slide15

Most systems have fragile entanglement

M

any LHs have highly entangled ground states

.

But have tensor product states that are “almost”

ground states

.

Approximating the ground energy is not “crucially” quantum.NLTS means: find (a family of ) LH’s for which the high-entanglement property is “robust”.

Ground-state

Spec(H)Slide16

Why are known LH’s not robust ? Topology !

Known systems are embedded on low-dimensional grids

Approximate by cutting out boxes B:

Fractional energy of

is

In a sense - question is unfair

!Mystery: recent results show that expanding topology per-se is insufficient for NLTS! [BH’13, AE’13, H’12].Slide17

The constructionSlide18

Our goal:

Define a property of quantum states that arise as ground states of LHs.

Show circuit lower bounds for this property.

Show it is “robust” against constant-fraction energy of the parent LH.Slide19

Lower

Bounds for Quantum

CircuitsSlide20

What is this property?Low vertex expansion

Canonical image to remember: Moses parting the red sea !

Essentially: two large-measure sets, separated by large distance, require divine intervention (quantum circuits of logarithmic depth) Slide21

Vertex expansion: analog of Cheeger’s constant for distributions

G = hypercube

{0,1}

n

with edges between vertices of

dist

≤ mV = {0,1}nE = { (x,y) : dist

(x,y) ≤ m}For each S, ∂(S): – the set of strings of S adjacent to

S

c

, union the set of strings in

S

c

adjacent to S.

T

he m-

th

vertex expansion:Slide22

Examples of vertex expansion

High expansion:

A single string (no S, with p(S)<1/2)

Uniform distribution on the hypercube (m at least √n)

Low expansion:

The cat state

Quantum code states

Uniform superpositions over classical codes.Slide23

Bounded-depth

quantum circuits induce high-expansion dist.

Claim: Let U be a quantum circuit of depth d, and

Consider its induced distribution on the first n

qubits

.

Its expansion is at least 1/2 for m > 2

1.5d√n.Slide24

Classical case:

Harper’s theorem

Claim:

Classical circuit C of depth d (bounded fan-in / fan-out), receives n uniform random bits, generates distribution D on n bits. Then D has no large separation.

Proof:

In D - identify two subsets S,T with measure >1/3.

Consider the pre-images under C: C

-1

(S), C-1

(T).

They are at most O(√n) apart. (Harper’s theorem).

So S,T are at distance 2

d

√n.Slide25

Toy proof: CAT state

CAT state

 has low expansion.

Let U be a circuit of depth d

that generates CAT : |CAT> = U |00…0>

Then H = U Σ

i

|0><0|i U+ distinguishes |CAT> from |cat> : |00…0> - |11…1> (because <00..0| U+|cat> = 0)H is 2

d local.Slide26

c

at metric explanation

Local Hamiltonians examine “pairs” |x><y|

Cannot affect pairs (

x,y

) if d(

x,y

) > locality

!

 d(U) > log(n).Slide27

Quantum circuits also limited by Harper

Start with

Ψ

= U |00…0>, depth(U) = d.

Ψ

’s expansion

minimized

at set A.Write Ψ = |A>+|Ac>.Consider the state

Ψ’ = |A>-|Ac>.

A

A

cSlide28

Distinguishing Hamiltonian

Put m = √n2

O(d

)

Consider H = U

Σ

i

|0><0|i U+.Take the √n-degree Chebyshev

polynomial of H, C(H). C(H) is m–local.

By assumption: <

Ψ

’,

Ψ

> is small.



Ψ

,

Ψ

’ have eigenvalues 1, and

<1

-1/poly in H.



Ψ

,

Ψ

’ have eigenvalues 1, <0.7 in C(H).C

laim: this is sufficient to conclude that the distribution of Ψhas high expansion. Why?Slide29

Recall CAT example:

Local Hamiltonians: examine “pairs” of strings in density matrix.

Cannot affect pairs whose distance exceeds locality !Slide30

Large energy gap  high expansion

T

A

A

c

(A,A):same sign in both

(

A

c

,A

c

):same sign in both

∂(A) – only place they differ

 ∂(A) must have large (relative) mass high expansionSlide31

Locally-Testable Codes

are classical

NLTSSlide32

Goal of this section

So far

: low expansion



high circuit depth

Next goal

: Hamiltonian where any

low-energy state has low expansion.Slide33

Local testability

Turns out that “local testability” is a “turnkey” property.

Definition: Locally-testable codes

A subspace

C

of GF(2)

n

.A local tester: set of check terms {Ci}

such that for every word w :Slide34

Example: Hadamard code

Encode x in {0,1}

n

as the function

c(w) = <

w,x

>Express as a truth table of length 2n.Rate is log(n).It is locally testable [BLR] !

Sample a,b at randomAccept if and only if c

(a)

+ c

(b)

= c

(

a+b

)

.

Reads only 3 bits from c

LTC with

ρ

≥ 1/3.Slide35

LTC = Classical NLTS

We would like a

locally defined

system that preserves

low expansion

in the presence of noise.

“

toy” example: distributions on LTCsUniform distribution on a code – low expansionNoisy uniform distribution on a code – could have high expansion.Noisy uniform distribution on an LTC – low expansion !Slide36

How do we make this property quantum ?

A

hypergraph

product [Tillich-

Zémor

‘09]

:Takes in two classical codes CProduces a quantum code Q = C xTZ C.We show:If C is LTC 

Q has a “residual” property of local testability.Informally, means clustering.Slide37

The hyper-graph

product code

[Tillich-

Zémor

‘09]

Parity-

check code

C, bi-partite graph G(A,B).Consider the transpose of C, CT – G(B,A).Generate two new linear codes:Bits are A x A, B x BChecks are X checks: B ⊗

I +I ⊗ A, Z checks: I ⊗

B+A

⊗ I]

AxA

BxB

New check:

query C, C

T

Slide38

Last tool: uncertainty principle

Suppose you have a quantum code with “residual” local testability.

Noisy words cluster around the original code.

But what if

they

all cluster around the same

codeword

?One cluster  no low-expansion !!

What is the answer ? The uncertainty principle ?There is a high-degree of uncertainty  at least two clusters.Slide39

Putting it all together

Take the

Hadamard

code C

Reduce its degree as a bi-partite graph: C

 C’.

Apply the

hypergraph-product to C’: derive a quantum CSS code from a classical code.Argue: the local constraints of the code are an NLTS local Hamiltonian.Slide40

Constructed LH is NLTS!

Fix a low-energy quantum state.

It

superposes

nontrivially on

at least two clusters.

These clusters correspond to different code-words of an LTC with large distance.

Distribution is low expansionSo any U approximating state has d(U) > log(n).Slide41

Clustering around affine spaces !

Live “footage” from F

2

n

_Slide42

Summary of NLTS-LH

Hamiltonian corresponds to quantum CSS code.

n

qubits

, O(n) local terms, each is 7-local.

Evades previous no-go’s:

Not expanding enough for [AE’13]

Not 2-local so doesn’t contradict [BH’13].Low girth so [H’12] doesn’t apply.Rate is only log(n).Distance is √n.Slide43

Take-home messageSlide44

Results

Entanglement is not “inherently fragile”.

Local

Hamiltonians

can “expel”

trivial states from the low side of the spectrum.

They need to have an expanding topology.

Expansion per-se is not enough !You need an extra structure. What is it ?

Local testability !Slide45

Future directions

Find NLTS Hamiltonians that actually do something useful.

To break the log(n) lower-bound, one needs to abandon “light-cone” arguments and encode computational problems…

If you can make it QMA-hard – it’s the

qPCP

conjecture.

Try to find

qLTCs – even with moderate locality.Slide46

Thank you!