On the energy landscape of 3D spin Hamiltonians with topological order

Sergey Bravyi (IBM) Jeongwan Haah (Caltech)

FRG Workshop,Cambridge, MAMay 18, 2011

• Classical and quantum self-correcting memory

• No-go theorem for quantum self-correction in 2D

• String-like logical operators

• 3D spin Hamiltonian with conjectured self-correction • Renormalization group lower bound on the energy barrier

Outline

L

Classical self-correcting memory: 2D Ising model

L

H = ¡ JP

(i ;j ) ¾i¾j

¡ hP

j ¾j

free

ene

rgy

magnetization

TTc

mag

netic

fiel

d h Stability region

Macroscopic energy barrier

J L

Stores a classical bit reliably when h=0 and T<Tc

Poor quantum memory: ground states are locally distinguishable

β=0.48

β =0.45

β =0.44

β =0.40

• If a spin flip would decrease energy, flip it

• Otherwise, flip it with probability

Lattice size L=256

Metropolis dynamics:

1. Qubits live at sites of a 2D or 3D lattice. O(1) qubits per site.

2. Stabilizers Ga live on cubes. O(1) stabilizers per cube.

Quantum memory Hamiltonians based on stabilizer codes

3. Stabilizers Ga are products of I, X, Y, and Z

4. Ground states are +1 eigenvectors of all stabilizers.

Use quantum system to store classical bits

Use quantum system to store quantum bits

En

cod

ing

Sto

rag

eD

eco

din

g

Interaction with a thermal bath kept at some fixed temperature over time t.

Error correction:(1) Measure eigenvalue of every stabilizer. It reveals error syndrome S.(2) Guess the most likely equivalence class of errors consistent with S. (3) Return the system to the ground state.

Only need macroscopic energybarrier for logical X operators.

Need macroscopic energybarrier for logical X, Y, Z operators.

Storage phase

Analogues of spin flips: Pauli operators X, Y, Z

Stabilizer Hamiltonians: Pauli operators map eigenvectors to eigenvectors.

Metropolis dynamics is well-defined.

Necessary conditions for quantum self-correction:

1. Ground states are locally indistinguishable (TQO)

Bacon, quant-ph/0506023S.B. and Terhal, arXiv:0810.1983 Pastawski et al, arXiv:0911.3843

2. Macroscopic energy barrier

Any sequence of single-qubit errors X,Y,Z mapping one ground state to another must cross an energy barrier. Infinite barrier in the thermodynamic limit.

An operator acting on a `microscopic’ subsystem has the same expectation value on all ground states.

h =Tc

Phase Diagram of Classical Ising model in D > 1 dimension. Stores a classical bit reliably when h=0 and T<Tc

0

h

Conjectured & proved phase diagrams for toric code Hamiltonians*

D = 2

Tc

D = 3

Tc

D =4

Degenerate ground state stores a qubit reliably at T=0,even for nonzero h. Unstable for T>0.

Stores a qubit at T=0. For T>0, stores a quantum-encoded classical bit, probably even when h is nonzero

Stores a quantum-encoded qubit even at nonzero T and h.

*Alicki, Fannes, Horodecki, 0811.0033… S.B., Hastings, Michalakis, 1001.0344

T

h

T

h

T

T

h

Kitaev 1997: toric code models

Star operators: Plaquette operators:

Spins ½ (qubits) live on links

0 1

1 0

1 0

0 -1

Stabilizers

Logical string-like operators

String

Star operators:

X X X

X

X

X

Applying a string-like operator to a ground state creates a pair oftopologically non-trivial excitations at the end-points of a string.

Contractible closed strings = product of plaquette operators.

Similar string-like operators of Z-type exist on the dual lattice.

X X X X X XZ

Z

Z

Z

Z

and commute with all stabilizers.

and have non-trivial action on the ground subspace

Logical-X operator Logical-Z operator

Logical string-like operators energy barrier is O(1)

X X X X X X

X X XXX X X X X

Sequence of local X-errors implementing logical-X operator.At each step at most 2 excitations are present.Energy barrier = 4 regardless of the lattice size.

XX

2D and 3D Hamiltonians with TQO: negative results

Any 2D stabilizer Hamiltonian has logical string-like operators. No-go result for quantum self-correction.Bravyi, Terhal 2008,Bravyi, Terhal, Poulin 2009,Kay, Colbeck 2008,Haah, Preskill 2011

Logical string-like operators were found for all 3D stabilizer Hamiltonians studied in the literature.

Chamon 2005,Bravyi, Terhal, Leemhuis 2010,Hamma, Zanardi, Wen 2005,Castelnovo, Chamon 2007,Bombin, Martin-Delgado 2007.

Recent breakthrough: 3D model which provably has nological string-like operators (Jeongwan Haah arXiv:1101.1962 )

Z1Z2Z1

Z1Z2

Z1Z2

Z2

X1

X1X2

X1X2

X2X1X2

Qubits live at sites of 3D cubic lattice (2 qubits per site)

Stabilizers live at cubes (2 generators per cube)

Topological quantum order

Not quite realistic: periodic boundary conditions for all coordinates

Suppose a stabilizer Hamiltonian has TQO but does nothave logical string-like operators.

What can we say about its energy barrier?

Main question for this talk:

First need to define rigorously TQO and string-like logical operators…

Definition: a cluster of excitations S is called neutral iff S can be created from the vacuum by acting on a finite number of qubits.Otherwise S is called charged.

X X X X X X X X X

Neutral cluster Charged cluster

Topological Quantum Order: 1. Ground states cannot be distinguished on cubes of

linear size Lb for some b>0. 2. Let S be a neutral cluster of excitations. Let C(S) be the smallest cube that contains S. Then S can be created from the vacuum by acting only on qubits of C(S).

Logical string segments (Haah arXiv:1101.1962)

- Pauli operator

XX

X

X

X

X

X

X

X

YY

Y

Y

Y

Y

Y

YY

YZ

Z

Z

Z Z

Z

Z

Z

e ee e

ee e

Applying P to the vacuum creates excitations only inside

P is a logical string segment. - anchors of the string

e ee e

ee e

A logical string segment is trivial iff the cluster of excitations inside each anchor region is neutral.

e ee e

ee e

for all ground states

Trivial logical string segments are ``fake strings”:

e ee e

ee e

Linear size of the anchors: w,Distance between anchors: R12

Aspect ratio:

TheoremConsider any stabilizer code Hamiltonian with TQO. Local stabilizers on a D-dimensional lattice with linear size L.

Suppose there exists a constant α such that any logical string segment with the aspect ratio greater than α is trivial.

Then the energy barrier for any logical operator is at least

Haah’s code: α=15. The lower bound is tight up to a constant.

The constant c depends only on α and spatial dimension D.

Lower bound on the energy barrier: renormalization group approach

Absence of long logical string segments = `no-strings rule’

Absence of long logical string segments = `no-strings rule’

No-strings rule implies that charged isolated clusters of excitations cannot be moved too far by local errors:

e

Absence of long logical string segments = `no-strings rule’

No-strings rule implies that charged isolated clusters of excitations cannot be moved too far by local errors:

e

ee

e

R

α R

Dynamics of neutral isolated clusters is irrelevant aslong as such clusters remain isolated.

The no-strings rule is scale-invariant. Use RG approachto analyze the dynamics of excitations.

Sparse and dense syndromes

Define a unit of length at a level-p of RG:

Definition: a syndrome S is called sparse at level p if the subset of cubes occupied by S can be partitioned into disjoint union of clusters such that

1. Each cluster has diameter at most r(p),

2. Any pair of clusters combined together has diametergreater than r(p+1)

Use distance. Elementary cubes have diameter =1.

Syndrome is a cluster of excitations created by some error.Excitations live on elementary cubes of the lattice.

Renormalization group method

0 = vacuum, S = sparse syndromes, D= dense syndromes

time

RG

leve

l

Level-0 syndrome history. Consecutive sydnromes are related by single-qubiterrors. Some syndromes are sparse (S), some syndromes are dense (D).

A sequence of local errors implementing a logical operator Pdefines level-0 syndrome history:

Renormalization group method

0 = vacuum, S = sparse syndromes, D= dense syndromes

time

RG

leve

l

Level-1 syndrome history includes only dense syndromes at level-0. Level-1 errors connect consecutive syndromes at level-0.

Renormalization group method

0 = vacuum, S = sparse syndromes, D= dense syndromes

time

RG

leve

l

Level-1 syndrome history includes only dense syndromes at level-0. Level-1 errors connect consecutive syndromes at level-0. Use level-1 sparsity to label level-1 syndromes as sparse and dense.

Renormalization group method

0 = vacuum, S = sparse states, D= dense states

time

RG

leve

l

Level-2 syndrome history includes only dense syndromes at level-1. Level-2 errors connect consecutive syndromes at level-1.

Renormalization group method

0 = vacuum, S = sparse states, D= dense states

time

RG

leve

l

Level-2 syndrome history includes only dense excited syndromes at level-1. Level-2 errors connect consecutive syndromes at level-1. Use level-2 sparsity to label level-2 syndromes as sparse and dense.

Renormalization group method

0 = vacuum, S = sparse syndromes, D= dense syndromes

time

RG

leve

l

At the highest RG level the syndrome history has no intermediate syndromes.

A single error at the level pmax implements a logical operator

Cluster Merging Lemma. Suppose a syndrome S is dense at all levels 0,…,p. Then S contains at least p+2 excitations.

Example: suppose non-zero S is dense at levels 0,1,2

e e e e

r(1)

r(2)

r(3)

Localization of level-p errors

time

RG

leve

l

No-strings rule can be used to `localize’ level-p errors by multiplying them by stabilizers.

Localized level-p errors connecting syndromes S and S’act on r(p)-neighborhood of S and S’.

Localization of level-p errors

Let S and S’ be a consecutive pair of level-p syndromes.Let E be the product of all level-0 errors between S and S’.Let m be the maximum number of excitations in the syndrome history. Suppose that p is sufficiently small:

Then E is equivalent modulo stabilizers to some errorsupported only on r(p)-neighborhood of S U S’.

Localization Lemma

A single error E at the highest RG level implements a logical operator.The smallness of p condition in Localization Lemma must be violated at this level. Otherwise the lemma says that E is a stabilizer. Therefore

We can assume that m=O(log L), otherwise we are done. Therefore

The history must contain a syndrome which is dense at all levels below pmax-1.

Cluster Merging Lemma implies that such syndrome has at least pmax defectsQED

Z1Z2Z1

Z1Z2

Z1Z2

Z2

Logical operators of the Haah’s code with the logarithmic energy barrier

Excitations created by a

single X1 error located in the

center of the cube (dual lattice)

Stabilizer of Z-type

e

e

e

e

This cluster of excitations is called level-0 pyramid

e

e

e

e

Level-p pyramidCorresponding error (level p=2)

X

X

X

X

X

X X X

X

X

XXX

XX

X

Suppose the lattice size is L=2p for some integer p.Then level-p pyramid = vacuum.The corresponding error is a logical operator of weight L2

Energy barrier for pyramid errors

6

5

8

7

2

1 3 9

10

11

12414

1513

16

Optimal error path creating level-p pyramid = concatenation of four optimal error paths creating level-(p-1) pyramids

Energy barrier:

Becomes a logical operator for p=log2(L)

Conclusions

Quantum self-correction requires two properties: - ground states are locally indistinguishable (TQO) - macroscopic energy barrier

All stabilizers Hamiltonians in 2D have string-like logicaloperators. The energy barrier does not grow with lattice size.

Some stabilizer Hamiltonians in 3D have no string-like logical operators. Their energy barrier grows logarithmicallywith the lattice size.