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

Sergey Bravyi (IBM Research) Jeongwan Haah (Caltech)

QEC 2011December 6, 2011

Phys.Rev.Lett. 107, 150504 (2011)and arXiv:1112.????

Main goal:

Store a quantum state reliably for a macroscopic time in a presence of hardware imperfections and thermal noise without active error correction.

Towards topological self-correcting memories

2D toric code [Kitaev 97]

Robust against small imperfections

Constant threshold with active EC[Dennis et al 2001]

No-go result for the thermal noise[Alicki, Fannes, Horodecki 2008]

No-go result for all 2D stabilizer code[S.B. and Terhal 2008]

No-go result for some 3D stabilizercodes [Yoshida 2011]

?Add one extra dimension to our space-time:[Alicki, Horodecki3 2008]

Most promising ideas:

2D + long range anyon-anyon interactions [Chesi et al 2009, Hamma et al 2009 ]

3D topological quantum spin glasses[Chamon 2005, Haah 2011, this work]

• Encoding, storage, and decoding for memory Hamiltonians based on stabilizer codes

• Memory time of the 3D Cubic Code: rigorous lower bound and numerical simulation

• Topological quantum order, string-like logical operators, and the no-strings rule • Logarithmic energy barrier for uncorrectable errors

Outline

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

Memory Hamiltonians based on stabilizer codes

Hamiltonian = sum of local commuting Pauli stabilizers

energy

0

1

2

3

[N,k,d] error correcting codeDistance d≈ L

Excited states with m=1,2,3… defects

Example: 3D Cubic Code [Haah 2011 ]

ZZZI

ZIIZ

ZIIZ

IZ

XI

XIIX

XIIX

IXXX

2 qubits per site, 2 stabilizers per cube

II

II

Each stabilizer acts on 8 qubits

Stabilizer code Hamiltonians with TQO: previous work

• 2D toric code and surface codes [Kitaev 97]

• 2D surface codes with twists [Bombin 2010]

• 2D topological color codes [Bombin and Martin-Delgado 2006]

• 3D toric code [Castelnovo, Chamon 2007]

• 3D topological spin glass model [Chamon 2005]

• 3D models with membrane condensation [Hamma,Zanardi, Wen 2004] Bombin, Martin-Delgado 2007]

• 4D toric code [Alicki, Horodecki3]The only example ofquantum self-correction

Storage: Markovian master equation

Must be local, trace preserving, completely positive

Evolution starts from a ground state of H.

Lindblad operators Lk act on O(1) qubits and have

norm O(1).

Each qubit is acted on by O(1) Lindblad operators.

Davies weak coupling limit

Lindblad operator transfers energy from the system to the bath (quantum jump).

The spectral density obeys detailed balance:

Heat bath

Memory system

Decoding

Syndrome measurement: perform non-destructive eigenvalue measurement for each stabilizer Ga.

Error correction algorithm

Measuredsyndrome

CorrectingPauli operator

The net action of the decoder:

is the projector onto the subspace with syndrome s

A list of all measured eigenvalues is called a syndrome.

Defect = spatial location of a violated stabilizer,

decoder’s task is to annihilate the defects in a way which is most likely to return the memory to its original state.

Defect diagrams will be used to represent syndromes.

Example:

2D surface code:

Z

Z

X

X

1

3

X-error Z-error

2

42

4

1

3

Creates defects at squares 1,3

Creates defects at squares 2,4

Renormalization Group (RG) decoder*

1. Find connected defect clusters

2. For each connected cluster C

Try to annihilate C by a Pauli operatoracting inside b(C). Record the annihilation operator.

3

4. Increase unit of length by factor 2.

5. Go to the first step

3. Stop if no defects are left.

1 2

4 5

*J. Harrington, PhD thesis (2004), Duclos-Cianci and Poulin (2009)

Measured syndrome

Find the minimum enclosing box b(C).

1

2

RG decoder

1. Find connected defect clusters

2. For each connected cluster C

Try to annihilate C by a Pauli operatoracting inside b(C). Record the annihilation operator.

3. Stop if no defects are left.

Find the minimum enclosing box b(C).

Syndrome after the 1st iteration

RG decoder

Failure 1: decoder has reached the maximum unit of length, but some defects are left.

The decoder stops whenever all defects have beenannihilated, or when the unit of length reached the lattice size.

The correcting operator is chosen as the productof all recorded annihilation operators.

Failure 2: all defects have been annihilated but the correcting operator does not return the system to the original state.

RG decoder can be implemented in time poly(L)

Main goal for this talk:

Derive an upper bound on the worst-case storage error:

Initialground state

Lindbladevolution

RGdecoder

Theorem 1

However, the lattice size cannot be too large:

If we are willing to tolerate error ε then the memory time is at least

Optimal memory time at a fixed temperature is exponential in β2

The storage error of the 3D Cubic Code decays polynomially with the lattice size L. Degree of the polynomial is proportional to β :

The theorem only provides a lower bound on the memory time. Is this bound tight ?

We observed the exponential decay:

Numerical estimate the memory time:

Monte-Carlo simulation

probability of the successful decoding on thetime-evolved state at time t.

Each data point = 400 Monte Carlo samples with fixed L and β

β=5.25β=5.1β=4.9 β=4.7β=4.5 β=4.3

Optimallattice size:

log(L*) as function of β

Exponent inthe power law

as function of β

log(memory time) vs linear lattice size for the 3D Cubic Code

1,000 CPU-days on Blue Gene P

Numerical test of the scaling

Main theorem: sketch of the proof

An error path implementing a Pauli operator P is a finite sequence of single-qubit Pauli errors whose combined action coincides with P.

Energy cost = maximum number of defects along the path.

vacuum

P1 P2 Pt

Energy barrier of a Pauli operator P is the smallest integer m such that P can be implemented by an error path withenergy cost m

Some terminology

Errors with high energy barrier can potentially confuse thedecoder. However, such errors are not likely to appear.

The thermal noise is likely to generate only errors with asmall energy barrier. Decoder must be able to correct them.

Basic intuition behind self-correction:

Lemma (storage error)Suppose the decoder corrects all errors whose energy barrier is smaller than m. Then for any constant 0<a<1 one has

Boltzmann factor Entropy

factor

= # physical qubits

= # logical qubits

Suppose we choose

Then the entropy factor can be neglected:

and

In order to have a non-trivial bound, we need at leastlogarithmic energy barrier for all uncorrectable errors:

More terminology [Haah 2011]

A logical string segment is a Pauli operator whose actionon the vacuum creates two well-separated clusters of defects.

vacuum

The smallest cubic boxes enclosing the two clusters of defects are called anchors

More terminology

A logical string segment is trivial iff its action on the vacuum can be reproduced by operators localized near the anchors:

vacuum

No-strings rule:There exist a constant α such that any logical string segment with aspect ratio > α is trivial.

Aspect ratio = Distance between the anchors Size of the acnhors

3D Cubic Code obeys the no-strings rule with α=15 [Haah 2011]

No 2D stabilizer code obeys the no-strings rule [S.B., Terhal 09]

Theorem 2

Consider any topological stabilizer code Hamiltonian on a D-dimensional lattice of linear size L. Suppose the code has TQO and obeys the no-strings rule with some constant α. Then the RG decoder corrects any error with the energy barrier at most c log(L).

The constant c depends only on α and D.

Haah’s 3D Cubic Code: α=15.

Recall that errors with energy barrier >clog(L) are exponentially suppressed due to the Boltzmann factor. We have shown that

Sketch of the proof: logarithmic lower bound on the energy barrier of logical operators

Idea 1: No-strings rule implies `localization’ of errors

S

E1

S1

E2

S2

E3

S’

E100

A stream of single-qubit errors:

Suppose however that all intermediate syndromes are sparse: the distance between any pair of defects is >>α.

Accumulated error: E= E1 E2 · · · E100 could be very non-local

A stream of local errors cannot move isolated topologically charged defectsmore than distance α away (the no-strings rule).

Localization: E=Eloc · S where S is a stabilizer and Eloc is supported on the α-neighborhood of S and S’

· · ·

Idea 1: No-strings rule implies `localization’ of errors

S

E1

S1

E2

S2

E3

S’

E100

A stream of single-qubit errors:

Accumulated error: E= E1 E2 · · · E100 could be very non-local

Localization: E=Eloc · S where S is a stabilizer and Eloc is supported on the α-neighborhood of S and S’

· · ·

In order for the accumulated error to have a largeweight at least one of the intermediate syndromes must be non-sparse (dense)

Idea 2: scale invariance and RG methods

A stream of local errors cannot move an isolated charged cluster of defects of size R by distance more than αR away.

In order for the accumulated error to have a large weight at least oneof the intermediate syndromes must be non-sparse (dense)

1. Define sparseness and denseness at different spatial scales.

2. Show that in order for the accumulated error to have a REALLY large weight (of order L), at least one intermediatesyndrome must be dense at roughly log(L) spatial scales.

3. Show that a syndrome which is dense at all spatial scalesmust contain at least clog(L) defects.

Definition: a syndrome S is called sparse at level p if it canbe partitioned into disjoint clusters of defects such that

1. Each cluster has diameter at most r(p)=(10 α)p,

2. Any pair of clusters merged together has diameter greater than r(p+1)

Otherwise, a syndrome is called dense at level p.

Lemma (Dense syndromes are expensive) Suppose a syndrome S is dense at all levels 0,…,p. Then S contains at least p+2 defects.

p

e

0 1 2 3

e

e

e

e

4

sparse

Renormalization group method

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

time

RG

leve

l

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

We are given an error path implementing a logical operator Pwhich maps a ground state to an orthogonal ground state.

Record intermediate syndrome after each step in the path.

It defines 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

pmax

Key technical result: 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’.

pmax

Localization of level-p errors

time

RG

leve

l

TQO implies that r(pmax) > L since any logical operator must

be very non-local. Therefore pmax is at least log(L).At least one syndrome must be dense at all levels. Such syndrome must contain at least log(L) defects.

pmax

Conclusions

The 3D Cubic Code Hamiltonian provides the first exampleof a (partially) quantum self-correcting memory.

Memory time of the encoded qubit(s) grows polynomially withthe lattice size. The degree of the polynomial is proportionalto the inverse temperature β.

The lattice size cannot be too big: L< L* ≈ exp(β).

For a fixed temperature the optimal memory time is roughlyexp(β2)