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

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

Sergey Bravyi (IBM Research) Jeongwan Haah (Caltech)

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

QEC 2011December 6, 2011

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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

Robust against small imperfections

Constant threshold with active EC[Dennis et al 2001]

2D toric code [Kitaev 97]

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]

Most promising ideas:

Add one extra dimension to our space-time:

[Alicki, Horodecki3 2008]

?

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

Memory Hamiltonians based on stabilizer codes

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

Hamiltonian = sum of local commuting Pauli stabilizers

energy

3

Excited states with m=1,2,3… defects

2

1

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

0

Example: 3D Cubic Code [Haah 2011 ]

2 qubits per site, 2 stabilizers per cube

IZ

IX

ZI

XI

ZI

ZZ

XI

II

II

XX

IZ

IX

IZ

ZI

IX

XI

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 Lkact on O(1) qubits and havenorm O(1).

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

Heat bath

Memory system

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

The spectral density obeys detailed balance:

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

A list of all measured eigenvalues is called a syndrome.

CorrectingPauli operator

Measuredsyndrome

Error correction algorithm

The net action of the decoder:

is the projector onto the subspace with syndrome s

Defect = spatial location of a violated stabilizer,

Defect diagrams will be used to represent syndromes.

Example:

2D surface code:

1

2

1

2

Z

X

4

3

4

3

Z

X

Z-error

X-error

Creates defects at squares 2,4

Creates defects at squares 1,3

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

Renormalization Group (RG) decoder*

Measured syndrome

1. Find connected defect clusters

1

2

3

2. For each connected cluster C

4

5

Find the minimum enclosing box b(C).

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

3. Stop if no defects are left.

4. Increase unit of length by factor 2.

5. Go to the first step

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

Syndrome after the 1st iteration

1. Find connected defect clusters

1

2. For each connected cluster C

Find the minimum enclosing box b(C).

2

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

3. Stop if no 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 1: decoder has reached the maximum unit of length, but some defects are left.

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)

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

Initialground state

RGdecoder

Lindbladevolution

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

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

Monte-Carlo simulation

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

We observed the exponential decay:

Numerical estimate the memory time:

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

β=5.25β=5.1

β=4.9β=4.7β=4.5β=4.3

Exponent in

the power law

as function of β

Optimallattice size:

log(L*) as function of β

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

1,000 CPU-days on Blue Gene P

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.

P1

P2

Pt

vacuum

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

Basic intuition behind self-correction:

The thermal noise is likely to generate only errors with a

small energy barrier. Decoder must be able to correct them.

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

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

= # physical qubits

Entropy factor

= # logical qubits

and

Then the entropy factor can be neglected:

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

A logicalstring segment is a Pauli operator whose action

on the vacuum creates two well-separated clusters of defects.

vacuum

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

A logical string segment is trivial iff its action on the vacuum

can be reproduced by operators localized near the anchors:

vacuum

There exist a constant α such that any logical string segment with aspect ratio > α is trivial.

Distance between the anchors Size of the acnhors

Aspect ratio =

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]

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 αand 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

A stream of single-qubit errors:

E1

E2

E3

E100

· ··

S’

S

S1

S2

could be very non-local

Accumulated error: E= E1 E2 · ··E100

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

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

A stream of single-qubit errors:

E1

E2

E3

E100

· ··

S’

S

S1

S2

could be very non-local

Accumulated error: E= E1 E2 · ··E100

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)

- Define sparseness and denseness at different spatial scales.
- 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.
- 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 can

- be partitioned into disjoint clusters of defects such that
- Each cluster has diameter at most r(p)=(10 α)p,
- 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.

e

e

sparse

e

e

e

0

1

2

3

4

p

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:

RG level

time

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

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

RG level

time

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

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

RG level

time

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

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.

RG level

time

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

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

Level-2 errors connect consecutive syndromes at level-1.

RG level

time

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

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.

pmax

RG level

time

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

At the highest RG level the syndrome history has no intermediate syndromes.A single error at the level pmax implements a logical operator

Key technical result: Localization of level-p errors

pmax

RG level

time

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

pmax

RG level

time

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.

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)

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