On the energy landscape of 3D spin Hamiltonians
with topological order
Sergey Bravyi
(IBM)
Jeongwan Haah (Caltech)
FRG Workshop,
Cambridge, MA
May 18, 2011
Outline
• 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
magnetic field h
Classical self-correcting memory: 2D Ising model
L
Stability region
Tc
T
L
H = ¡ J
¡ h
P
( i ;j )
P
j
¾j
¾i ¾j
free energy
Macroscopic energy barrier
JL
magnetization
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
Lattice size
L=256
Metropolis dynamics:
• If a spin flip would decrease energy, flip it
• Otherwise, flip it with probability
Quantum memory Hamiltonians based on stabilizer codes
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.
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
quantum bits
Storage
Interaction with a thermal bath kept at some fixed temperature
over time t.
Decoding
Encoding
Use quantum system to store
classical bits
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 energy
barrier for logical X operators.
Need macroscopic energy
barrier 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:
Bacon, quant-ph/0506023
S.B. and Terhal, arXiv:0810.1983
Pastawski et al, arXiv:0911.3843
1. Ground states are locally indistinguishable (TQO)
An operator acting on a `microscopic’ subsystem has the same expectation
value on all ground states.
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.
h
h= 0
Tc
T
Phase Diagram of Classical Ising model
in D > 1 dimension. Stores a classical
bit reliably when h=0 and T<Tc
Conjectured & proved phase diagrams for toric code Hamiltonians*
h
h
D=2
T
Degenerate ground
state stores a qubit
reliably at T=0,
even for nonzero h.
Unstable for T>0.
h
D=3
Tc
D =4
T
Tc
T
Stores a qubit at
Stores a quantumT=0. For T>0,
encoded qubit even
stores a quantumat nonzero T and h.
encoded classical *Alicki, Fannes, Horodecki,
bit, probably even
0811.0033…
S.B., Hastings, Michalakis,
when h is nonzero
1001.0344
Kitaev 1997: toric code models
Spins ½ (qubits) live on links
0
1
1
0
1
0
0
-1
Stabilizers
Star operators:
Plaquette operators:
Logical string-like operators
Star operators:
X
X
X
X
X
X
String
Applying a string-like operator to a ground state creates a pair of
topologically 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.
Logical-Z operator
Logical-X operator
X
Z
X
X
X
X
X
Z
Z
Z
Z
and
commute with all stabilizers.
and
have non-trivial action on the ground subspace
Logical string-like operators
X
energy barrier is O(1)
X X
X X X X X
X X X
X
X 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.
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 no
logical string-like operators (Jeongwan Haah arXiv:1101.1962 )
Z2
Z1
Z1
Z 1Z 2
Z1
X1
X1X2
X2
X1
Z2
Z2
X2
X2
X1
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
Main question for this talk:
Suppose a stabilizer Hamiltonian has TQO but does not
have logical string-like operators.
What can we say about its energy barrier?
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
Neutral cluster
X X
X
X X
X X
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)
X
X
Z
X
Z
Y
Y
Z
Y
Z
Z
Z
X
X
Y
Y
Y
Y Z
X
Y
X
X
Z
X
Y
Y
- Pauli operator
e e
e e
e
e e
Applying P to the vacuum creates excitations only inside
P is a logical string segment.
- anchors of the string
A logical string segment is trivial iff the cluster of excitations
inside each anchor region is neutral.
Trivial logical string segments are ``fake strings”:
e e
e e
e
e e
e e
e e
e
e e
for all ground states
e
e e
e e
e e
Linear size of the anchors: w,
Distance between anchors: R12
Aspect ratio:
Theorem
Consider 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
The constant c depends only on α and spatial dimension D.
Haah’s code: α=15. The lower bound is tight up to a constant.
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
e
e
e
R
αR
Dynamics of neutral isolated clusters is irrelevant as
long as such clusters remain isolated.
The no-strings rule is scale-invariant. Use RG approach
to analyze the dynamics of excitations.
Sparse and dense syndromes
Syndrome is a cluster of excitations created by some error.
Excitations live on elementary cubes of the lattice.
Define a unit of length at a level-p of RG:
Use
distance. Elementary cubes have diameter =1.
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 diameter
greater than r(p+1)
Renormalization group method
RG level
A sequence of local errors implementing a logical operator P
defines level-0 syndrome history:
time
0 = vacuum, S = sparse syndromes, D= dense syndromes
Level-0 syndrome history. Consecutive sydnromes are related by single-qubit
errors. Some syndromes are sparse (S), some syndromes are dense (D).
RG level
Renormalization group method
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
Renormalization group method
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
Renormalization group method
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
Renormalization group method
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.
RG level
Renormalization group method
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
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
r(1)
r(2)
r(3)
e
e
e
RG level
Localization of level-p errors
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
Localization Lemma
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 error
supported only on r(p)-neighborhood of S U S’.
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 defects
QED
Logical operators of the Haah’s code
with the logarithmic energy barrier
Z2
Z1
Z2
Z1
e
Z2
e
Z 1Z 2
Z1
Stabilizer of Z-type
e
e
Excitations created by a
single X1 error located in the
center of the cube (dual lattice)
This cluster of excitations is called level-0 pyramid
e
X
X
X
e
X
e
X
X
e
X
X
Level-p pyramid
X
X
X
X
X
X
X
X
Corresponding error
(level p=2)
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
16
Optimal error path creating
level-p pyramid = concatenation
of four optimal error paths
creating level-(p-1) pyramids
13
14
4
1
8
2
5
Energy barrier:
6
Becomes a logical operator for p=log2(L)
15
12
3
9
10
7
11
Conclusions
Quantum self-correction requires two properties:
- ground states are locally indistinguishable (TQO)
- macroscopic energy barrier
All stabilizers Hamiltonians in 2D have string-like logical
operators. The energy barrier does not grow with lattice size.
Some stabilizer Hamiltonians in 3D have no string-like
logical operators. Their energy barrier grows logarithmically
with the lattice size.