Barriers in Hamiltonian Complexity
Umesh V. Vazirani
U.C. Berkeley
Quantum
Complexity
Theory
Condensed
Matter
Theory
Hamiltonian Complexity
Classical
Quantum
Constraint Satisfaction
Problem
Local Hamiltonian
Solution
Ground State
Condensed Matter Theory
• Describing ground states of local Hamiltonians,
and understanding their properties.
• Problem: n qubit state described by 2n complex
numbers.
• Ground states of realistic systems have concise
descriptions.
• Reason: Limited entanglement.
Area Laws
Assuming area laws, beautiful sequence of results showing how
to simulate quantum systems efficiently using tensor networks,
MERAs and PEPs. [Vidal; Verstraete & Cirac, ...]
Quantum Complexity Theory
Quantum
H  H1  L  H m
Local Hamiltonian
[Kitaev] QMA-hard
to find ground state
Classical
f (x1 ,L , xn )  c1  L  cm
k-SAT
NP-hard to find
assignment min
# UNSAT clauses
Quantum Complexity Theory
Quantum
H  H1  L  H m
Local Hamiltonian
[Kitaev] QMA-hard
to find ground state
Classical
f (x1 ,L , xn )  c1  L  cm
k-SAT
NP-hard to find
assignment min
# UNSAT clauses
[Gottesman, Irani 09] Ground states of
translationally invariant 1-D Hamiltonians
hard unless BQEXP = QMAEXP
Quantum Complexity Theory
Quantum
H  H1  L  H m
Local Hamiltonian
??
QMA-hard to
find any low energy
state?
Classical
f (x1 ,L , xn )  c1  L  cm
k-SAT
NP-hard to find
assignment min approx
# UNSAT clauses
PCP Theorem
Two Major Challenges in Hamiltonian
Complexity
• Prove or disprove a quantum PCP theorem.
• Prove the area law for 2-D and 3-D gapped
local Hamiltonians.
Local Hamiltonians
• n qubit system
• Hamiltonian: H = 2n x 2n hermitian matrix.
• Energy operator: eigenstates of H are states with definite
energy. Energy = eigenvalue.
H  H1  L  H m
•
k-local if each term acts non-trivially on k qubits.
• Each term assigns energy penalty to state.
• Interested in structure and eigenvalue (energy) associated
with lowest eigenstate (ground state).
3SAT as a local Hamiltonian Problem
f (x1 ,L , xn )  c1  L  cm
•
•
n bits ---> n qubits
Clause ci = x1 v x2 v x3 corresponds to 8x8 Hamiltonian
 1

matrix acting on first 3 qubits:


0



hi  





•
•
0



0

0

0


0

0 
Satisfying assignment is eigenvector with evalue 0.
All truth assignments are eigenvectors with
eigenvalue = # unsat clauses.
[Kitaev] Given a local hamiltonian H = H1 + ... + Hm
it is QMA-hard to determine the minimum eigenvalue
(ground state energy) of H to within 1/poly(n)
PCP Theorem: Polynomial time procedure to
convert 3-SAT formula f into g:
• f satisfiable implies g satisfiable
• f unsatisfiable implies g < 1-c satisfiable.
Quantum PCP Formulation
[Kitaev] Given a local hamiltonian H = H1 + ... + Hm it is QMA-hard to
determine the minimum eigenvalue (ground state energy) of H to
within 1/poly(n).
PCP Theorem: Polynomial time procedure to convert 3-SAT
formula f into g:
• f satisfiable implies g satisfiable
• f unsatisfiable implies g < 1-c satisfiable.
Quantum PCP: Given a local hamiltonian H = H1 + ... + Hm
is it QMA-hard to determine the minimum eigenvalue (ground
state energy) of 1/m H to within c for some constant c?
i.e. Is there a quantum poly time alg that converts any local
hamiltonian H into H’ such that |H’| = O(1) and if H has promise
gap 1/poly(n) then H’ has promise gap constant c.
•
Well balanced question: No strong intuition to
call it a quantum PCP conjecture.
•
[Aharonov, Arad, Landau, V 2008] Proof of
quantum gap amplification using the detectability
lemma. Dinur’s proof uses GA + degree reduction +
alphabet reduction.
•
Can define quantum PCP in terms of proof
checking. i.e. is there a quantum state that can be
checked by accessing only constant number of
qubits. The two definitions are equivalent.
Area Law
For gapped local Hamiltonians H = H1 + ... + Hm , ground
state has low entanglement. Gapped = e1 - e0 > c.
[Hastings 2007] Proof of area law for 1-D systems.
[Aharonov, Arad, Landau, V 2010] Simplified proof for
frustration-free 1-D systems, using detectability lemma.
Area Law
For gapped local Hamiltonians H = H1 + ... + Hm , ground
state has low entanglement. Gapped = e1 - e0 > c.
How to quantify entanglement?
Quantifying Entanglement
A
B
   ci ai  bi
Schmidt decomposition: {|ai>}, {|bi>} orthonormal sets
Entanglement rank = number of non-zero terms
Entanglement rank = 1 iff product state.
Entanglement entropy = classical entropy of {ci2}
Detectability Lemma
• H = H1 + ... + Hm
Assume Hi = I - Pi where Pi is a
projection matrix. Assume gap = e1 - e0.
• Frustration-free: Assume ground energy = 0. i.e.
ground state satisfies all m constraints.
• The normalized operator G = (I - 1/m H) fixes
the ground state, but shrinks all other evectors by
a factor of (1 - gap/m). So if H is gapped,
i.e. gap = constant, then shrinkage ~ 1/m.
•
Can a local operator achieve constant shrinkage?
Overall Idea of Proof (of 1D area law)
• Step 1: Show that there is a product state
|a> x |b> which has constant overlap (inner
product) with the ground state.
• Step 2: Show that this implies that the ground
state has constant entanglement.
Overall Idea of Proof (of 1D area law)
• Step 1: Show that there is a product state |a> x |b> which
has constant overlap (inner product) with the ground state.
• Step 2: Show that this implies that the ground state has
constant entanglement.
To prove step 2, repeatedly apply a transformation
to |a> x |b> that moves it closer to the ground state
without increasing its entanglement entropy much.
The detectability lemma gives exactly such a
transformation.
Overlap r implies entropy = O(1/e log 1/re log D)
A
B
To prove Step 1: Assume for contradiction that the maximum overlap
between ground state and a product state is at most 2-l for some large
constant l.
Consider the product state above corresponding to the ground state.
Since it has small overlap with the ground state, there is a measurement
that can distinguish the two with probability at least 1 - 2-l.
Use the detectability lemma to show that such a measurement can be done
locally (on O(l) qudits).
Conclude that the entanglement across the boundary is proportional to l.
The Numbers
• Measurement on 2l qudits distinguishes product state from
ground state with probability 1 - exp(-el), where e = gap.
• This implies entanglement entropy of el across this
boundary.
•
Now by monogamy of entanglement:
el
el/2
• e l log l < l log D
el/2
=> l < exp(1/e log D)
• So overlap > exp(-el), with l < exp(1/e log D)
• Overlap r implies entropy = O(1/e log 1/re log D)
• So entanglement entropy = O(1/e log D exp(1/e log D))
Conclusions
•
Proving area law in more than 1-D and quantum
PCP theorem are two major challenges.
•
To prove 2-D case sufficient to consider
frustration-free Hamiltonians. i.e. detectability
lemma applies.
•
Would be interesting to know if area law breaks
down for any interaction graph.