1. 2nd Lecture: QMA & The local Hamiltonian problem

2. Classical and Quantum Computation
Turing Machine
Circuits Cn A Q B C ≈
time
Quantum Turing Machine…
[Deutch’85]
Quantum circuits [Yao’89]
● Input:
● Gates
● Measure U1 …. U5 U4 U3 U2
Very different than the turing machine…
time
Running time: number of gates L.

3. A Computational complexity mapNP
BQP: Class of problems solvable in
polynomial time by quantum computers
QNP
BPP: Class of problems solvable in
polynomial time by classical computers
BPP
BQP P
factoring 3 3
Computational point of view about the difference between quantum and classical
How can the computational point of view on physics enlighten us when it comes to understanding physics?
All physically realizable
computational models can be
simulated in poly time by a Turing
machine” (Extended CTT)
Widely believed:
QC violates ECTT
BQP is strictly larger than BPP,
Quantum Systems can in principle
physically implement BQP

4. Dawn of Quantum Hamiltonian Complexity…
K-SAT
NP-Completeness theory
Quantum SAT?
Quantum NP?
Quantum hardness?
Cook-Levin’71: k-SAT is NP-complete
Quantum Cook-Levin:
Kitaev’98: Local Hamiltonian problem is quantum NP-complete.
Since then: QNP-hardness for a variety of physical systems. 9

5. Computer Science Condensed Matter Physics CSP is a special case!
Major CS problem:
Constraint Satisfaction
Problem (CSP)
Major CMP problem:
The Local
Hamiltonian
Problem:
Given: CSP formula
Objectives:
Min. # of Violations
Optimal assignment
Approximations
Given: Local Hamiltonian
Objective: Ground value,
ground state(s)
Reductions between Hamiltonians
Gadgets
On the board:
Recall what’s a Hamiltonian (Hermitian, energies, eigenstates)
Define local Hamiltonian
How a kSAT can be viewed as a special case of k-Local Ham.
The # number of violations  groundenergy.

6. Example for CSPs: Spin glass 
Which  spin distribution minimizes
red green
(1 violation.)
Want to be different
Want to be the same
violation  Energy Penalty: Project on unsatisfying values of x
The lowest energy state (ground state) of the spin glass is the solution to our optimization problem.

7. More generally: Quantum constraints 
Now the terms need not be diagonal in the comp. basis
Energy Penalty: Project on an unsatisfying subspace
The lowest energy state (ground state) of the system is the
solution to our quantum optimization problem.
#violations  Energy

8. Computer Science Condensed Matter Physics   CSP is a special case!
Multiparticle
Entanglement
Major CS problem:
Constraint Satisfaction
Problem (CSP)
Major CMP problem:
The Local
Hamiltonian
(LH) Problem:
Given: CSP formula
Objectives:
Min. # of Violations
Optimal assignment
Approximations
Given: Local Hamiltonian
Objective: Ground state(s)
Reductions between Hamiltonians
Gadgets
Strings – not interesting from a
comp complexity perspective  
Apply the computer science questions to
the extremely difficult physics situations

9. Quantum NP (QMA) NP Quantum NP (QMA) (Completeness & soundness)
Q Verifier
Verifier
Why focus on this problem? Know a lot about the structure of multi-particle entanglement in this case, structure theorems, Topological order, etc.
Quantum error correcting codes
X in L: Exists Ψ s.t. Pr(Q accepts)>2/3
X not in L: for all Ψ, Pr(Q accepts)<1/3
(Completeness
& soundness)
On the board:
Amplification when |c-s|>1/poly
Mention QCMA

10. The Local Hamiltonian problem
Given: Local Hamiltonian H on n qubits,
Terms are Projections, or PSD
b-a>1/poly(n)
Objective: Is min. eigenvalue of H<a or >b?
The Quantum-Cook-Levin Theorem [Kitaev’98]
The local Hamiltonian problem is QMA complete
Quantum NP (QMA)
Why focus on this problem? Know a lot about the structure of multi-particle entanglement in this case, structure theorems, Topological order, etc.
Quantum error correcting codes
X in L:
Exists Ψ s.t. Pr(Q accepts)>2/3
X not in L:
for all Ψ, Pr(Q accepts)<1/3
Q Verifier

11. The Easier direction: LH is in QMA
The Local Hamiltonian problem (LH):
Given: Local Hamiltonian H on n qubits,
Terms are Projections, or PSD
b-a>1/poly(n)
Objective: Is min. eigenvalue of H<a or >b?
Quantum NP (QMA)
X in L:
Exists Ψ s.t. Pr(Q accepts)>c
X not in L:
for all Ψ, Pr(Q accepts)<s
c-s>1/poly
Q Verifier
Why focus on this problem? Know a lot about the structure of multi-particle entanglement in this case, structure theorems, Topological order, etc.
Quantum error correcting codes
Claim: Local Hamiltonian is in QMA
On the board: the proof