1. 3rd Lecture: QMA & The local Hamiltonian problem (CNT’D)

2. What did we do Yesterday…
Recap:
What did we do Yesterday…

3. The Quantum Circuit model+ a b 1 1
Qubits, states, measurements, gates
Quantum circuits
● Input:
● Gates
● Measure U1 …. U5 U4 U3 U2
Very different than the turing machine…
time
Running time: number of gates L.

4. A Computational complexity mapNP
BQP: Class of problems solvable in
polynomial time by quantum computers
BPP: Class of problems solvable in
polynomial time by classical computers
BPP
BQP
factoring P 4 4
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

5. N=PQ N Q P Quantum algorithms using Fourier sampling Prime factors
of Integers
N=PQ
Deutsch
Josza [‘92] N
Bernstein
Vazirani[‘93]
Shor’94
Simon
[‘94] Q P
Computational point of view about the difference between quantum and classical
Bpp VERSUS bqp

6. Interference between exponentially many paths
Classical computation: moving from one configuration to the next. Quantum: moving to superpositions of configurations. :
Least positive solutions for Pell’s equation x^2-dy^2=1
Fourier transform discovers periodicity
Can work in parallel with a huge number of computational paths which “communicate” between themselves
Using interference we can make sure that paths leading to wrong answers will cancel each other

7. Quantum Computational Hardness?
K-SAT
NP-Completeness theory
Quantum SAT?
Quantum NP?
Quantum hardness?
Cook-Levin’71: k-SAT is NP-complete 9

8. Classical CSPs as Local Hamiltonians: Spin glass 
Which  spin distribution minimizes
red green
(1 violation.)
Want to be different
Want to be the same
The groundstate of H is the solution of optimization problem.

9. 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  
Which groundstates have efficient description?

10. Towards proving LH is QMA complete
The Local Hamiltonian problem (LH):
Given: Local Hamiltonian H on n qubits,
Terms are Projections
b-a>1/poly(n)
Objective: Is min. eigenvalue of H>b or <a?
Quantum NP (QMA)
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
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
We proved the easy direction: LH is in QMA with c-s>1/poly, by having
the verifier pick a random constraint and measure it,
and accepting if the state happens to be in its groundspace.

11. Kitaev’s Ciruit-to-Hamiltonian
Today:
QMA hardness of LH
&
Kitaev’s Ciruit-to-Hamiltonian
Construction.

12. The interesting direction: LH is QMA hard
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>b or <a?
Quantum NP (QMA)
X in L:
Exists Ψ s.t. Pr(Q accepts)>1-1/exp(n)
X not in L:
for all Ψ, Pr(Q accepts)<1/exp(n)
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
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
We now want to show how every quantum verification circuit V
Can be mapped into a local Hamiltonian whose ground-energy will
Indicate whether there exists a state which V accepts with good prob,
or V rejects all states with good probability

13. Why not mimic Cook-Levin’s proof?
Cook-Levin: K-SAT is NP complete.
Q-Cook-Levin: LH is QMA complete [Kitaev’99]
Q Verifier
Verifier
Time
steps
Can’t compare states locally
But can compare states which are as complicated as we want
Reductions between Hamiltonians
Put CSP and LH on the board :
Computation is local
Problem!

14. Entanglement, Ex. II: Inner Product Estimation
Two distributions over n bit strings.
Are they equal or
their supports do not intersect?
need exp(n) many samples.
Nucleos spin – in a large constant magnetic field, apply a perturbation of an oscilating magnetic field in resonant frequency. Generates rotations of the spin.
Can estimate <P|Q> efficiently
(by measuring the left qubit)
On the board –
Compute the prob above. Extend to show how to locally test propagation of one step

15. Using entanglement for local tests
Given: Local Hamiltonian H on n qubits ,
a,b s.t. b-a>1/poly(n)
Objective: Is min. eigenvalue of H <a or >b
Time
steps :
Computation is local
Can’t compare states locally
But can compare states which are as complicated as we want
Put CSP and LH on the board

16. A B Circuit to Hamiltonian construction: WHY? Dynamics to Statics
Universality of Adiabatic evolution [A’KempeLandauLloydRegevVanDam’04]
(QMA hardness often goes together w/ universality. see David’s lecture)
2. Hardness of the Physics “Density functional theory” [SchuchVerstraete’09]
3. Creation of Hamiltonians with “adversarially” highly entangled Gstates
[Irani’09, GottesmanHastings’09, A’HarrowLandauNagajSzegedyVazirani’14]
Creation of approximate quantum codes with local constraints [NirkheVaziraniYuen’18] …. A B
Deep: Physics and CS; Computational hardness, local versus global
Reductions between Hamiltonians
Gadgets
WHY? Dynamics to Statics

17. The reduction: LH is QMA hard
Reductions between Hamiltonians
Gadgets

18. A B Applications of Circuit to Hamiltonian construction:
Universality of Adiabatic evolution [A’KempeLandauLloydRegevVanDam’04]
(QMA hardness often goes together w/ universality. see David’s lecture)
2. Hardness of the Physics “Density functional theory” [SchuchVerstraete’09]
3. Creation of Hamiltonians with “adversarially” highly entangled Gstates
[Irani’09, GottesmanHastings’09, A’HarrowLandauNagajSzegedyVazirani’14]
Creation of approximate quantum codes with local constraints [NirkheVaziraniYuen’18] …. A B
Deep: Physics and CS; Computational hardness, local versus global
Reductions between Hamiltonians
Gadgets
WHY? Dynamics to Statics

19.  Quantum Hamiltonian complexity
Easy/
tractable
Computationally
Hard
QMA hardness:
Many results…
1D systems are QMA hard [AharonovGottesmanIraniKempe’07,
1D Trans Invariant systems can be hard computationally [GottesmanIrani’09]
Useful object (also for Q simulations): gadgets
[KempeKitaevRegev’04,OliveiraTerhal’05]
Reductions between Hamiltonians
Gadgets
A class of Hamiltonians is QMA hard 
No classical description of GSs (unless QMA=NP).
QMA hardness often goes together with
BQP universality… See David’s lecture A B