Dorit Aharonov

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Why is it interesting?
What are The implications?
What is it?
Quantum Hamiltonian Complexity
Dorit Aharonov School of Computer Science and
Engineering The Hebrew University, Jerusalem,
Israel
Ground states
Entanglement
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Modern Church Turing Thesis

Post-
Corner stone of theoretical computer Science

• All physically reasonable computational
• models can be simulated in polynomial time by
  a Turing machine

Probabilistic
Quantum


Quantum computation Only Model which threatens
this thesis Seem to have exponential power
Polynomial time, Equivalence up to Polynomial
reductions
Computational properties of Quantum are different
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Quantum computation ? Physics

• Quantum Universality (BQP) The question of the
• computational power of the system Is it fully
  
• quantum?
• Reductions Equivalence between systems
• from a Computational point of view

• Multiscale Entanglement (examples QECCs)

• Quantum error correction Meta stability
• out of equilibrium

Q. Hamiltonian complexity apply to Cond. matter
physics
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Condensed Matter Physics
Local Hamiltonian, (e.g.AKLT)
Ground states What are their properties?
Expectation values of various observables? How
do two-point correlations behave? And what about
the spectral gap?
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Constraint Satisfaction problem (CSP)
n variables, constraints on k-tuples
K-SAT formulas
3-coloring of a graph Mathematical proofs..
(1 violation.)
J1 (red) J-1 green
NP completeness Reductions!!! (Polynomial
time) Probabilistically checkable proofs
(PCP) Inapproximability
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Quantum Hamiltonian Complexity

Deep connection between these two major Problems
Local Hamiltonians can be viewed as quantum
CSPs Similar questions plus more complications
enter entanglement
Power of various Hamiltonian classes Entanglement
properties of ground states
Provides a whole new lance through which to look
at Quantum many body physics The computational
point of view.
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CSP Hamiltonian
Constraints ? Energy Penalties Solution ? Ground
state
CSP ? Is ground energy 0 or at least 1? CSP ?
estimate the ground energy of H.
Cook-Levin69ish CSP is NP-complete?
Estimating the ground energy is at least as hard
as NP
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The Local Hamiltonian Problem Input Output
ground energy of H lta or gta1/poly(n).
Theorem Kitaev98 The k-local Hamiltonian
problem is QMA complete
QMA Like NP, Except both Verifier Circuit
and Witness are quantum.
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The Cook-Levin Theorem Computation is local
Cook-Levin79
History of a computation can be checked locally ?
can associate a CSP with the local dynamics
The verifier is mapped to a SAT formula ? SAT is
NP-complete
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The Circuit-to-Hamiltonian constructionKitaev98,
Following Feynman82
Feynmans particle on a line
Reduction from any Qcircuit to a local
Hamiltonian ? Quantum Universality

• Hamiltonian whose ground
• state is the History.

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Adiabatic Computation
FarhiGodstoneGutmanSipser00
Ground state of H(0)
ground state of H(T)
Adiabatic Computation Quantum Computation
AvanDamKempeLandauLloydRegev04

• Want adiabatic computation with ?(t)gt1/Lc
  from which to deduce answer.

Instead of , use a local
Hamiltonian H(T) whose ground state is the
History.
Reduction ? Quantum Universality
Spectral gap
H(t) random walk on time steps! Markov chain
techniques.
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Adiabatic Computation is Quantum Universal (
estimating the ground energy is QMA complete) for
much stricter families of Hamiltonians
2D, 2-local Hams (6-states) AvanDamKempeLandauR
egevLloyd04 2-local Hams, in general geometry
(qubits) (using Gadgets)

KempeKitaevRegev06 2D, 2-local Hams (Using
Gadgets) OliveiraTerhal05 1D,
2-local Hams ! (using 12 states)
AIraniKempeGottesman07
1Dim result is surprising
Perturbation Gadgets
Reductions

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Quantum Hamiltonian Complexity
Open Correlations entanglement?
Hard BQP complete, QMA complete, NP hard, etc.
Easy In P, gs is MPS
Efficient simulation of 1D gapped adiabatic
Hastings09 Open Can the ground state be found
classically efficienlty?
In 1D Limited entanglement too (area law).
Hastings07. MPS description of ground state
of 1D gapped systems
Constant gap Correlations decay exponentially
for all D Hastings05.
Small correlations ? Little Entanglement! (data
Hiding, Q expander states)
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Back to the Hardness side Some examples
1. Hardness for interesting physical systems
Approximating ground energy of Hubbard model
QMA complete Solving Schrodingers eq. for
interacting electrons QMA-hard
SchuchVerstraete07
2. Ruling out various physical attempts Universa
l density functional cannot be efficiently
computable unless NPQMA.SchuchVerstraete07
3. How hard is local Hamiltonians for restricted
Hamiltonians? For a 1/poly(n) gapped 1D system
QCMA hard ABen-OrBrandaoSattath08. What if
we know the ground state is an MPS? The
classical analog (solving 1D CSPs) is easy
Quantumly NP-hard SchuchCiracVerstraete08
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The PCP theorem

Verifier
PCP
Verifier
NP

Witness\Proof
X
Slightly longer Witness\Proof
X
Gap amplification version Dinur07
CSP Y ? CSP Z
Y satisfiable Z is satisfiable Y is not
Z violated gt 10.
(Hardness of approximation!!!)
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Quantum PCP theorem?
Quantum Ground Energy amplification?
Hamiltonian H ? Hamiltonian H
H Frustration free So is H H is not Ground
energy of H large and detectable.
What would be the implications? Hardness of
Quantum approximations.. Ways to manipulate
ground energies, Maybe spectral gaps
(adiabatic Fault-Tolerance?)
Mainly Attempts to follow Dinurs proof seem to
encounter conceptual difficulties No cloning
theorem. No go for QPCP sophisticated no
cloning theorem On the other hand, a proof might
constitute a sophisticated version of QECCs.
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Quantum Gap Amplification AAradLandauVazirani0
9 (A proof of an important ingredient in
Dinurs proof, but without handling the
no-cloning issue)

Larger constraints, defined by walks on the graph
Local terms
Analyzing the ground energy of the new
Hamiltonian H Requires a sophisticated
reduction to a commuting case (The XY
decomposition, pyramids, the detectability lemma)
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Open problems
Computational power of commuting Hamiltonians?
Quantum PCP? Relations to adiabatic fault
tolerance? or Can we rule out quantum PCP (a
sophisticated No-Cloning theorem?) Extending
other important classical results, e.g. 1.
Remove degeneracy? (Q Valiant-Vazirani) see
ABenOrBrandaoSattath09? 2. Frustration
freeness? (QMA1 vs. QMA?)see Aaronson08 Rule
out other physics programs similar to the
universal density functional? Identify the
complexity of other types of systems?
(interacting electrons), Check the Post-Modern CT
thesis for other known systems (field theory)?
Much more on the computationally easy side
Area laws and entanglement vs. correlations in
Dimgt1? Finding the ground state for gapped 1D
Hamiltonians?
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Thanks!