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Dorit AharonovDorit AharonovSchool of Computer Science and EngineeringSchool of Computer Science and Engineering

The Hebrew University, The Hebrew University, Jerusalem,Jerusalem,

IsraelIsrael

Quantum Hamiltonian Quantum Hamiltonian ComplexityComplexity

Quantum Hamiltonian Quantum Hamiltonian ComplexityComplexity

What is it?

What is it?

What are

What are The The implications

implications??

Ground states

Ground states Entanglemen

Entanglemen

tt

Why is it

Why is it

interesting

interesting

??

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Modern Church Turing ThesisModern Church Turing Thesis

“ All physically reasonable computational models can be simulated in polynomial time

by a Turing machine”

Quantum computation: Quantum computation: Only Model which threatens this thesis: Only Model which threatens this thesis: Seem to have Seem to have exponentialexponential power power

Corner stone of theoretical computer Science:Corner stone of theoretical computer Science:

≈≈ ≈≈

Computational properties of Quantum are differentComputational properties of Quantum are different

ProbabilisticProbabilisticQuantumQuantum

Polynomial time, Polynomial time, Equivalence up to Equivalence up to

Polynomial reductionsPolynomial reductions

Post-Post-

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Quantum computation Quantum computation PhysicsPhysics

0,1|1,0|2

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

• Multiscale EntanglementMultiscale Entanglement (examples: QECCs) (examples: QECCs)

Q. Hamiltonian complexity: Q. Hamiltonian complexity: apply to Cond. matter apply to Cond. matter physicsphysics

•Quantum error correction: Quantum error correction: Meta stabilityMeta stability out of equilibriumout of equilibrium

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Condensed Matter PhysicsCondensed Matter Physics

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),(31

),(

)( jji

ijji

i SSSSH

Local Hamiltonian, (e.g.AKLT)

Ground states: What are their properties? Ground states: What are their properties? Expectation values of various observables?Expectation values of various observables?How do two-point correlations behave? How do two-point correlations behave? And what about the spectral gap? And what about the spectral gap?

)(|)()(| ttHi dt

td

5

21)( e

EeeJE

J=1 (red) J=-1 green

(1 violation.)

...) ( )()...( 7423211 xxxxxxxxF n

NP completeness – NP completeness – Reductions!!! (Polynomial time)Reductions!!! (Polynomial time)Probabilistically checkable proofs (PCP)Probabilistically checkable proofs (PCP)InapproximabilityInapproximability

K-SAT formulasK-SAT formulas

3-coloring of a graph…3-coloring of a graph…Mathematical proofs..Mathematical proofs..

Constraint Satisfaction problem (CSP): Constraint Satisfaction problem (CSP):

n variables, constraints on k-tuples n variables, constraints on k-tuples

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Quantum Hamiltonian Quantum Hamiltonian

ComplexityComplexityDeep connection between these two major Problems: Deep connection between these two major Problems: Local Hamiltonians can be viewed as quantum CSPsLocal Hamiltonians can be viewed as quantum CSPsSimilar questions plus more complications: enter entanglementSimilar questions plus more complications: enter entanglement

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),(31

),(

)( jji

ijji

i SSSSH

Power of various Hamiltonian classes…Power of various Hamiltonian classes…Entanglement properties of ground states…Entanglement properties of ground states…

Provides a whole new lance through which to look at Provides a whole new lance through which to look at Quantum many body physics: The computational Quantum many body physics: The computational point of view. point of view.

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...) ( )()...( 7423211 xxxxxxxxCSP n

c sConstraint

cHH .... 7,4,23,2,1 101|000|

Constraints Constraints Energy Penalties Energy PenaltiesSolution Solution Ground state Ground state

CSP CSP Is ground energy 0 or at least 1? Is ground energy 0 or at least 1?CSP CSP estimate the ground energy of H. estimate the ground energy of H.

[Cook-Levin[‘69ish]: CSP is NP-complete[Cook-Levin[‘69ish]: CSP is NP-complete Estimating the ground energy is at least as hard as NPEstimating the ground energy is at least as hard as NP

CSP & HamiltonianCSP & Hamiltonian

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i

iHHThe Local Hamiltonian Problem The Local Hamiltonian Problem InputInput::OutputOutput: ground energy of H <a or : ground energy of H <a or >a>a+1/poly(n).+1/poly(n).

[QMA:[QMA: Like NP, Like NP,

Except both Verifier Except both Verifier

Circuit and Witness Circuit and Witness

are quantum.are quantum.

Input Input WitnessWitness]:]:

UU11

…….. UU55UU44 UU33 UU22

Theorem [Kitaev’98]: The k-local Hamiltonian Theorem [Kitaev’98]: The k-local Hamiltonian problem is QMA completeproblem is QMA complete

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The Cook-Levin Theorem:The Cook-Levin Theorem: Computation is local Computation is local [Cook-Levin’79][Cook-Levin’79]

History of a computation can History of a computation can be checked locally be checked locally can associate a CSP with the can associate a CSP with the local dynamics local dynamics

TimeTimestepssteps

The verifier is mapped to a SAT formulaThe verifier is mapped to a SAT formula SAT is NP-completeSAT is NP-complete

... )...(...) (),...,( 1,1,,3,12,11,1,1,1 jijijiTn xxxxxxxxF

jix ,

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The Circuit-to-Hamiltonian constructionThe Circuit-to-Hamiltonian construction[Kitaev98, Following Feynman82][Kitaev98, Following Feynman82]

Hamiltonian whose ground

state is the History.

kLk

k 0..001..11||

kkhistory

L

kL

|)(||0

11

TimeTimestepssteps

)(| k

)0(|)1(|

::

LL00 k-1k-1k+1k+1kk

Feynman’s particle on a lineFeynman’s particle on a line

Reduction from any

Reduction from any

Qcircuit to a local

Qcircuit to a local

Hamiltonian

Hamiltonian

Quantum Quantum

UniversalityUniversality

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H(0)H(0)

H(T)H(T)

UU11

…….. UU55UU44 UU33 UU22

Adiabatic Computation ≈ Quantum ComputationAdiabatic Computation ≈ Quantum Computation [A’vanDamKempeLandauLloydRegev’04][A’vanDamKempeLandauLloydRegev’04]

H(t) ≈ random walk on time steps! Markov chain techniques. LL00 k-1k-1k+1k+1kk

Spectral gap:Spectral gap:

Instead of , use a local Hamiltonian H(T) whose ground state is the History.

)(| L

Reduction Reduction

Quantum Quantum

UniversalityUniversality

Want adiabatic computation with γ(t)>1/Lc from which to deduce answer.

)(| TH(0)H(0) H(T)H(T)

)0(|

Adiabatic Computation: [FarhiGodstoneGutmanSipser’00] [FarhiGodstoneGutmanSipser’00]

Ground state of H(0) ground state of H(T)

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2D, 2-local Ham’s (6-states) 2D, 2-local Ham’s (6-states) [A’vanDamKempeLandauRegevLloyd’04][A’vanDamKempeLandauRegevLloyd’04]

2-local Ham’s, in general geometry (qubits) (using Gadgets) 2-local Ham’s, in general geometry (qubits) (using Gadgets) [KempeKitaevRegev’06][KempeKitaevRegev’06]

2D, 2-local Ham’s (Using Gadgets) 2D, 2-local Ham’s (Using Gadgets) [OliveiraTerhal’05][OliveiraTerhal’05]

1D1D, 2-local Ham’s ! (using 12 states) , 2-local Ham’s ! (using 12 states) [A’IraniKempeGottesman’07][A’IraniKempeGottesman’07]

Adiabatic Computation is Quantum Universal Adiabatic Computation is Quantum Universal (& estimating the ground energy is QMA (& estimating the ground energy is QMA complete) complete) for much stricter families of for much stricter families of

Hamiltonians:Hamiltonians:

Perturbation Gadgets:Perturbation Gadgets:

** ****** ****ReductionsReductions

≈≈

1Dim result is surprising…1Dim result is surprising…

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Quantum Hamiltonian Complexity: Quantum Hamiltonian Complexity:

Easy:Easy:In P,In P,

gs is MPSgs is MPS

Hard: Hard: BQP complete, BQP complete, QMA complete, QMA complete,

NP hard, etc.NP hard, etc.

Constant gapConstant gap: : CorrelationsCorrelations decay exponentially for all D [Hastings’05]. decay exponentially for all D [Hastings’05].Small correlations Small correlations Little Entanglement! (data Hiding, Q expander states) Little Entanglement! (data Hiding, Q expander states)

In 1DIn 1D: Limited : Limited entanglement entanglement too (area law).too (area law). [Hastings’07].[Hastings’07].

MPS description of ground state of 1D gapped systemsMPS description of ground state of 1D gapped systems

Efficient simulation of 1D Efficient simulation of 1D gappedgapped adiabatic adiabatic [Hastings’09][Hastings’09]

OpenOpen: Can the ground state be found classically : Can the ground state be found classically efficienlty?efficienlty?

Open: Open: Correlations & entanglement?Correlations & entanglement?

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3. How hard is local Hamiltonians for restricted 3. How hard is local Hamiltonians for restricted Hamiltonians?Hamiltonians?

For a 1/poly(n) gapped 1D system: QCMA hard For a 1/poly(n) gapped 1D system: QCMA hard [A’Ben-OrBrandaoSattath’08].[A’Ben-OrBrandaoSattath’08].What if we know the ground state is an MPS? What if we know the ground state is an MPS? The classical analog (solving 1D CSPs) is easy…The classical analog (solving 1D CSPs) is easy… Quantumly: NP-hard Quantumly: NP-hard [SchuchCiracVerstraete’08][SchuchCiracVerstraete’08]

1.1. Hardness for interesting physical systems: Hardness for interesting physical systems: Approximating ground energy of Hubbard model: QMA complete Approximating ground energy of Hubbard model: QMA complete Solving Schrodinger’s eq. for interacting electrons: QMA-hard Solving Schrodinger’s eq. for interacting electrons: QMA-hard [SchuchVerstraete’07][SchuchVerstraete’07]

2. Ruling out various physical attempts:2. Ruling out various physical attempts:““Universal density functional” cannot be efficiently Universal density functional” cannot be efficiently computable unless NP=QMAcomputable unless NP=QMA.[SchuchVerstraete’07].[SchuchVerstraete’07]

Back to the Hardness side… Some examplesBack to the Hardness side… Some examples

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The PCP theorem The PCP theorem

XX

≈≈ VerifierVerifier

VerifierVerifier

XX

NPNPPCPPCP

Witness\ProofWitness\ProofSlightlySlightly longer longerWitness\ProofWitness\Proof

Gap amplification version Gap amplification version [Dinur’07][Dinur’07]

CSP Y CSP Y CSP Z CSP ZY satisfiableY satisfiable: Z is satisfiable : Z is satisfiable Y is notY is not : Z violated > 10%. : Z violated > 10%.

(Hardness of approximation!!!) (Hardness of approximation!!!)

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Quantum PCP theorem? Quantum PCP theorem? Quantum Ground Energy amplification? Quantum Ground Energy amplification?

What would be the implications?What would be the implications? Hardness of Quantum approximations..Hardness of Quantum approximations.. Ways to manipulate ground energies, Ways to manipulate ground energies, Maybe spectral gaps (adiabatic Fault-Maybe spectral gaps (adiabatic Fault-Tolerance?) Tolerance?)

Mainly:Mainly: Attempts to follow Dinur’s proof seem to encounter Attempts to follow Dinur’s proof seem to encounter

conceptual difficulties: No cloning theorem.conceptual difficulties: No cloning theorem. No go for QPCP – No go for QPCP – sophisticated no cloning theorem… On the other hand, a proof sophisticated no cloning theorem… On the other hand, a proof might constitute a sophisticated version of QECCs. might constitute a sophisticated version of QECCs.

Hamiltonian H Hamiltonian H Hamiltonian H’Hamiltonian H’

H Frustration freeH Frustration free: So is H’ : So is H’ H is notH is not : Ground energy of H’ large : Ground energy of H’ large and detectable. and detectable.

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Quantum Gap AmplificationQuantum Gap Amplification [A’AradLandauVazirani’09][A’AradLandauVazirani’09]

((A proof of an important ingredient in Dinur’s proof, but without A proof of an important ingredient in Dinur’s proof, but without handling the no-cloning issue)handling the no-cloning issue)

Local termsLocal terms Larger constraints, Larger constraints, defined by walks on the graphdefined by walks on the graph

Analyzing the ground energy of the new Hamiltonian H’: Analyzing the ground energy of the new Hamiltonian H’: Requires a sophisticated Requires a sophisticated reductionreduction to a commuting case to a commuting case (The XY decomposition, pyramids, the detectability lemma) (The XY decomposition, pyramids, the detectability lemma)

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Open problems: Open problems:

Quantum Quantum PCPPCP? Relations to adiabatic fault tolerance? or: Can we rule ? Relations to adiabatic fault tolerance? or: Can we rule out quantum PCP (a sophisticated No-Cloning theorem?) out quantum PCP (a sophisticated No-Cloning theorem?)

Extending other important classical results, e.g.: 1. Remove degeneracy? (Q Valiant-Vazirani) [see A’BenOrBrandaoSattath’09]?2. Frustration freeness? (QMA1 vs. QMA?)[see Aaronson’08]

Rule out other physics programs similar to the universal density Rule out other physics programs similar to the universal density functional? Identify the complexity of other types of systems? functional? Identify the complexity of other types of systems? (interacting electrons), Check the (interacting electrons), Check the Post-ModernPost-Modern CTCT thesis for other known thesis for other known systems (field theory)? systems (field theory)?

Much more on the computationally “easy” side: Much more on the computationally “easy” side: Area laws and Area laws and entanglemententanglement vs. vs. correlationscorrelations in Dim>1? in Dim>1? Finding the ground state for gapped 1D Hamiltonians?Finding the ground state for gapped 1D Hamiltonians?

Computational power of commuting Hamiltonians?

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Thanks!Thanks!