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FSTTCS, Chennai, December 18 th , 2004 . The Complexity of the Local Hamiltonian Problem. Julia Kempe CNRS & LRI, Univ. of Paris-Sud Alexei Kitaev Caltech Oded Regev Tel-Aviv University. Also implies:. 2-local adiabatic computation is equivalent to standard quantum computation. Results.

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julia kempe cnrs lri univ of paris sud alexei kitaev caltech oded regev tel aviv university

FSTTCS, Chennai, December 18th, 2004

The Complexity of the Local Hamiltonian Problem

Julia Kempe

CNRS & LRI, Univ. of Paris-Sud

Alexei Kitaev

Caltech

Oded Regev

Tel-Aviv University

results

Also implies:

2-local adiabatic computation is equivalent to standard quantum computation

Results
  • Result:
    • 2-local Hamiltonian is QMA complete
outline
Outline
  • Introduction
    • Quantum computing
    • QMA
    • Local Hamiltonians
  • Previous Constructions
  • The 3-qubit Gadget
  • Implications
    • Adiabatic computation
    • Other applications of the technique
quantum computation

U

|’ = U |

|

Quantum Computation
  • A qubit is described by a unit vector in two-dimensional space:
    • | = c0|0 + c1|1 such that |c0|2+|c1|2=1
    • |0 and |1 are simply two orthogonal vectors
  • An n-qubit system is described by a unit vector in a 2n dimensional space:
    • |C{0,1}nsuch that|| |||2=1
  • An operation on an n-qubit system is described by a unitary matrix:
    • U C2n2nsuch that UU†=I(i.e., unitary)
quantum computation5
Quantum Computation
  • The model of quantum computation is as strong as classical computation
  • Moreover, there exists a small set of quantum gates that are universal

[Deutsch ’95, Barenco et al. ’95, DiVincenzo’95]

|0

|0

|1

U

CNOT

H

“Hadamard”

  • Quantum complexity theory is born!
the class np

V

V

1 (accept)

exists witness: y

The class NP
  • NP – Nondeterministic Polynomial Time
  • Def: L  NP if there is a poly-time verifier V and a polynomial p s.t.

“yes” instance: x  L

“no” instance: x  L

0 (reject)

for all y

  • Cook-Levin Theorem: SAT is NP-complete
the class qma

Def: L  QMA if there is a poly-time quantum

  • verifier U and a polynomial p s.t.

U

U

exists witness |

prob 

0 (reject)

for all |

prob 

1 (accept)

The class QMA
  • QMA – Quantum Merlin Arthur

“yes” instance: x  L

1 (accept)

prob 1-

“no” instance: x  L

0 (reject)

prob 1-

local hamiltonian problem
Local Hamiltonian Problem

Kitaev’s quantum Cook-Levin Theorem (’99):

Local Hamiltonian is QMA-complete.

  • Def. k-local Hamiltonian problem:
  • Input: k-local Hamiltonian , , Hi acts on k
  • qubits, a<b constants
  • Promise:
      • Smallest eigenvalue of H either  a or  b (b-a const.)
  • Output:
      • 1 if H has eigenvalue  a
      • 0 if all eigenvalues of H  b

“witness = ground state”

local hamiltonian problem9

Intuition:

Formula:

,

Hamiltonians:

H2

local Hamiltonians

H1

Satisfying assignment is groundstate of

  • Energy-penalty 1 for each unsatisfied constraint.
    • x1x2 … xn| H |x1x2 … xn  = #unsatisfied constraints
Local Hamiltonian Problem

Penalties for: x1x2x3 = 010 x3x4x5 = 100 …

results10

MAX2SAT is NP-complete

  • 2-local Hamiltonian is NP-hard
Results

Classical

Quantum

  • log|x|-local Hamiltonian is QMA-compl.
  • [Kitaev’99]
  • 5-local Hamiltonian is QMA-complete
  • [Kitaev’99]
  • 3-local Hamiltonian is QMA-complete
  • [KempeRegev’02]
  • MAX3SAT is NP-complete

2-local Hamiltonian ??

New result:

2-local Hamiltonian is QMA-complete

  • 1-local Hamiltonian is in P
adiabatic computation
Adiabatic Computation
  • Quantum computers can simulate adiabatic computation [Farhi et al. 00]
  • Adiabatic computation can simulate quantum computers [ADKLLR 04]
  • In fact, 3-local adiabatic computation is enough [ADKLLR 04]
  • New result: 2-local adiabatic computation can simulate quantum computers
outline12
Outline
  • Introduction
    • Quantum Computation
    • QMA
    • Local Hamiltonians
  • Previous Constructions
  • The 3-qubit Gadget
  • Implications
    • Adiabatic computation
    • Other applications of the technique
classical cook levin theorem

V

Vx

Classical Cook-Levin Theorem
  • Thm: SAT is NP-complete
  • Proof: First, given a verifier V, encode the input into V

x1

x2

y1

y2

0

0

y1

y2

0

0

input x

1

1

witness y

witness y

ancilla 0

ancilla 0

classical cook levin theorem14

propagationclauses

output clause

Classical Cook-Levin Theorem
  • Thm: SAT is NP-complete
  • Proof: Next, create a tableau of variables and 3 kinds of clauses.

z01

ancilla clauses

z02

z03

z04

ancilla

z0N

z1N

z2N

zTN

time = 0 1 2 3 4 … T

quantum cook levin theorem

|

|0

|0

|1

Ux

ancilla

qubits

Quantum Cook-Levin Theorem
  • Let us try to extend this to the quantum setting
quantum cook levin theorem16

output clause

Quantum Cook-Levin Theorem
  • Let us try to extend this to the quantum setting
  • The naïve attempt does not work
  • There is no local way to check local consistency

z01

ancilla clauses

z02

z03

propagationclauses

z04

ancilla qubits

z0N

zTN

|0|1 |2 … |T

quantum cook levin theorem17

output clause

Quantum Cook-Levin Theorem
  • Instead of tensoring the columns, we put them in superposition
  • So the witness is a sum over history

|0|1|2 … |T

z01

ancilla clauses

z02

z03

propagationclauses

z04

ancilla qubits

z0N

zTN

| |0|0+|1|1+…+|T|T

quantum cook levin theorem18

input

Time register {|0, |1,…, |T}

Computation qubits

  • propagation
  • output
Quantum Cook-Levin Theorem

Thm [Kitaev]:

Local Hamiltonian is QMA-complete

Proof:

Expect the witness described before.

Construct the following Hamiltonians.

H= Jin Hin + Jprop Hprop + Hout

reducing locality

t

T-t

Reducing Locality

Notice that we have log-local terms:

Thm [Kitaev]: 5-local Hamiltonian is QMA-complete

Proof idea:

Use unary encoding |t | 11…100…0

Penalise illegal time states:

Sclock - space of legal time-states is preserved (invariant) ▪

Thm [KempeRegev]: 3-local Hamiltonian is QMA-complt.

|tt|  |1010|t,t+1

|tt-1|  |110100|t-1,t,t+1

outline20
Outline
  • Introduction
    • Quantum Computation
    • QMA
    • Local Hamiltonians
  • Previous Constructions
  • The 3-qubit Gadget
  • Implications
    • Adiabatic computation
    • Other applications of the technique
three qubit gadget

Spectrum: H

H’ = H + V

Energy

gap:

||H||>>||V||

0 groundspace S

Three-qubit gadget

Idea: use perturbation theory to obtain effective 3-local Hamiltonians from 2-local ones by restricting to subspaces

What is the effective Hamiltonian in the lower part of the spectrum?

perturbation theory

Energy gap> ||V||

V--- restriction of V to S

V++ - restriction of V to S

S

S

Perturbation Theory

Spectrum: H

H’ = H + V

S

Energy

gap:

||H||>>||V||

0 groundspace S

What is the effective Hamiltonian in the lower part of the spectrum?

perturbation theory23

Energy gap> ||V||

V--- restriction of V to S

V++ - restriction of V to S

S

S

Theorem:

Perturbation Theory

Spectrum: H

H’ = H + V

S

Energy

gap:

||H||>>||V||

0 groundspace S

The lower spectrum of H’ is close to the spectrum of Heff (under certain conditions).

three qubit gadget24

Heff=P1P2P3

3-local

H=P1P2P3

3-local

2

2

3

3

P2XB

P3XC

ZZ

B

C

ZZ

ZZ

A

P1XA

1

1

Terms in H’=H+V

are 2-local

Fine-tune the energy gap = -3

Three-qubit gadget
three qubit gadget25

ZZ

B

C

ZZ

ZZ

A

Three-qubit gadget

S={|001,|010,|100,

|110,|101,|011}

=-3

Energy

gap:

S={|000, |111}

0

three qubit gadget26

P2XB

|100  |110

Ex.:

P1XA

P3XC

|000

|111

Three-qubit gadget

2

3

P2XB

P3XC

B

C

S={|001,|010,|100,

|110,|101,|011}

=-3

Energy

gap:

A

P1XA

S={|000, |111}

0

1

V++

S 

S

V-+

V+-

Third order: S S

Theorem:

2 local hamiltonian is qma complete
2-local Hamiltonian is QMA-complete
  • Start with the QMA-complete 3-local Hamiltonian
  • Replace each 3-local term by a 3-qubit gadget
outline28
Outline
  • Introduction
    • Quantum Computation
    • QMA
    • Local Hamiltonians
  • Previous Constructions
  • The 3-qubit Gadget
  • Implications
    • Adiabatic computation
    • Other applications of the technique
idea of adiabatic computation

Adiabatic theorem:

g(t) gap between ground- and first excitedstate

If

then the final state arbitrarily close to groundstate of HP.

Idea of Adiabatic Computation*
  • Start in the groundstate of a Hamiltonian H0 (easy to prepare)
  • Encode problem as a Hamiltonian HP (groundstate gives solution)
  • Adiabatically (slowly!) evolve from H0 to HP

*E. Farhi, J. Goldstone, S. Gutmann, M. Sipser:“Quantum Computation by Adiabatic Evolution”, q-p/’00

adiabatic computation simulates quantum computation

U

Adiabatic simulation*:

  • Hfinal
  • groundstate =
  • Kitaev’s “history state”
  • Hinitial
  • groundstate
  • |0…0 |0

T’=poly(T):

The gap between groundstate and first excited state is 1/poly(T) at all times.

H(t) = (1-t/T’)Hinitial +t/T’ Hfinal

Adiabatic Computation simulates quantum computation

Standard quantum circuit:

|0…0

|T

T gates

*D. Aharonov, W.  van Dam, J. Kempe, Z. Landau, S. Lloyd, O. Regev: "Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation", FOCS’04, p.42-51

adiabatic computation simulates quantum computation31

Adiabatic simulation*:

  • Hfinal
  • groundstate =
  • Kitaev’s “history state”
  • Hinitial
  • groundstate
  • |0…0 |0

H(t) = (1-t/T’)Hinitial +t/T’ Hfinal

Adiabatic Computation simulates quantum computation

Result: 2-local adiabatic computation is equivalent to standard quantum computation

Use the gadget to replace everything by 2-local terms.

other applications of the gadget work in progress

-2ZA

-1P1XA

-1P2XA

H=P1P2

Heff=P1P2

“Proxy Interaction”:

(with A. Landahl)

-1Z1ZA

-1X2XB

-2YAYB

Heff=Z1X2

Other applications of the gadget(work in progress)

“Interaction at a distance”:

H=Z1X2

only XX,YY,ZZ available

Useful for Hamiltonian-based quantum architectures