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

Quantum Computation[Deutsch ’95, Barenco et al. ’95, DiVincenzo’95]
0
0
1
U
CNOT
H
“Hadamard”
V
1 (accept)
exists witness: y
The class NP“yes” instance: x L
“no” instance: x L
0 (reject)
for all y
Def: L QMA if there is a polytime quantum
U
U
exists witness 
prob
0 (reject)
for all 
prob
1 (accept)
The class QMA“yes” instance: x L
1 (accept)
prob 1
“no” instance: x L
0 (reject)
prob 1
Kitaev’s quantum CookLevin Theorem (’99):
Local Hamiltonian is QMAcomplete.
“witness = ground state”
Formula:
,
Hamiltonians:
H2
local Hamiltonians
H1
Satisfying assignment is groundstate of
Penalties for: x1x2x3 = 010 x3x4x5 = 100 …
Classical
Quantum
2local Hamiltonian ??
New result:
2local Hamiltonian is QMAcomplete
Vx
Classical CookLevin Theoremx1
x2
…
y1
y2
…
0
0
…
y1
y2
…
0
0
…
input x
1
1
witness y
witness y
ancilla 0
ancilla 0
output clause
Classical CookLevin Theoremz01
ancilla clauses
z02
z03
z04
ancilla
z0N
z1N
z2N
zTN
time = 0 1 2 3 4 … T
0
0
…
1
Ux
ancilla
qubits
Quantum CookLevin Theoremz01
ancilla clauses
z02
z03
propagationclauses
z04
ancilla qubits
z0N
zTN
01 2 … T
012 … T
z01
ancilla clauses
z02
z03
propagationclauses
z04
ancilla qubits
z0N
zTN
 00+11+…+TT
Time register {0, 1,…, T}
Computation qubits
Thm [Kitaev]:
Local Hamiltonian is QMAcomplete
Proof:
Expect the witness described before.
Construct the following Hamiltonians.
H= Jin Hin + Jprop Hprop + Hout
Tt
Reducing LocalityNotice that we have loglocal terms:
Thm [Kitaev]: 5local Hamiltonian is QMAcomplete
Proof idea:
Use unary encoding t  11…100…0
Penalise illegal time states:
Sclock  space of legal timestates is preserved (invariant) ▪
Thm [KempeRegev]: 3local Hamiltonian is QMAcomplt.
tt 1010t,t+1
tt1 110100t1,t,t+1
Spectrum: H
H’ = H + V
Energy
gap:
H>>V
0 groundspace S
Threequbit gadgetIdea: use perturbation theory to obtain effective 3local Hamiltonians from 2local ones by restricting to subspaces
What is the effective Hamiltonian in the lower part of the spectrum?
V restriction of V to S
V++  restriction of V to S
S
S
Perturbation TheorySpectrum: H
H’ = H + V
S
Energy
gap:
H>>V
0 groundspace S
What is the effective Hamiltonian in the lower part of the spectrum?
V restriction of V to S
V++  restriction of V to S
S
S
Theorem:
Perturbation TheorySpectrum: 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).
3local
H=P1P2P3
3local
2
2
3
3
P2XB
P3XC
ZZ
B
C
ZZ
ZZ
A
P1XA
1
1
Terms in H’=H+V
are 2local
Finetune the energy gap = 3
Threequbit gadget100 110
Ex.:
P1XA
P3XC
000
111
Threequbit gadget2
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:
g(t) gap between ground and first excitedstate
If
then the final state arbitrarily close to groundstate of HP.
Idea of Adiabatic Computation**E. Farhi, J. Goldstone, S. Gutmann, M. Sipser:“Quantum Computation by Adiabatic Evolution”, qp/’00
Adiabatic simulation*:
T’=poly(T):
The gap between groundstate and first excited state is 1/poly(T) at all times.
H(t) = (1t/T’)Hinitial +t/T’ Hfinal
Adiabatic Computation simulates quantum computationStandard 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.4251
H(t) = (1t/T’)Hinitial +t/T’ Hfinal
Adiabatic Computation simulates quantum computationResult: 2local adiabatic computation is equivalent to standard quantum computation
Use the gadget to replace everything by 2local terms.
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 Hamiltonianbased quantum architectures