quantum information theory: present status and future directions
DESCRIPTION
Newton Institute, Cambridge, August 24 th , 2004. Quantum Information Theory: Present Status and Future Directions. The Complexity of Local Hamiltonians. Julia Kempe CNRS & LRI, Univ. de Paris-Sud, Orsay, France. Also implies:. - PowerPoint PPT PresentationTRANSCRIPT
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Quantum Information Theory: Present Status and Future Directions
Julia KempeJulia KempeCNRS & LRI, Univ. de Paris-Sud, Orsay, France
Newton Institute, Cambridge, August 24th, 2004
The Complexity of Local HamiltoniansThe Complexity of Local Hamiltonians
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Joint work with Joint work with Oded Oded RegevRegev and and Alexei KitaevAlexei Kitaev
Result: 2-local Hamiltonian is QMA complete
J. K., Alexei Kitaev and Oded Regev, quant-ph/0406180
2-local adiabatic computation is equivalent to standard quantum computation
Also implies:
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OutlineOutline
A Bit of History• QMA • Local Hamiltonians
Previous Constructions
The 3-qubit Gadget
Implications• Adiabatic computation• Other applications of the technique
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A Bit of (ancient) A Bit of (ancient) HistoryHistory
Complexity Theory:
• classify “easy” and “hard”
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A Bit of (ancient) A Bit of (ancient) HistoryHistory
NP – Nondeterministic Polynomial Time:
Def. L NP if there is a poly-time verifier V and a polynomial p s.t.
p( x )
p( x )
y {0,1} V(x,y)=1
y {0,1} V(x,y)=0
x L
x L
V“yes” instance: x L
witness: y1 (accept)
V“no” instance: x L
for all “witnesses” y0 (reject)
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Example: SATExample: SAT 1 1 2 3 3 4 5,..., ...nx x x x x x x x Formula:
SAT iff there is a satisfying assignment for x1,…,xn
(i.e. all clauses true simultaneously).
0 1 1
10 0 0
1
0 - false1 - true0 = 1, 1 = 0
V= (y)“yes” instance: SAT
witness: y=011000…1 (true, accept)
“no” instance: SAT
for all “witnesses” y=010110…
0 (false, reject)V= (y)
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NP completeNP completeA language is NP complete if it is in NP and as hard as any other problem in NP.
Cook-Levin Theorem: SAT is NP-complete
L SAT
y=011000…y1 1V
x0
SAT
y=010110…0
L
y
x NP NP-complete
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NP completeNP complete
Cook-Levin Theorem: 3SAT is NP-complete
1 2 3 3 4 5 ...x x x x x x
3SAT: 3 variables per clause
3 variables
2SAT is in P (there is a poly time algorithm).
MAX2SAT is NP-complete
MAX2SAT:Input: Formula with 2 variables per clause, number mOutput: 1 (accept) if there is an assignment that violates m clauses
0 (reject) all assignments violate >m clauses
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QMAQMA
V“yes” instance: x Lyes 1 (accept)
V“no” instance: x Lno
witness: |
for all “witnesses” | 0 (reject)
prob 1-
0 (reject) prob
1 (accept) prob
prob 1-
QMA – Quantum Merlin Artur = BQNP = “Quantum NP”
Def. L QMA if there is a poly-time quantum verifier V and
a polynomial p s.t.
p( x )
p( x )
prob V x, =1 1
prob V x, =1
x L
x L
2
2
C
C
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More recent (quantum) HistoryMore recent (quantum) HistoryQMA – Quantum Merlin Artur = BQNP
Def. L QMA if there is a poly-time quantum verifier V and
a polynomial p s.th.
p( x )
p( x )
prob V x, =1 1
prob V x, =1
x L
x L
2
2
C
C
•First studied in [Knill’96] and [Kitaev’99] – called it BQNP• “QMA” coined by [Watrous’00] – also: group-nonmembership QMA
Kitaev’s quantum Cook-Levin Theorem (’99): Local Hamiltonian is QMA-complete.
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Local HamiltoniansLocal Hamiltonians
Def. k-local Hamiltonian problem:
Input: k-local Hamiltonian , , Hi acts on k qubits, a<b constantsPromise:
• 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
( )
1
poly n
ii
H H
iH ( )poly n
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Local HamiltoniansLocal Hamiltonians
1 2 3 3 4 5 ...x x x x x x
Intuition:
Formula:
Penalties for: x1x2x3 = 010 x3x4x5 = 100 …
Satisfying assignment is groundstate of
ii
H HEnergy-penalty 1 for each unsatisfied constraint.
x1x2 … xn| H |x1x2 … xn = #unsatisfied constraints
Hamiltonians: 1,2,3010 010
3,4,5100 100,
H1 H2 local Hamiltonians
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NP and QMANP and QMANP-completeness: QMA-completeness?
x1
x2
…y1
y2
…00…
1x
y
0
Verifier V:
input
witness
ancilla
…
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NP and QMANP and QMANP-completeness: QMA-completeness?
y1
y2
…
00…
1
…
y
0
Verifier Vx :
…
…
3-clauses check:
• propagation
• output
z01
z02
z03
z04
z0N z1N z2NzTN
t = 0 1 2 3 4 … T
|
|0 |0…
C
H
|1
…
C H
|0|1 … |T
?
Verifier Ux :
ancilla qubits
witness
ancilla
• input
ancilla
No local way to check!
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NP and QMANP and QMANP-completeness: QMA-completeness?
y1
y2
…
00…
1
…
y
0
Verifier Vx :
3-clauses check:
• propagation
• output
|
|0 |0…
C
H
|1
…
C H
?
Verifier Ux :
ancilla qubits
witness
ancilla
• input
||0=|0 |1 … |T
+ + ++
|0 |1 |2 |T
| |0|0+|1|1+…+ |T|T
witness = sum over history
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NP-completeness: QMA-completeness:
• 3SAT is NP-complete
• 2SAT is in P
• log|x|-local Hamiltonian is QMA-compl. [Kitaev’99]• 5-local Hamiltonian is QMA-compl. [Kitaev’99]
• 3-local Hamiltonian is QMA-compl. [KempeRegev’02]
• but: MAX2SAT is NP-complete • 2-local Hamiltonian is NP-hard
2-local Hamiltonian????
• 1-local Hamiltonian is in P
More recent (quantum) HistoryMore recent (quantum) History
Is 2-local Hamiltonian QMA-complete??
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OutlineOutline
A Bit of History• QMA • Local Hamiltonians
Previous Constructions
The 3-qubit Gadget
Implications• Adiabatic computation• Other applications of the technique
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Kitaev’s log-local ConstructionKitaev’s log-local Construction
Local Hamiltonians check: H= Jin Hin + Jprop Hprop + Hout
| |1
Verifier Ux :
witness = sum over historym
N-m
TT=poly(N)
• input
• propagation
• output
1
1 1 0 0N
in ii m
H
†
1
1-1 -1 -1 -1
2
T
prop t tt
H I t t I t t U t t U t t
11 0 0outH T T T
Computation qubits
Time register {|0, |1,…, |T}
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Kitaev’s log-local ConstructionKitaev’s log-local ConstructionH= Jin Hin + Jprop Hprop + HoutVerifier: Ux=UTUT-1…U1
To show: If Ux accepts with prob. 1- on input |,0, then H has eigenvalue . If Ux accepts with prob. on all |,0, then all eigenvalues of H ½-.
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Completeness Completeness H= Jin Hin + Jprop Hprop + HoutVerifier: Ux=UTUT-1…U1
To show: If Ux accepts with prob. 1- on input |,0, then H has eigenvalue . If Ux accepts with prob. on all |,0, then all eigenvalues of H ½-.
1
1 1 0 0N
in ii m
H
†,
1 1
1-1 -1 -1 -1
2
T T
prop t t prop tt t
H I t t I t t U t t U t t H
11 0 0outH T T T
|Hin| =0
†, 1 -1 1 1 -1 1... ... -1 ... ... -1 0prop t t t t t t tH U U t U U t U U t U U U U t
|Hprop| =0
|Hout| 0 11 1
10 1 1 ...
1T TT T
T
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5-local Hamiltonians5-local HamiltoniansLog-local terms: , -1 , -1t t t t t t
Idea (Kitaev): unary |t | 11…100…0 t T-t
|tt| |1010|t,t+1
|tt-1| |110100|t-1,t,t+1
Penalise illegal time states: ,01 01clock i j
i j
H I
clock - space of legal time-states is preserved (invariant)
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3-local Hamiltonians3-local Hamiltonians5-local terms: |tt-1| |110100|t-1,t,t+1 ,
01 01clock i ji j
H I
Idea [KR’02]: |110100|t-1,t,t+1 |10|t
clock clock KitaevH J H H
(|10|t)|clock = |tt-1|
Give a high energy penalty to illegal time statesto effectively prevent transitions outside clock :
clock
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OutlineOutline
A Bit of History• QMA • Local Hamiltonians
Previous Constructions
The 3-qubit Gadget
Implications• Adiabatic computation• Other applications of the technique
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Idea: use perturbation theory to obtain effective 3-local Hamiltonians from 2-local ones by restricting
to subspaces
H’ = H + V
Spectrum: H…
0 groundspace S
Energy gap: ||H||>>||V||
What is the effective Hamiltonian in the lower part of the spectrum?
Three-qubit gadgetThree-qubit gadget
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Perturbation TheoryPerturbation Theory
H’ = H + V
Spectrum: H
0 groundspace S
Energy gap: ||H||>>||V||
S
Case 1: Energy gap >>> ||V|| V V
VV V
S
S
V-- - restriction of V to S
V++ - restriction of V to S
What is the effective Hamiltonian in the lower part of the spectrum?
Projection Lemma: Heff = V-- (same spectrum) =O(||V||2/)
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Perturbation TheoryPerturbation Theory
H’ = H + V
Spectrum: H
0 groundspace S
Energy gap: ||H||>>||V||
Theorem:
2 3 1
1 1 1...
n
eff n
VH V V V V V V V V V V O
S
What is the effective Hamiltonian in the lower part of the spectrum?
Case 2: Fine tune the energy gap > ||V|| V V
VV V
S
S
V-- - restriction of V to S
V++ - restriction of V to S
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Perturbation TheoryPerturbation Theory
H
0 groundspace S
Energy gap:
Theorem:
2 3 1
1 1 1...
n
eff n
VH V V V V V V V V V V O
S
First orderSecond order
Third order
The lower spectrum of H’ is close to the spectrum of Heff (under certain conditions).
H’ = H + V
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Perturbation TheoryPerturbation Theory
H
0 groundspace S
Energy gap:
Theorem: 21
...eff
VH V V V V O
S
First order: ||V||2 <<
The lower eigenvalues (<||V||) of H’ are close to the eigenvalues of Heff (under certain conditions).
Projection Lemma
H’ = H + V
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Three-qubit gadgetThree-qubit gadget
H=P1P2P3
3-local
1
32
1
32
B
A
C
ZZ
ZZ
ZZ
P1XA
P2XB P3XC
Terms in H’ are 2-local
Heff=P1P2P3
3-local
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Three-qubit gadgetThree-qubit gadgetH’ = H + V
Energy gap:
S={|000, |111}
S={|001,|010,|100, |110,|101,|011}
0
=-3B
A
C
ZZ
ZZ
ZZ
3
34 A B B C A CH Z Z Z Z Z Z I
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Three-qubit gadgetThree-qubit gadget
B
A
C
H’ = H + V
Energy gap:
S={|000, |111}
S={|001,|010,|100, |110,|101,|011}
0
=-3
2 P2XB3P3XC
1P1XA
Theorem: 4 32
1 1effH V V V V V V O V
Second order: S S
SV-+ V+-
Third order: S S
S V-+ V+-
V++S
First order: S SV--
V VV
V V
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Three-qubit gadgetThree-qubit gadget
B
A
C
Energy gap:
S={|000, |111}
S={|001,|010,|100, |110,|101,|011}
0
=-3
2 P2XB3P3XC
1P1XA
Theorem:
Second order: S S
SV-+ V+-Ex.: P1XA P1XA
|000
|100
|000
1 000 100 ...V P 2
1 000 000 ...V V P
4 32
1 1effH V V V V V V O V
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Three-qubit gadgetThree-qubit gadget
B
A
C
H’ = H + V
Energy gap:
S={|000, |111}
S={|001,|010,|100, |110,|101,|011}
0
=-3
2 P2XB3P3XC
1P1XA
Theorem:
Third order: S S
S V-+ V+-
V++S
Ex.: P1XA P3XC
|000
|100 |110
|111
P2XB
1 2 3 000 111 ...V V V PP P
4 32
1 1effH V V V V V V O V
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Three-qubit gadgetThree-qubit gadget
B
A
CH’ = H + V
2 P2XB3P3XC
1P1XA
1 2 2 21 2 3P P P 2
1 2 3A B CV PX P X P X
0V
1 12
2
000 100 111 011
000 010 ...
P PV
P
21 2010 110 100 110 ...V P P
Theorem:
4 32
1 1 effH V V V V V V O V
3
34 A B B C A CH Z Z Z Z Z Z I
4
2 2 21 2 33
0 SP P P I
6
1 2 36
3 000 111 111 000PP P O
1 2 2 21 2 3 SP P P I
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Three-qubit gadgetThree-qubit gadget
B
A
CH’ = H + V
2 P2XB3P3XC
1P1XA
1 2 2 21 2 3P P P 2
1 2 3A B CV PX P X P X
V VV
V V
0V
1 12
2
000 100 111 011
000 010 ...
P PV
P
21 2010 110 100 110 ...V P P
Theorem:
3
34 A B B C A CH Z Z Z Z Z Z I
1 2 2 21 2 3 SP P P I
4 32
1 1 effH V V V V V V O V
6
1 2 36
3 000 111 111 000PP P O
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Three-qubit gadgetThree-qubit gadget
B
A
CH’ = H + V
2 P2XB3P3XC
1P1XA
1 2 2 21 2 3P P P 2
1 2 3A B CV PX P X P X
V VV
V V
0V
1 12
2
000 100 111 011
000 010 ...
P PV
P
21 2010 110 100 110 ...V P P
Theorem:
4 32
1 1 effH V V V V V V O V
3
34 A B B C A CH Z Z Z Z Z Z I
1 2 33 SPP P X O
1 2 2 21 2 3 SP P P I
effH
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Three-qubit gadgetThree-qubit gadget
B
A
CH’ = H + V
2 P2XB3P3XC
1P1XA
1 2 2 21 2 3P P P 2
1 2 3A B CV PX P X P X
Theorem:
4 32
1 1 effH V V V V V V O V
3
34 A B B C A CH Z Z Z Z Z Z I
1 2 33 SPP P X O effH
=-3
0
H
0
-1V
Heff
const.
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2-local Hamiltonian is QMA-complete2-local Hamiltonian is QMA-complete
• start with the QMA-complete 3-local Hamiltonian
• replace each 3-local term by 3-qubit gadget
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OutlineOutline
A Bit of History• QMA • Local Hamiltonians
Previous Constructions
The 3-qubit Gadget
Implications• Adiabatic computation• Other applications of the technique
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Implications for Adiabatic ComputationImplications for Adiabatic ComputationAdiabatic computation [Farhi et al.’00]:
• “track” the groundstate of a slowly varying Hamiltonian
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", lanl-report quant-ph/0405098
Adiabatic simulation*:
Hinitial
•groundstate |0…0 |0
Hfinal
•groundstate
H(t) = (1-t/T’)Hinitial +t/T’ Hfinal
T’=poly(T):
If gap 0(H(t))-1(H(t)) between groundstate and first excited state is 1/poly(T)
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Implications for Adiabatic ComputationImplications for Adiabatic Computation
2-local adiabatic computation is equivalent to standard quantum computation
Our result also implies:
*D. Aharonov, W. van Dam, J. Kempe, Z. Landau, S. Lloyd, O. Regev: "Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation", lanl-report quant-ph/0405098
|0…0 |0 adiabat
H(t) = (1-t/T’) Hin + t/T’ HpropLog-local*:
Replace with 2-local: H(t) = (1-t/T’)(Hin+Jclock Hclock) + t/T’(Hprop
gadget+Jclock Hclock)
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Other applications of the Other applications of the gadgetgadget
(work in progress)(work in progress)“Interaction at a distance”:
H=P1P2Heff=P1P2
-1P1XA -1P2XA
-2ZA
“Proxy Interaction”: (with A. Landahl)
H=Z1X2
only XX,YY,ZZ availableHeff=Z1X2
-2YAYB-1Z1ZA -1X2XB
Useful for Hamiltonian-based quantum architectures
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ReferencesReferences
Quantum Complexity :J. Kempe, A. Kitaev, O. Regev: “The Complexity of the local
Hamiltonian Problem”, quant-ph/0406180, to appear in Proc. FSTTCS’04
J. Kempe and O. Regev: "3-Local Hamiltonian is QMA-complete", Quantum Information and Computation, Vol. 3 (3), p.258-64 (2003), lanl-report quant-ph/0302079
Adiabatic Computation :D. Aharonov, W. van Dam, J. Kempe, Z. Landau, S. Lloyd, O.
Regev: "Adiabatic Quantum Computationis Equivalent to Standard Quantum Computation", lanl-report quant-ph/0405098, to appear in FOCS’04
*Photo: Oded Regev: Ladybug reading “3-local Hamiltonian” paper
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MAMA
V“yes” instance: x Lyes
witness: y1 (accept)
V“no” instance: x Lno
for all “witnesses” y
0 (reject)
MA – Merlin-Artur:Def. L MA if there is a poly-time verifier V and a polynomial p s.t.
p( x )
p( x )
y {0,1} prob V(x,y)=1 1
y {0,1} prob V(x,y)=1
yes
no
x L
x L
0 (reject)
prob 1-prob
1 (accept) prob
prob 1-