Roadmap for Coatings Approval Current Status and Future Directions.
Quantum Information Theory: Present Status and Future Directions
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Transcript of Quantum Information Theory: Present Status and Future Directions
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
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:
OutlineOutline
A Bit of History• QMA • Local Hamiltonians
Previous Constructions
The 3-qubit Gadget
Implications• Adiabatic computation• Other applications of the technique
A Bit of (ancient) A Bit of (ancient) HistoryHistory
Complexity Theory:
• classify “easy” and “hard”
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)
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)
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
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
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
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.
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
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
NP and QMANP and QMANP-completeness: QMA-completeness?
x1
x2
…y1
y2
…00…
1x
y
0
Verifier V:
input
witness
ancilla
…
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!
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
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??
OutlineOutline
A Bit of History• QMA • Local Hamiltonians
Previous Constructions
The 3-qubit Gadget
Implications• Adiabatic computation• Other applications of the technique
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}
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-.
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
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)
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
OutlineOutline
A Bit of History• QMA • Local Hamiltonians
Previous Constructions
The 3-qubit Gadget
Implications• Adiabatic computation• Other applications of the technique
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
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/)
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
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
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
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
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
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
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
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
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
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
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
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.
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
OutlineOutline
A Bit of History• QMA • Local Hamiltonians
Previous Constructions
The 3-qubit Gadget
Implications• Adiabatic computation• Other applications of the technique
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)
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)
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
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
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-