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

…


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!


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



The 3-qubit Gadget


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



The 3-qubit Gadget


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


S

Case 1: Energy gap >>> ||V|| V V

VV V

S

S

V-- - restriction of V to S

V++ - restriction of V to S


Projection Lemma: Heff = V-- (same spectrum) =O(||V||2/)


H’ = H + V

Spectrum: H

0 groundspace S


Theorem:

2 3 1

1 1 1...

n

eff n

VH V V V V V V V V V V O

S


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


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


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


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


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


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



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



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


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


B

A

CH’ = H + V

2 P2XB3P3XC

1P1XA

1 2 2 21 2 3P P P 2


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


1 2 2 21 2 3 SP P P I

4 32


6

1 2 36

3 000 111 111 000PP P O


B

A

CH’ = H + V

2 P2XB3P3XC

1P1XA

1 2 2 21 2 3P P P 2


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


3


1 2 33 SPP P X O

1 2 2 21 2 3 SP P P I

effH


B

A

CH’ = H + V

2 P2XB3P3XC

1P1XA

1 2 2 21 2 3P P P 2


Theorem:

4 32


3


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



The 3-qubit Gadget


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-

quantum information theory: present status and future directions

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