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5年弱前 (2013/02/05)にアップロードin学び

Presented at GCOE interdisciplinary workshop on numerical methods for many-body correlations, htt...

Presented at GCOE interdisciplinary workshop on numerical methods for many-body correlations, https://sites.google.com/a/cns.s.u-tokyo.ac.jp/shimizu/gcoe

- Direct variational calculation of

second-order reduced density matrix :

application to the two-dimensional

Hubbard model

中田 「アングリーバード」 真秀

maho@riken.jp

http://nakatamaho.riken.jp/

RIKEN, Advanced Center for Computing and Communication

GCOE interdisciplinary workshop on numerical methods for many-body correlations, Faculity of

Science Building 4, Room 1320, Hongo Campus, Univ of Tokyo, 14:10-14:50

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Hubbard model - Today’s Angry Birds Score

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Hubbard model - Overview

Introduction of reduced density matrix method

Introduction of semidefinite programming

Application to two dimensional Hubbard model

The fundamental theoretical limitations of

calculating the ground and excited state on

computers

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Hubbard model - Introduction of reduced density matrix method

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Hubbard model - 98% of chemistry is explained by obtaining the ground state

wavefunction and the total energy by solving Schr ¨odinger

equation with Born-Oppenheimer approximation (+ basis set expansion).

Hamiltonian H

N

∑ 2

K

∑ Z

∑

Ae2

e2

H =

−

∇2 −

+

2m

j

4π

4π

j=1

e

A

0 r A j

i> j

0 ri j

Schr ¨odinger equation

HΨ(1, 2, · · · N) = EΨ(1, 2, · · · N)

Pauli exclusion principle

Ψ(· · · , i, · · · , j, · · · ) = −Ψ(· · · , j, · · · , i, · · · )

Problem statement

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Hamiltonian H

N

∑ 2

K

∑ Z

∑

Ae2

e2

H =

−

∇2 −

+

2m

j

4π

4π

j=1

e

A

0 r A j

i> j

0 ri j

Schr ¨odinger equation

HΨ(1, 2, · · · N) = EΨ(1, 2, · · · N)

Pauli exclusion principle

Ψ(· · · , i, · · · , j, · · · ) = −Ψ(· · · , j, · · · , i, · · · )

Problem statement

98% of chemistry is explained by obtaining the ground state

wavefunction and the total energy by solving Schr ¨odinger

equation with Born-Oppenheimer approximation (+ basis set expansion).

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Schr ¨odinger equation

HΨ(1, 2, · · · N) = EΨ(1, 2, · · · N)

Pauli exclusion principle

Ψ(· · · , i, · · · , j, · · · ) = −Ψ(· · · , j, · · · , i, · · · )

Problem statement

98% of chemistry is explained by obtaining the ground state

wavefunction and the total energy by solving Schr ¨odinger

equation with Born-Oppenheimer approximation (+ basis set expansion).

Hamiltonian H

N

∑ 2

K

∑ Z

∑

Ae2

e2

H =

−

∇2 −

+

2m

j

4π

4π

j=1

e

A

0 r A j

i> j

0 ri j

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Hubbard model - HΨ(1, 2, · · · N) = EΨ(1, 2, · · · N)

Pauli exclusion principle

Ψ(· · · , i, · · · , j, · · · ) = −Ψ(· · · , j, · · · , i, · · · )

Problem statement

98% of chemistry is explained by obtaining the ground state

wavefunction and the total energy by solving Schr ¨odinger

equation with Born-Oppenheimer approximation (+ basis set expansion).

Hamiltonian H

N

∑ 2

K

∑ Z

∑

Ae2

e2

H =

−

∇2 −

+

2m

j

4π

4π

j=1

e

A

0 r A j

i> j

0 ri j

Schr ¨odinger equation

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Pauli exclusion principle

Ψ(· · · , i, · · · , j, · · · ) = −Ψ(· · · , j, · · · , i, · · · )

Problem statement

98% of chemistry is explained by obtaining the ground state

wavefunction and the total energy by solving Schr ¨odinger

equation with Born-Oppenheimer approximation (+ basis set expansion).

Hamiltonian H

N

∑ 2

K

∑ Z

∑

Ae2

e2

H =

−

∇2 −

+

2m

j

4π

4π

j=1

e

A

0 r A j

i> j

0 ri j

Schr ¨odinger equation

HΨ(1, 2, · · · N) = EΨ(1, 2, · · · N)

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Ψ(· · · , i, · · · , j, · · · ) = −Ψ(· · · , j, · · · , i, · · · )

Problem statement

98% of chemistry is explained by obtaining the ground state

wavefunction and the total energy by solving Schr ¨odinger

equation with Born-Oppenheimer approximation (+ basis set expansion).

Hamiltonian H

N

∑ 2

K

∑ Z

∑

Ae2

e2

H =

−

∇2 −

+

2m

j

4π

4π

j=1

e

A

0 r A j

i> j

0 ri j

Schr ¨odinger equation

HΨ(1, 2, · · · N) = EΨ(1, 2, · · · N)

Pauli exclusion principle

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Problem statement

98% of chemistry is explained by obtaining the ground state

wavefunction and the total energy by solving Schr ¨odinger

equation with Born-Oppenheimer approximation (+ basis set expansion).

Hamiltonian H

N

∑ 2

K

∑ Z

∑

Ae2

e2

H =

−

∇2 −

+

2m

j

4π

4π

j=1

e

A

0 r A j

i> j

0 ri j

Schr ¨odinger equation

HΨ(1, 2, · · · N) = EΨ(1, 2, · · · N)

Pauli exclusion principle

Ψ(· · · , i, · · · , j, · · · ) = −Ψ(· · · , j, · · · , i, · · · )

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - The general theory of quantum

mechanics is now almost com-

plete. · · · the whole of chemistry

are thus completely known, and

the difficultly is only that the exact

application of these laws leads to

equations much too complected

to be soluble.

[Dirac 1929]

“Quantum Mechanics of Many-Electron Systems.”

✞

☎

However, Dirac did’t say how difficult it is!

✝

✆

“The Schr ¨odinger equation is much too complected to be soluble.”

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Hubbard model - ✞

☎

However, Dirac did’t say how difficult it is!

✝

✆

“The Schr ¨odinger equation is much too complected to be soluble.”

The general theory of quantum

mechanics is now almost com-

plete. · · · the whole of chemistry

are thus completely known, and

the difficultly is only that the exact

application of these laws leads to

equations much too complected

to be soluble.

[Dirac 1929]

“Quantum Mechanics of Many-Electron Systems.”

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - “The Schr ¨odinger equation is much too complected to be soluble.”

The general theory of quantum

mechanics is now almost com-

plete. · · · the whole of chemistry

are thus completely known, and

the difficultly is only that the exact

application of these laws leads to

equations much too complected

to be soluble.

[Dirac 1929]

“Quantum Mechanics of Many-Electron Systems.”

✞

☎

However, Dirac did’t say how difficult it is!

✝

✆

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Classical answers to sigh of Dirac (I)

✞

☎

✝Density Functional Theory (DFT) ✆

“Electron” density can be used fundamental variable, instead of wavefunction:

∫

∫

∫

ρ(r) = N

d3 r2

d3 r3 · · ·

d3 rNΨ∗(r, r2, . . . , rN)Ψ(r, r2, . . . rN)

Hohenberg-Kohn theorem states that there exists universal functional

F[ρ]

∫

E[ρ] = F[ρ] +

V(r)ρ(r)d3r

Kohn-Sham equation.

(

)

2

−

∇2 + veﬀ(r) φi(r) = εiφi(r)

2m

v-representability : simply we ignore.

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Hubbard model - ☛

✟

No, I don’t think so.

✡

✠

Classical answers to sigh of Dirac (I)

“Electron” density can be used fundamental variable, instead of wavefunction:

∫

∫

∫

ρ(r) = N

d3 r2

d3 r3 · · ·

d3 rNΨ∗(r, r2, . . . , rN)Ψ(r, r2, . . . rN)

Why this believed to answer to Dirac?

Now the basic variable is ρ instead of complicated Ψ.

B3LYP rules!

However:

Quite semi-empirical,and very hard to understand why B3LYP works for

molecules!

systematic improvement is not possible.

✄

✂So what? Is it an answer to Dirac? ✁

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Hubbard model - Classical answers to sigh of Dirac (I)

“Electron” density can be used fundamental variable, instead of wavefunction:

∫

∫

∫

ρ(r) = N

d3 r2

d3 r3 · · ·

d3 rNΨ∗(r, r2, . . . , rN)Ψ(r, r2, . . . rN)

Why this believed to answer to Dirac?

Now the basic variable is ρ instead of complicated Ψ.

B3LYP rules!

However:

Quite semi-empirical,and very hard to understand why B3LYP works for

molecules!

systematic improvement is not possible.

✄

✂So what? Is it an answer to Dirac?

☛

✁ ✟

No, I don’t think so.

✡

✠

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Hubbard model - [Coulson 1960] “wave functions tell us more than we need to

know...All the necessary information required for energy and

calculating properties of molecules is embodied in the first and

second order density matrices.”

[Husimi 1940], [L¨owdin 1954], [Mayer 1955], [Coulson 1960], [Coleman 1963], [Rosina 1968]

( ) ∫

Γ

N

(12|1 2 ) =

Ψ(12 · · · N)Ψ∗(1 2 3 · · · N)d3 · · · dN

2

∫

γ(1|1 ) = N

Ψ(12 · · · N)Ψ∗(1 2 · · · N)d2d3 · · · dN

Coulson was a leading (old) quantum chemist, and supervisor of Peter Higgs.

Husimi K ˆodi was a leading physist who first defined the RDM of general order.

Classical answers to sigh of Dirac (II)

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Hubbard model - [Husimi 1940], [L¨owdin 1954], [Mayer 1955], [Coulson 1960], [Coleman 1963], [Rosina 1968]

( ) ∫

Γ

N

(12|1 2 ) =

Ψ(12 · · · N)Ψ∗(1 2 3 · · · N)d3 · · · dN

2

∫

γ(1|1 ) = N

Ψ(12 · · · N)Ψ∗(1 2 · · · N)d2d3 · · · dN

Coulson was a leading (old) quantum chemist, and supervisor of Peter Higgs.

Husimi K ˆodi was a leading physist who first defined the RDM of general order.

Classical answers to sigh of Dirac (II)

[Coulson 1960] “wave functions tell us more than we need to

know...All the necessary information required for energy and

calculating properties of molecules is embodied in the first and

second order density matrices.”

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

GCOE

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Hubbard model - Classical answers to sigh of Dirac (II)

[Coulson 1960] “wave functions tell us more than we need to

know...All the necessary information required for energy and

calculating properties of molecules is embodied in the first and

second order density matrices.”

[Husimi 1940], [L¨owdin 1954], [Mayer 1955], [Coulson 1960], [Coleman 1963], [Rosina 1968]

( ) ∫

Γ

N

(12|1 2 ) =

Ψ(12 · · · N)Ψ∗(1 2 3 · · · N)d3 · · · dN

2

∫

γ(1|1 ) = N

Ψ(12 · · · N)Ψ∗(1 2 · · · N)d2d3 · · · dN

Coulson was a leading (old) quantum chemist, and supervisor of Peter Higgs.

Husimi K ˆodi was a leading physist who first defined the RDM of general order.

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - 2-RDM has only four variables.

Equivalent to solve Schr ¨odinger equation.

Minimization of linear functional.

Classical answers to sigh of Dirac (II)

The first- and second-order reduced density matrices is defined as:

( ) ∫

Γ

N

(12|1 2 ) =

Ψ(12 · · · N)Ψ∗(1 2 3 · · · N)d3 · · · dN

2

∫

γ(1|1 ) = N

Ψ(12 · · · N)Ψ∗(1 2 · · · N)d2d3 · · · dN.

Then, minimize

Eg = Min TrHΓ

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Equivalent to solve Schr ¨odinger equation.

Minimization of linear functional.

Classical answers to sigh of Dirac (II)

The first- and second-order reduced density matrices is defined as:

( ) ∫

Γ

N

(12|1 2 ) =

Ψ(12 · · · N)Ψ∗(1 2 3 · · · N)d3 · · · dN

2

∫

γ(1|1 ) = N

Ψ(12 · · · N)Ψ∗(1 2 · · · N)d2d3 · · · dN.

Then, minimize

Eg = Min TrHΓ

2-RDM has only four variables.

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Minimization of linear functional.

Classical answers to sigh of Dirac (II)

The first- and second-order reduced density matrices is defined as:

( ) ∫

Γ

N

(12|1 2 ) =

Ψ(12 · · · N)Ψ∗(1 2 3 · · · N)d3 · · · dN

2

∫

γ(1|1 ) = N

Ψ(12 · · · N)Ψ∗(1 2 · · · N)d2d3 · · · dN.

Then, minimize

Eg = Min TrHΓ

2-RDM has only four variables.

Equivalent to solve Schr ¨odinger equation.

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Classical answers to sigh of Dirac (II)

The first- and second-order reduced density matrices is defined as:

( ) ∫

Γ

N

(12|1 2 ) =

Ψ(12 · · · N)Ψ∗(1 2 3 · · · N)d3 · · · dN

2

∫

γ(1|1 ) = N

Ψ(12 · · · N)Ψ∗(1 2 · · · N)d2d3 · · · dN.

Then, minimize

Eg = Min TrHΓ

2-RDM has only four variables.

Equivalent to solve Schr ¨odinger equation.

Minimization of linear functional.

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

GCOE

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Hubbard model - Classical answers to sigh of Dirac (II)

[Husimi 1940], [L ¨owdin 1954], [Mayer 1955], [Coulson 1960], [Coleman 1963], [Rosina 1968]

( ) ∫

Γ

N

(12|1 2 ) =

Ψ(12 · · · N)Ψ∗(1 2 3 · · · N)d3 · · · dN

2

∫

γ(1|1 ) = N

Ψ(12 · · · N)Ψ∗(1 2 · · · N)d2d3 · · · dN

How this approach is different from the DFT?

We can apply to any general Hamiltonian. In DFT F[ρ] should be

recalculated when two-particle interaction has changed.

All observables up to two-particles including kinetic energy can be

evaluated exactly, while it’s impossible in DFT.

We have N-representability to 2-RDM instead of v-rep., and it is much

more systematic.

The number of variables becomes four instead of one.

Just minimize linear functional.

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Hubbard model - ✞

☎

✝Is it an answer to Dirac?

☛

✆

✟

Yes, partially.

✡

✠

Classical answers to sigh of Dirac (II)

Our approach: To obtain the Ground state energy and property, use 1, 2-RDMs

instead of the wavefunction,

( ) ∫

Γ

N

(12|1 2 ) =

Ψ(12 · · · N)Ψ∗(1 2 3 · · · N)d3 · · · dN

2

∫

γ(1|1 ) = N

Ψ(12 · · · N)Ψ∗(1 2 · · · N)d2d3 · · · dN.

Then, just minimize the Hamiltonian

Eg = Min TrHΓ

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Hubbard model - ☛

✟

Yes, partially.

✡

✠

Classical answers to sigh of Dirac (II)

Our approach: To obtain the Ground state energy and property, use 1, 2-RDMs

instead of the wavefunction,

( ) ∫

Γ

N

(12|1 2 ) =

Ψ(12 · · · N)Ψ∗(1 2 3 · · · N)d3 · · · dN

2

∫

γ(1|1 ) = N

Ψ(12 · · · N)Ψ∗(1 2 · · · N)d2d3 · · · dN.

Then, just minimize the Hamiltonian

Eg = Min TrHΓ

✞

☎

✝Is it an answer to Dirac? ✆

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

GCOE

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Hubbard model - Classical answers to sigh of Dirac (II)

Our approach: To obtain the Ground state energy and property, use 1, 2-RDMs

instead of the wavefunction,

( ) ∫

Γ

N

(12|1 2 ) =

Ψ(12 · · · N)Ψ∗(1 2 3 · · · N)d3 · · · dN

2

∫

γ(1|1 ) = N

Ψ(12 · · · N)Ψ∗(1 2 · · · N)d2d3 · · · dN.

Then, just minimize the Hamiltonian

Eg = Min TrHΓ

✞

☎

✝Is it an answer to Dirac?

☛

✆

✟

Yes, partially.

✡

✠

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - ∑

∑

H =

vi a† a

wi1i2 a† a† a a

j

j + 1

j

j

i

2

j

2

1

1 j2

i1 i2

i j

i1i2 j1 j2

The ground stat energy becomes:

Eg = min Ψ|H|Ψ

∑

∑

= min

vi Ψ|a†a

wi1i2 Ψ|a† a† a a |Ψ

j

j|Ψ + 1

j

j

i

2

j

2

1

1 j2

i1 i2

i j

i1i2 j1 j2

∑

∑

= min{

vi γi +

wi1i2 Γi1i2 }

j

j

j1 j2

j1 j2

i j

i1i2 j1 j2

Second quantized version of 1, 2-RDM.

Γi1i2 = 1 Ψ|a† a† aj aj |Ψ , γi = Ψ|a†aj|Ψ .

j

2

1

1 j2

2

i1 i2

j

i

The RDM method : ground state calculation using 2-RDM

[Husimi 1940], [L ¨owdin 1954], [Mayer 1955], [Coulson 1960], [Coleman 1963], [Rosina 1968]

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Hubbard model - The ground stat energy becomes:

Eg = min Ψ|H|Ψ

∑

∑

= min

vi Ψ|a†a

wi1i2 Ψ|a† a† a a |Ψ

j

j|Ψ + 1

j

j

i

2

j

2

1

1 j2

i1 i2

i j

i1i2 j1 j2

∑

∑

= min{

vi γi +

wi1i2 Γi1i2 }

j

j

j1 j2

j1 j2

i j

i1i2 j1 j2

Second quantized version of 1, 2-RDM.

Γi1i2 = 1 Ψ|a† a† aj aj |Ψ , γi = Ψ|a†aj|Ψ .

j

2

1

1 j2

2

i1 i2

j

i

The RDM method : ground state calculation using 2-RDM

[Husimi 1940], [L ¨owdin 1954], [Mayer 1955], [Coulson 1960], [Coleman 1963], [Rosina 1968]

∑

∑

H =

vi a† a

wi1i2 a† a† a a

j

j + 1

j

j

i

2

j

2

1

1 j2

i1 i2

i j

i1i2 j1 j2

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Hubbard model - Second quantized version of 1, 2-RDM.

Γi1i2 = 1 Ψ|a† a† aj aj |Ψ , γi = Ψ|a†aj|Ψ .

j

2

1

1 j2

2

i1 i2

j

i

The RDM method : ground state calculation using 2-RDM

[Husimi 1940], [L ¨owdin 1954], [Mayer 1955], [Coulson 1960], [Coleman 1963], [Rosina 1968]

∑

∑

H =

vi a† a

wi1i2 a† a† a a

j

j + 1

j

j

i

2

j

2

1

1 j2

i1 i2

i j

i1i2 j1 j2

The ground stat energy becomes:

Eg = min Ψ|H|Ψ

∑

∑

= min

vi Ψ|a†a

wi1i2 Ψ|a† a† a a |Ψ

j

j|Ψ + 1

j

j

i

2

j

2

1

1 j2

i1 i2

i j

i1i2 j1 j2

∑

∑

= min{

vi γi +

wi1i2 Γi1i2 }

j

j

j1 j2

j1 j2

i j

i1i2 j1 j2

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Hubbard model - The RDM method : ground state calculation using 2-RDM

[Husimi 1940], [L ¨owdin 1954], [Mayer 1955], [Coulson 1960], [Coleman 1963], [Rosina 1968]

∑

∑

H =

vi a† a

wi1i2 a† a† a a

j

j + 1

j

j

i

2

j

2

1

1 j2

i1 i2

i j

i1i2 j1 j2

The ground stat energy becomes:

Eg = min Ψ|H|Ψ

∑

∑

= min

vi Ψ|a†a

wi1i2 Ψ|a† a† a a |Ψ

j

j|Ψ + 1

j

j

i

2

j

2

1

1 j2

i1 i2

i j

i1i2 j1 j2

∑

∑

= min{

vi γi +

wi1i2 Γi1i2 }

j

j

j1 j2

j1 j2

i j

i1i2 j1 j2

Second quantized version of 1, 2-RDM.

Γi1i2 = 1 Ψ|a† a† aj aj |Ψ , γi = Ψ|a†aj|Ψ .

j

2

1

1 j2

2

i1 i2

j

i

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - ✞

☎

Looks too simple, but what is the barter?

✝

✆

The RDM method : ground state calculation using 2-RDM

Eg = min Ψ|H|Ψ

∑

∑

= min{

vi γi +

wi1i2 Γi1i2 }

j

j

j1 j2

j1 j2

i j

i1i2 j1 j2

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - The RDM method : ground state calculation using 2-RDM

Eg = min Ψ|H|Ψ

∑

∑

= min{

vi γi +

wi1i2 Γi1i2 }

j

j

j1 j2

j1 j2

i j

i1i2 j1 j2

✞

☎

Looks too simple, but what is the barter?

✝

✆

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - [Mayers 1955], [Tredgold 1957]: early attempts failed with too low

energies from the exact value.

N-representability condition coined by [Coleman 1963].

∑

∑

Eg = min{

vi γi +

wi1i2 Γi1i2 }

P

j

j

j1 j2

j1 j2

i j

i1i2 j1 j2

γ, Γ ∈ P should satisfy N-representability condition.

Γ(12|1 2 ) → Ψ(123 · · · N)

γ(1|1 ) → Ψ(123 · · · N).

The N-representability conditions

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - N-representability condition coined by [Coleman 1963].

∑

∑

Eg = min{

vi γi +

wi1i2 Γi1i2 }

P

j

j

j1 j2

j1 j2

i j

i1i2 j1 j2

γ, Γ ∈ P should satisfy N-representability condition.

Γ(12|1 2 ) → Ψ(123 · · · N)

γ(1|1 ) → Ψ(123 · · · N).

The N-representability conditions

[Mayers 1955], [Tredgold 1957]: early attempts failed with too low

energies from the exact value.

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - The N-representability conditions

[Mayers 1955], [Tredgold 1957]: early attempts failed with too low

energies from the exact value.

N-representability condition coined by [Coleman 1963].

∑

∑

Eg = min{

vi γi +

wi1i2 Γi1i2 }

P

j

j

j1 j2

j1 j2

i j

i1i2 j1 j2

γ, Γ ∈ P should satisfy N-representability condition.

Γ(12|1 2 ) → Ψ(123 · · · N)

γ(1|1 ) → Ψ(123 · · · N).

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Trace condition

∑

∑

N(N − 1)

γi = N,

Γij =

i

i j

2

i=1

i, j=1

Trace condition (higher to lower)

N − 1

∑

γi =

Γik

2

j

jk

k=1

1, 2-RDM should be Hermitian

γi = (γj)∗, Γi1i2 = (Γj2 j1)∗, Γi1i2 = −Γi2i1 = −Γi1i2 ...

j

i

j1 j2

i1i2

j1 j2

j1 j2

j2 j1

The N-representability condition

Some explict forms of N-representability conditions:

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Trace condition (higher to lower)

N − 1

∑

γi =

Γik

2

j

jk

k=1

Trace condition

∑

∑

N(N − 1)

γi = N,

Γij =

i

i j

2

i=1

i, j=1

The N-representability condition

Some explict forms of N-representability conditions:

1, 2-RDM should be Hermitian

γi = (γj)∗, Γi1i2 = (Γj2 j1)∗, Γi1i2 = −Γi2i1 = −Γi1i2 ...

j

i

j1 j2

i1i2

j1 j2

j1 j2

j2 j1

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Trace condition (higher to lower)

N − 1

∑

γi =

Γik

2

j

jk

k=1

The N-representability condition

Some explict forms of N-representability conditions:

1, 2-RDM should be Hermitian

γi = (γj)∗, Γi1i2 = (Γj2 j1)∗, Γi1i2 = −Γi2i1 = −Γi1i2 ...

j

i

j1 j2

i1i2

j1 j2

j1 j2

j2 j1

Trace condition

∑

∑

N(N − 1)

γi = N,

Γij =

i

i j

2

i=1

i, j=1

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - The N-representability condition

Some explict forms of N-representability conditions:

1, 2-RDM should be Hermitian

γi = (γj)∗, Γi1i2 = (Γj2 j1)∗, Γi1i2 = −Γi2i1 = −Γi1i2 ...

j

i

j1 j2

i1i2

j1 j2

j1 j2

j2 j1

Trace condition

∑

∑

N(N − 1)

γi = N,

Γij =

i

i j

2

i=1

i, j=1

Trace condition (higher to lower)

N − 1

∑

γi =

Γik

2

j

jk

k=1

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Hubbard model - G-condition [Garrod and Percus 1964]

T1, T2, T2 , ( ¯

T2)-condition [Zhao et al. 2004], [Erdahl 1978]

[Braams et al 2007] [Mazziotti 2006, 2007]

Complete N-rep. condition (not practical) [Garrod-Percus

1964].

Computational complexity of complete N-rep. condition :

QMA-complete [Liu 2007].

P, Q-condition, complete condition for 1-RDM [Coleman

1963]

✞

☎

✝How effective and how to calculate 2-RDM systematically had been not known ✆

Some known facts about N-representability conditions

Approximate (necessity) condtion : some well known conditios.

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Hubbard model - T1, T2, T2 , ( ¯

T2)-condition [Zhao et al. 2004], [Erdahl 1978]

[Braams et al 2007] [Mazziotti 2006, 2007]

Complete N-rep. condition (not practical) [Garrod-Percus

1964].

Computational complexity of complete N-rep. condition :

QMA-complete [Liu 2007].

G-condition [Garrod and Percus 1964]

✞

☎

✝How effective and how to calculate 2-RDM systematically had been not known ✆

Some known facts about N-representability conditions

Approximate (necessity) condtion : some well known conditios.

P, Q-condition, complete condition for 1-RDM [Coleman

1963]

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Complete N-rep. condition (not practical) [Garrod-Percus

1964].

Computational complexity of complete N-rep. condition :

QMA-complete [Liu 2007].

T1, T2, T2 , ( ¯

T2)-condition [Zhao et al. 2004], [Erdahl 1978]

[Braams et al 2007] [Mazziotti 2006, 2007]

✞

☎

✝How effective and how to calculate 2-RDM systematically had been not known ✆

Some known facts about N-representability conditions

Approximate (necessity) condtion : some well known conditios.

P, Q-condition, complete condition for 1-RDM [Coleman

1963]

G-condition [Garrod and Percus 1964]

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Computational complexity of complete N-rep. condition :

QMA-complete [Liu 2007].

Complete N-rep. condition (not practical) [Garrod-Percus

1964].

✞

☎

✝How effective and how to calculate 2-RDM systematically had been not known ✆

Some known facts about N-representability conditions

Approximate (necessity) condtion : some well known conditios.

P, Q-condition, complete condition for 1-RDM [Coleman

1963]

G-condition [Garrod and Percus 1964]

T1, T2, T2 , ( ¯

T2)-condition [Zhao et al. 2004], [Erdahl 1978]

[Braams et al 2007] [Mazziotti 2006, 2007]

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Computational complexity of complete N-rep. condition :

QMA-complete [Liu 2007].

✞

☎

✝How effective and how to calculate 2-RDM systematically had been not known ✆

Some known facts about N-representability conditions

Approximate (necessity) condtion : some well known conditios.

P, Q-condition, complete condition for 1-RDM [Coleman

1963]

G-condition [Garrod and Percus 1964]

T1, T2, T2 , ( ¯

T2)-condition [Zhao et al. 2004], [Erdahl 1978]

[Braams et al 2007] [Mazziotti 2006, 2007]

Complete N-rep. condition (not practical) [Garrod-Percus

1964].

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - ✞

☎

✝How effective and how to calculate 2-RDM systematically had been not known ✆

Some known facts about N-representability conditions

Approximate (necessity) condtion : some well known conditios.

P, Q-condition, complete condition for 1-RDM [Coleman

1963]

G-condition [Garrod and Percus 1964]

T1, T2, T2 , ( ¯

T2)-condition [Zhao et al. 2004], [Erdahl 1978]

[Braams et al 2007] [Mazziotti 2006, 2007]

Complete N-rep. condition (not practical) [Garrod-Percus

1964].

Computational complexity of complete N-rep. condition :

QMA-complete [Liu 2007].

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

GCOE

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17 o-dimensional

/ 45

Hubbard model - Some known facts about N-representability conditions

Approximate (necessity) condtion : some well known conditios.

P, Q-condition, complete condition for 1-RDM [Coleman

1963]

G-condition [Garrod and Percus 1964]

T1, T2, T2 , ( ¯

T2)-condition [Zhao et al. 2004], [Erdahl 1978]

[Braams et al 2007] [Mazziotti 2006, 2007]

Complete N-rep. condition (not practical) [Garrod-Percus

1964].

Computational complexity of complete N-rep. condition :

QMA-complete [Liu 2007].

✞

☎

✝How effective and how to calculate 2-RDM systematically had been not known ✆

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Q-condition: eigenvalues of Γ should be non-negative [Coleman 1963]

Ψ|aiaja†a†|Ψ

0

l

k

G-condition: eigenvalues of Γ should be non-negative [Garrod-Perucs

1964]

Ψ|a†aja†ak|Ψ

0

i

l

T1-condition: [Zhao et al 2004] [Eradahl 1978]

Ψ|(a†a†a† anama + anama a†a†a†|Ψ

0

i

j

k

i

j

k

T2-condition: [Zhao et al 2004] [Eradahl 1978]

Ψ|(a†a†aka† ama + a† ama a†a†ak|Ψ

0

i

j

n

n

i

j

P-condition: eigenvalues of Γ should be non-negative [Coleman 1963]

Ψ|a†a†alak|Ψ

0

i

j

Positive semidefinite type of N-representability conditions

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Hubbard model - G-condition: eigenvalues of Γ should be non-negative [Garrod-Perucs

1964]

Ψ|a†aja†ak|Ψ

0

i

l

T1-condition: [Zhao et al 2004] [Eradahl 1978]

Ψ|(a†a†a† anama + anama a†a†a†|Ψ

0

i

j

k

i

j

k

T2-condition: [Zhao et al 2004] [Eradahl 1978]

Ψ|(a†a†aka† ama + a† ama a†a†ak|Ψ

0

i

j

n

n

i

j

Q-condition: eigenvalues of Γ should be non-negative [Coleman 1963]

Ψ|aiaja†a†|Ψ

0

l

k

Positive semidefinite type of N-representability conditions

P-condition: eigenvalues of Γ should be non-negative [Coleman 1963]

Ψ|a†a†alak|Ψ

0

i

j

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - T1-condition: [Zhao et al 2004] [Eradahl 1978]

Ψ|(a†a†a† anama + anama a†a†a†|Ψ

0

i

j

k

i

j

k

T2-condition: [Zhao et al 2004] [Eradahl 1978]

Ψ|(a†a†aka† ama + a† ama a†a†ak|Ψ

0

i

j

n

n

i

j

G-condition: eigenvalues of Γ should be non-negative [Garrod-Perucs

1964]

Ψ|a†aja†ak|Ψ

0

i

l

Positive semidefinite type of N-representability conditions

P-condition: eigenvalues of Γ should be non-negative [Coleman 1963]

Ψ|a†a†alak|Ψ

0

i

j

Q-condition: eigenvalues of Γ should be non-negative [Coleman 1963]

Ψ|aiaja†a†|Ψ

0

l

k

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - T2-condition: [Zhao et al 2004] [Eradahl 1978]

Ψ|(a†a†aka† ama + a† ama a†a†ak|Ψ

0

i

j

n

n

i

j

T1-condition: [Zhao et al 2004] [Eradahl 1978]

Ψ|(a†a†a† anama + anama a†a†a†|Ψ

0

i

j

k

i

j

k

Positive semidefinite type of N-representability conditions

P-condition: eigenvalues of Γ should be non-negative [Coleman 1963]

Ψ|a†a†alak|Ψ

0

i

j

Q-condition: eigenvalues of Γ should be non-negative [Coleman 1963]

Ψ|aiaja†a†|Ψ

0

l

k

G-condition: eigenvalues of Γ should be non-negative [Garrod-Perucs

1964]

Ψ|a†aja†ak|Ψ

0

i

l

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - T2-condition: [Zhao et al 2004] [Eradahl 1978]

Ψ|(a†a†aka† ama + a† ama a†a†ak|Ψ

0

i

j

n

n

i

j

Positive semidefinite type of N-representability conditions

P-condition: eigenvalues of Γ should be non-negative [Coleman 1963]

Ψ|a†a†alak|Ψ

0

i

j

Q-condition: eigenvalues of Γ should be non-negative [Coleman 1963]

Ψ|aiaja†a†|Ψ

0

l

k

G-condition: eigenvalues of Γ should be non-negative [Garrod-Perucs

1964]

Ψ|a†aja†ak|Ψ

0

i

l

T1-condition: [Zhao et al 2004] [Eradahl 1978]

Ψ|(a†a†a† anama + anama a†a†a†|Ψ

0

i

j

k

i

j

k

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Positive semidefinite type of N-representability conditions

P-condition: eigenvalues of Γ should be non-negative [Coleman 1963]

Ψ|a†a†alak|Ψ

0

i

j

Q-condition: eigenvalues of Γ should be non-negative [Coleman 1963]

Ψ|aiaja†a†|Ψ

0

l

k

G-condition: eigenvalues of Γ should be non-negative [Garrod-Perucs

1964]

Ψ|a†aja†ak|Ψ

0

i

l

T1-condition: [Zhao et al 2004] [Eradahl 1978]

Ψ|(a†a†a† anama + anama a†a†a†|Ψ

0

i

j

k

i

j

k

T2-condition: [Zhao et al 2004] [Eradahl 1978]

Ψ|(a†a†aka† ama + a† ama a†a†ak|Ψ

0

i

j

n

n

i

j

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - P,Q,G,T1,T2-matrices are all positive semidefinite ↔

eigenvalues are all non-negatieve.

λ

1

0

λ

2

U†ΓU =

0

.

.

.

0

λn

Positive semidefinite N-representability conditions

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Positive semidefinite N-representability conditions

P,Q,G,T1,T2-matrices are all positive semidefinite ↔

eigenvalues are all non-negatieve.

λ

1

0

λ

2

U†ΓU =

0

.

.

.

0

λn

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Geometical interpretation of approximate

N-representability condition

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - The first application to atoms: Be atom

[Garrod et al 1975, 1976]

Some how disappeared...

Reasons might be lack of computer resources, poor

results on nucleon systems, etc.

The first application to atoms

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Reasons might be lack of computer resources, poor

results on nucleon systems, etc.

The first application to atoms

The first application to atoms: Be atom

[Garrod et al 1975, 1976]

Some how disappeared...

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - The first application to atoms

The first application to atoms: Be atom

[Garrod et al 1975, 1976]

Some how disappeared...

Reasons might be lack of computer resources, poor

results on nucleon systems, etc.

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

GCOE

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Hubbard model - Eg = Min trHΓ

Γ∈P

P = {Γ : { N-rep.condition P, Q, G } ∈ Γ}

Positive semidefinite programming

The first results solved exactly and the first results to

molecules.

[Nakata-Nakatsuji-Ehara-Fukuda-Nakata-Fujisawa 2001]

[Nakata-Nakatsuji-Ehara 2002]

Small enough “primal dual gap, feasibility” values show that total energies etc are MATHEMATICALLY correct

Revival: introduction of the semidefinite programming

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Positive semidefinite programming

The first results solved exactly and the first results to

molecules.

[Nakata-Nakatsuji-Ehara-Fukuda-Nakata-Fujisawa 2001]

[Nakata-Nakatsuji-Ehara 2002]

Small enough “primal dual gap, feasibility” values show that total energies etc are MATHEMATICALLY correct

Revival: introduction of the semidefinite programming

Eg = Min trHΓ

Γ∈P

P = {Γ : { N-rep.condition P, Q, G } ∈ Γ}

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Small enough “primal dual gap, feasibility” values show that total energies etc are MATHEMATICALLY correct

Revival: introduction of the semidefinite programming

Eg = Min trHΓ

Γ∈P

P = {Γ : { N-rep.condition P, Q, G } ∈ Γ}

Positive semidefinite programming

The first results solved exactly and the first results to

molecules.

[Nakata-Nakatsuji-Ehara-Fukuda-Nakata-Fujisawa 2001]

[Nakata-Nakatsuji-Ehara 2002]

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Revival: introduction of the semidefinite programming

Eg = Min trHΓ

Γ∈P

P = {Γ : { N-rep.condition P, Q, G } ∈ Γ}

Positive semidefinite programming

The first results solved exactly and the first results to

molecules.

[Nakata-Nakatsuji-Ehara-Fukuda-Nakata-Fujisawa 2001]

[Nakata-Nakatsuji-Ehara 2002]

Small enough “primal dual gap, feasibility” values show that total energies etc are MATHEMATICALLY correct

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - The ground state energy of atoms and molecules [Nakata et al 2008]

System State N

r

∆ EGT1T2 ∆ EGT1T2

∆ ECCSD(T) ∆ EHF

EFCI

C

3 P

6 20

−0.0004

−0.0001

+0.00016 +0.05202 −37.73653

O

1 D

8 20

−0.0013

−0.0012

+0.00279 +0.10878 −74.78733

Ne

1S

10 20

−0.0002

−0.0001

−0.00005 +0.11645 −128.63881

O+

2Πg 15 20

−0.0022

−0.0020

+0.00325 +0.17074 −148.79339

2

BH

1Σ+

6 24

−0.0001

−0.0001

+0.00030 +0.07398 −25.18766

CH

2Πr

7 24

−0.0008

−0.0003

+0.00031 +0.07895 −38.33735

NH

1∆

8 24

−0.0005

−0.0004

+0.00437 +0.11495 −54.96440

HF

1Σ+ 14 24

−0.0003

−0.0003

+0.00032 +0.13834 −100.16031

SiH

1

4

A1

18 26

−0.0002

−0.0002

+0.00018 +0.07311 −290.28490

F−

1S

10 26

−0.0003

−0.0003

+0.00067 +0.15427 −99.59712

P

4S

15 26

−0.0001

−0.0000

+0.00003 +0.01908 −340.70802

H2O

1 A1 10 28

−0.0004

−0.0004

+0.00055 +0.14645 −76.15576

GT1T2

:

The RDM method (P, Q, G, T1 and T2 conditions)

GT1T2

:

The RDM method (P, Q, G, T1 and T2 conditions)

CCSD(T)

:

Coupled cluster singles and doubles with perturbation treatment of triples

HF

:

Hartree-Fock

FCI

:

FullCI

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Introduction of semidefinite programming

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Hubbard model - What is the Semidefinite programming (SDP)?

primal

minimize

A0 • X

s.t.:

Ai • X = bi

(i = 1, 2, · · · , m)

X

0

m

∑

dual

maximize

bi zi

i=1

m

∑

s.t.:

Ai zi + Y = A0

i=1

Y

0

Ai: n × n symmetric matrices, X: variable matrix of n × n symmetric mat, zi :

∑

m-dim. vector, bi: real values, X • Y :=

Xi jYi j. X

0: X is positive

semidefinite.

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Primal Dual Interior Point Method

Established algorithm and there are many good implementations

Step 0: Choose initial points:x0, X0, Y0, X0

0, Y0

0. Set h = 0 and

choose γ ∈ (0, 1)

Step 1: Calculate Shur complementary matrix B ∈ Sn:

Bi j = ((Xh)−1FiYh) • Fj

Step 2: Solve the linear eq. Bdx = r and calculate the search direction

(dx, dX, dY)

Step 3: Compute the max step length α to keep the positive

semidefiniteness

α = max{α ∈ [0, 1] : Xh + αdX

0, Yh + αdY

0}.

Step 4: Update the current point

(xh+1, Xh+1, Yh+1) = (xh, Xh, Yh) + γα(dx, dX, dY).

Step 5: Stop if (xh+1, Xh+1, Yh+1) satisfies criteria, otherwise, h := h + 1

and return to step 1.

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Semidefinite programming

Good news

We never stucked to local minima: convex optimization.

Quantum many body problems can be casted exactly to semidefinite

programming.

Good quantum numbers are conserved via linear constraints.

The number of iterations needed to achieve minima is polynomial (Note:

not known in Hartree-Fock).

A very fast massively parallel implimentation is available on GPU and

CPU [Fujisawa-Endo-Sato-Yamashita-Matsuoka-Nakata Supercomputing

2012]

Bad news

Still not so large calculation is possible.

Attaining very accurate solution is theoretically difficult therefore multiple

precision calculation is needed [Nakata 2008].

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Application to two dimensional Hubbard model

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Hubbard model - Application to the two-dimensional Hubbard model

The Hamiltonian of the Hubbard model:

N

∑ ∑

L

∑

H = −t

a† a j,σ + U

a† a j,↑a† aj,↓

i,σ

j,↑

j,↓

i, j σ=↑,↓

j=1

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model

We choose 4 × 4 lattice with 16 electrons and S = 0

(largest configuration with this lattice), and calculations

are done with various U/t, and N-representability

conditions. - It is believed that two-dimensional Hubbard model is the simplest model

that exhibits the high-Tc superconductivity of copper oxide.

The simplest and a theoretically important model for strongly correlated

electronic systems is the Hubbard model; Hartree-Fock nor CCSD, don’t

work.

The RDM method can be applied very easily (at least formally).

Large size calculation on the 2D hubbard model is challenging : DMRG

fails as the number of basis required exponentailly grows!

Quantum monte carlo method doesn’t give wavefunction (but the RDM

method gives 2-RDM).

The FullCI (aka ED; exact diagonalization) is not feasible. The largest

one is 20 lattices [Tohyama 2007]

Application to the two-dimensional Hubbard model

Why and what is the two-dimensional Hubbard model?

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - The simplest and a theoretically important model for strongly correlated

electronic systems is the Hubbard model; Hartree-Fock nor CCSD, don’t

work.

The RDM method can be applied very easily (at least formally).

Large size calculation on the 2D hubbard model is challenging : DMRG

fails as the number of basis required exponentailly grows!

Quantum monte carlo method doesn’t give wavefunction (but the RDM

method gives 2-RDM).

The FullCI (aka ED; exact diagonalization) is not feasible. The largest

one is 20 lattices [Tohyama 2007]

Application to the two-dimensional Hubbard model

Why and what is the two-dimensional Hubbard model?

It is believed that two-dimensional Hubbard model is the simplest model

that exhibits the high-Tc superconductivity of copper oxide.

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - The RDM method can be applied very easily (at least formally).

Large size calculation on the 2D hubbard model is challenging : DMRG

fails as the number of basis required exponentailly grows!

Quantum monte carlo method doesn’t give wavefunction (but the RDM

method gives 2-RDM).

The FullCI (aka ED; exact diagonalization) is not feasible. The largest

one is 20 lattices [Tohyama 2007]

Application to the two-dimensional Hubbard model

Why and what is the two-dimensional Hubbard model?

It is believed that two-dimensional Hubbard model is the simplest model

that exhibits the high-Tc superconductivity of copper oxide.

The simplest and a theoretically important model for strongly correlated

electronic systems is the Hubbard model; Hartree-Fock nor CCSD, don’t

work.

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

GCOE

to

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30 o-dimensional

/ 45

Hubbard model - Large size calculation on the 2D hubbard model is challenging : DMRG

fails as the number of basis required exponentailly grows!

Quantum monte carlo method doesn’t give wavefunction (but the RDM

method gives 2-RDM).

The FullCI (aka ED; exact diagonalization) is not feasible. The largest

one is 20 lattices [Tohyama 2007]

Application to the two-dimensional Hubbard model

Why and what is the two-dimensional Hubbard model?

It is believed that two-dimensional Hubbard model is the simplest model

that exhibits the high-Tc superconductivity of copper oxide.

The simplest and a theoretically important model for strongly correlated

electronic systems is the Hubbard model; Hartree-Fock nor CCSD, don’t

work.

The RDM method can be applied very easily (at least formally).

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

GCOE

to

CMSI the tw

30 o-dimensional

/ 45

Hubbard model - Quantum monte carlo method doesn’t give wavefunction (but the RDM

method gives 2-RDM).

The FullCI (aka ED; exact diagonalization) is not feasible. The largest

one is 20 lattices [Tohyama 2007]

Application to the two-dimensional Hubbard model

Why and what is the two-dimensional Hubbard model?

It is believed that two-dimensional Hubbard model is the simplest model

that exhibits the high-Tc superconductivity of copper oxide.

The simplest and a theoretically important model for strongly correlated

electronic systems is the Hubbard model; Hartree-Fock nor CCSD, don’t

work.

The RDM method can be applied very easily (at least formally).

Large size calculation on the 2D hubbard model is challenging : DMRG

fails as the number of basis required exponentailly grows!

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

GCOE

to

CMSI the tw

30 o-dimensional

/ 45

Hubbard model - The FullCI (aka ED; exact diagonalization) is not feasible. The largest

one is 20 lattices [Tohyama 2007]

Application to the two-dimensional Hubbard model

Why and what is the two-dimensional Hubbard model?

It is believed that two-dimensional Hubbard model is the simplest model

that exhibits the high-Tc superconductivity of copper oxide.

The simplest and a theoretically important model for strongly correlated

electronic systems is the Hubbard model; Hartree-Fock nor CCSD, don’t

work.

The RDM method can be applied very easily (at least formally).

Large size calculation on the 2D hubbard model is challenging : DMRG

fails as the number of basis required exponentailly grows!

Quantum monte carlo method doesn’t give wavefunction (but the RDM

method gives 2-RDM).

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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30 o-dimensional

/ 45

Hubbard model - Application to the two-dimensional Hubbard model

Why and what is the two-dimensional Hubbard model?

It is believed that two-dimensional Hubbard model is the simplest model

that exhibits the high-Tc superconductivity of copper oxide.

The simplest and a theoretically important model for strongly correlated

electronic systems is the Hubbard model; Hartree-Fock nor CCSD, don’t

work.

The RDM method can be applied very easily (at least formally).

Large size calculation on the 2D hubbard model is challenging : DMRG

fails as the number of basis required exponentailly grows!

Quantum monte carlo method doesn’t give wavefunction (but the RDM

method gives 2-RDM).

The FullCI (aka ED; exact diagonalization) is not feasible. The largest

one is 20 lattices [Tohyama 2007]

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Application to the two-dimensional Hubbard model

We choose 4 × 4 lattice with 16 electrons and S = 0 (largest configuration with

this lattice), and calculations are done with various U/t, and N-representability

conditions.

[Anderson-Nakata-Igarashi-Fujisawa-Yamashita THEOCHEM 1003, 22-27 (2013)]

U/t

EED

EPQG

EPQGT1

EPQGT1T2

∆EPQG

∆EPQGT1

∆EPQGT1T2

0.01

−23.9656

−23.9657 −23.9657

−23.9657

−2.39 × 10−5 −1.59 × 10−5 −1 × 10−7

0.1

−23.6587

−23.6606 −23.6599

−23.6587

−1.98 × 10−3 −1.22 × 10−3 −1 × 10−5

0.2

−23.3221

−23.3298 −23.3268

−23.3221

−7.74 × 10−3 −4.74 × 10−3 −1 × 10−5

0.5

−22.3402

−22.3858 −22.3682

−22.3411

−4.55 × 10−2 −2.79 × 10−2 −8.64 × 10−4

0.8

−21.3991

−21.5090 −21.4666

−21.4024

−1.10 × 10−1 −6.75 × 10−2 −3.31 × 10−3

1

−20.7936

−20.9584 −20.8953

−20.7998

−1.65 × 10−1 −1.02 × 10−1 −6.13 × 10−3

2

−18.0176

−18.5478 −18.3522

−18.0535

−5.30 × 10−1 −3.35 × 10−1 −3.60 × 10−2

3

−15.6367

−16.5790 −16.2473

−15.7243

−9.42 × 10−1 −6.11 × 10−1 −8.77 × 10−2

4

−13.6219

−14.9454 −14.4941

−13.7711

−1.32

−8.72 × 10−1 −1.49 × 10−1

5

−11.9405

−13.5745 −13.0214

−12.1479

−1.63

−1.08

−2.07 × 10−1

6

−10.5522

−12.4134 −11.7728

−10.8045

−1.86

−1.22

−2.52 × 10−1

7

−9.41048

−11.4208 −10.7033

−9.69084

−2.01

−1.29

−2.80 × 10−1

8

−8.46888

−10.5641 −9.77809

−8.76192

−2.10

−1.31

−2.93 × 10−1

9

−7.68624

−9.81887 −8.97982

−7.98034

−2.13

−1.29

−2.94 × 10−1

10

−7.02900

−9.16556 −8.28788

−7.31630

−2.14

−1.26

−2.87 × 10−1

100

−7.68192 × 10−1 −1.21923 −8.75302 × 10−1 −7.83706 × 10−1 −4.51 × 10−1 −1.07 × 10−1 −1.55 × 10−2

✞

☎

✝The RDM method with PQGT1T2 N-rep. gave good energies. ✆

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Hubbard model - Summary

Good news

The RDM method with positive semidefinite type N-representability

conditions is quantum mechanical method which scales polynomially.

The RDM method doesn’t have empirical parameters, and we do not

have local minima in the energy.

The RDM method doesn’t depend on Hartree-Fock solutions unlike post

Hartree-Fock methods.

The PQGT1T2 N-representability conditions are quite good even for

high correlation system like two-dimensional Hubbard model: the largest

deviation was 0.03 per site.

The PQGT1T2 N-representability conditions gives good energies for

atoms and molecules :100.1% of correation energy.

Bad news

This method is size-consistent nor extensive.

Complete N-representability conditions may not scale like polynomially.

Still computational cost is somewhat disappointing.

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - The fundamental theoretical limitations of calculating

the ground and excited state on computers.

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Do I answer to the Dirac’s question?

Actually we avoid to employ difficult algorithms.

and we used algorithms which scale polynomially.

Then, how difficult acutally the problem is?

In modern language, what is the complexity for solving Schr ¨odinger

equation in general?

What if when we use a quantum computer?

The fundamental theoretical limitations of calculating the ground and excited state on computers

Now we turn to some fundamental questions.

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Actually we avoid to employ difficult algorithms.

and we used algorithms which scale polynomially.

Then, how difficult acutally the problem is?

In modern language, what is the complexity for solving Schr ¨odinger

equation in general?

What if when we use a quantum computer?

The fundamental theoretical limitations of calculating the ground and excited state on computers

Now we turn to some fundamental questions.

Do I answer to the Dirac’s question?

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - and we used algorithms which scale polynomially.

Then, how difficult acutally the problem is?

In modern language, what is the complexity for solving Schr ¨odinger

equation in general?

What if when we use a quantum computer?

The fundamental theoretical limitations of calculating the ground and excited state on computers

Now we turn to some fundamental questions.

Do I answer to the Dirac’s question?

Actually we avoid to employ difficult algorithms.

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Then, how difficult acutally the problem is?

In modern language, what is the complexity for solving Schr ¨odinger

equation in general?

What if when we use a quantum computer?

The fundamental theoretical limitations of calculating the ground and excited state on computers

Now we turn to some fundamental questions.

Do I answer to the Dirac’s question?

Actually we avoid to employ difficult algorithms.

and we used algorithms which scale polynomially.

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - In modern language, what is the complexity for solving Schr ¨odinger

equation in general?

What if when we use a quantum computer?

The fundamental theoretical limitations of calculating the ground and excited state on computers

Now we turn to some fundamental questions.

Do I answer to the Dirac’s question?

Actually we avoid to employ difficult algorithms.

and we used algorithms which scale polynomially.

Then, how difficult acutally the problem is?

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - What if when we use a quantum computer?

The fundamental theoretical limitations of calculating the ground and excited state on computers

Now we turn to some fundamental questions.

Do I answer to the Dirac’s question?

Actually we avoid to employ difficult algorithms.

and we used algorithms which scale polynomially.

Then, how difficult acutally the problem is?

In modern language, what is the complexity for solving Schr ¨odinger

equation in general?

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - The fundamental theoretical limitations of calculating the ground and excited state on computers

Now we turn to some fundamental questions.

Do I answer to the Dirac’s question?

Actually we avoid to employ difficult algorithms.

and we used algorithms which scale polynomially.

Then, how difficult acutally the problem is?

In modern language, what is the complexity for solving Schr ¨odinger

equation in general?

What if when we use a quantum computer?

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - What is computational complexity?

From wikipedia:

Computational complexity theory is a branch of the theory of

computation in theoretical computer science and mathematics that

focuses on classifying computational problems according to their

inherent difficulty, and relating those classes to each other. In this

context, a computational problem is understood to be a task that is

in principle amenable to being solved by a computer (i.e. the

problem can be stated by a set of mathematical instructions).

Informally, a computational problem consists of problem instances

and solutions to these problem instances. For example, primality

testing is the problem of determining whether a given number is

prime or not. The instances of this problem are natural numbers,

and the solution to an instance is yes or no based on whether the

number is prime or not.

Restate: Dirac’s question:

✞

☎

✝What is the computational complexity of solving Sch¨odinger equation? ✆

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Hubbard model - Traveling salesperson problem (TSP)

Max-cut problem

Determining the ground state of the spin glass Hamiltonian

∑

H = −

Ji jSiS j.

i j

Some known NP-complete problems

NP-complete is a subset of NP, the set of all decision

problems whose solutions can be verified in polynomial

time; NP may be equivalently defined as the set of

decision problems that can be solved in polynomial

time on a nondeterministic Turing machine.

Some problems which are hard to solve are known:

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Hubbard model - Max-cut problem

Determining the ground state of the spin glass Hamiltonian

∑

H = −

Ji jSiS j.

i j

Some known NP-complete problems

NP-complete is a subset of NP, the set of all decision

problems whose solutions can be verified in polynomial

time; NP may be equivalently defined as the set of

decision problems that can be solved in polynomial

time on a nondeterministic Turing machine.

Some problems which are hard to solve are known:

Traveling salesperson problem (TSP)

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Determining the ground state of the spin glass Hamiltonian

∑

H = −

Ji jSiS j.

i j

Some known NP-complete problems

NP-complete is a subset of NP, the set of all decision

problems whose solutions can be verified in polynomial

time; NP may be equivalently defined as the set of

decision problems that can be solved in polynomial

time on a nondeterministic Turing machine.

Some problems which are hard to solve are known:

Traveling salesperson problem (TSP)

Max-cut problem

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Some known NP-complete problems

NP-complete is a subset of NP, the set of all decision

problems whose solutions can be verified in polynomial

time; NP may be equivalently defined as the set of

decision problems that can be solved in polynomial

time on a nondeterministic Turing machine.

Some problems which are hard to solve are known:

Traveling salesperson problem (TSP)

Max-cut problem

Determining the ground state of the spin glass Hamiltonian

∑

H = −

Ji jSiS j.

i j

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Hubbard model - Traveling salesperson problem (TSP)

Problem: given the costs and a number x, decide

whether there is a round-trip route cheaper than x.

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Hubbard model - Max-cut problem

A decision problem

Given a graph G and an integer k, determine whether there is a

cut of size at least k

The polynomial-time approximation algorithm for max-cut with the best known

approximation ratio is a method by Goemans and Williamson using semidefinite

programming and randomized rounding that achieves an approximation ratio

α ≈ 0.878.

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Hubbard model - What if when we use a quantum computer?

From wikipedia:

A quantum computer is a computation device that makes direct

use of quantum mechanical phenomena, such as superposition

and entanglement, to perform operations on data. Quantum

computers are different from digital computers based on

transistors. Whereas digital computers require data to be encoded

into binary digits (bits), quantum computation uses quantum

properties to represent data and perform operations on these data.

On Complexity:

NP-complete analogues of quantum computer is known as QMA-complete. If a

problem is QMA-complete, then NP-complete in classical computer. Complexity

class which classical computer cannot solve efficiently but quantum computer

can is called “Bounded-error Quantum Polynomial time” (BQP). Unfortunately

relationship between BQP and NP is not known.

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Hubbard model - Decision problem for N-representability of 2-RDM is

QMA-complete [http://arxiv.org/pdf/quant-ph/0609125.pdf]

Decision problem for universal density functional is

QMA-complete [Nature Physics, 5, pp. 732-735, 2009]

QMA-complete even for interacting fermions under two-body

interactions [http://arxiv.org/abs/1208.3334]

Hartreep-Fock is NP-complete

[http://arxiv.org/abs/1208.3334].

✞

☎

✝No fundamental speedups even if we use a quantum computer.

✞

☎

✆

✝The answer to Dirac is :

✞

✆

☎

✝Solving Schr¨odinger equation in general seems to be very difficult. ✆

What if when we use a quantum computer?

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Decision problem for universal density functional is

QMA-complete [Nature Physics, 5, pp. 732-735, 2009]

QMA-complete even for interacting fermions under two-body

interactions [http://arxiv.org/abs/1208.3334]

Hartreep-Fock is NP-complete

[http://arxiv.org/abs/1208.3334].

✞

☎

✝No fundamental speedups even if we use a quantum computer.

✞

☎

✆

✝The answer to Dirac is :

✞

✆

☎

✝Solving Schr¨odinger equation in general seems to be very difficult. ✆

What if when we use a quantum computer?

Decision problem for N-representability of 2-RDM is

QMA-complete [http://arxiv.org/pdf/quant-ph/0609125.pdf]

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - QMA-complete even for interacting fermions under two-body

interactions [http://arxiv.org/abs/1208.3334]

Hartreep-Fock is NP-complete

[http://arxiv.org/abs/1208.3334].

✞

☎

✝No fundamental speedups even if we use a quantum computer.

✞

☎

✆

✝The answer to Dirac is :

✞

✆

☎

✝Solving Schr¨odinger equation in general seems to be very difficult. ✆

What if when we use a quantum computer?

Decision problem for N-representability of 2-RDM is

QMA-complete [http://arxiv.org/pdf/quant-ph/0609125.pdf]

Decision problem for universal density functional is

QMA-complete [Nature Physics, 5, pp. 732-735, 2009]

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

GCOE

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Hubbard model - Hartreep-Fock is NP-complete

[http://arxiv.org/abs/1208.3334].

✞

☎

✝No fundamental speedups even if we use a quantum computer.

✞

☎

✆

✝The answer to Dirac is :

✞

✆

☎

✝Solving Schr¨odinger equation in general seems to be very difficult. ✆

What if when we use a quantum computer?

Decision problem for N-representability of 2-RDM is

QMA-complete [http://arxiv.org/pdf/quant-ph/0609125.pdf]

Decision problem for universal density functional is

QMA-complete [Nature Physics, 5, pp. 732-735, 2009]

QMA-complete even for interacting fermions under two-body

interactions [http://arxiv.org/abs/1208.3334]

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - ✞

☎

✝No fundamental speedups even if we use a quantum computer.

✞

☎

✆

✝The answer to Dirac is :

✞

✆

☎

✝Solving Schr¨odinger equation in general seems to be very difficult. ✆

What if when we use a quantum computer?

Decision problem for N-representability of 2-RDM is

QMA-complete [http://arxiv.org/pdf/quant-ph/0609125.pdf]

Decision problem for universal density functional is

QMA-complete [Nature Physics, 5, pp. 732-735, 2009]

QMA-complete even for interacting fermions under two-body

interactions [http://arxiv.org/abs/1208.3334]

Hartreep-Fock is NP-complete

[http://arxiv.org/abs/1208.3334].

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

GCOE

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Hubbard model - ✞

☎

✝The answer to Dirac is :

✞

✆

☎

✝Solving Schr¨odinger equation in general seems to be very difficult. ✆

What if when we use a quantum computer?

Decision problem for N-representability of 2-RDM is

QMA-complete [http://arxiv.org/pdf/quant-ph/0609125.pdf]

Decision problem for universal density functional is

QMA-complete [Nature Physics, 5, pp. 732-735, 2009]

QMA-complete even for interacting fermions under two-body

interactions [http://arxiv.org/abs/1208.3334]

Hartreep-Fock is NP-complete

[http://arxiv.org/abs/1208.3334].

✞

☎

✝No fundamental speedups even if we use a quantum computer. ✆

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

GCOE

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CMSI the tw

40 o-dimensional

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Hubbard model - ✞

☎

✝Solving Schr¨odinger equation in general seems to be very difficult. ✆

What if when we use a quantum computer?

Decision problem for N-representability of 2-RDM is

QMA-complete [http://arxiv.org/pdf/quant-ph/0609125.pdf]

Decision problem for universal density functional is

QMA-complete [Nature Physics, 5, pp. 732-735, 2009]

QMA-complete even for interacting fermions under two-body

interactions [http://arxiv.org/abs/1208.3334]

Hartreep-Fock is NP-complete

[http://arxiv.org/abs/1208.3334].

✞

☎

✝No fundamental speedups even if we use a quantum computer.

✞

☎

✆

✝The answer to Dirac is : ✆

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

GCOE

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Hubbard model - What if when we use a quantum computer?

Decision problem for N-representability of 2-RDM is

QMA-complete [http://arxiv.org/pdf/quant-ph/0609125.pdf]

Decision problem for universal density functional is

QMA-complete [Nature Physics, 5, pp. 732-735, 2009]

QMA-complete even for interacting fermions under two-body

interactions [http://arxiv.org/abs/1208.3334]

Hartreep-Fock is NP-complete

[http://arxiv.org/abs/1208.3334].

✞

☎

✝No fundamental speedups even if we use a quantum computer.

✞

☎

✆

✝The answer to Dirac is :

✞

✆

☎

✝Solving Schr¨odinger equation in general seems to be very difficult. ✆

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Complementary slides

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Computational cost and complexity

Approximate number of floating-point operations per iteration (FPOI) and

theoretical number of iterations for primal-dual interior-point methods (PDIMP)

and for RRSDP applied to primal and dual SDP formulations.

P, Q, and G

P, Q, G, and T1 or

P, Q, G, T1, and T2

algorithm

FPOI

# iterations

FPOI

# iterations

Primal SDP Formulation

PDIPM

r12

r ln ε−1

r18

r3/2 ln ε−1

RRSDP

at least r10?

none

at least r15?

none

Dual SDP Formulation

PDIPM

r10

r ln ε−1

r12

r3/2 ln ε−1

RRSDP

at least r8?

none

at least r10?

none

Note that obtaining lowest energy in Hartree-Fock approximation is

NP-complete.

The RDM method is a method which scales polynomially.

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Hubbard model - Q-condition: Q-matrix is positive semidefinite

Qi1i2

= (δi1 δi2 − δi1 δi2 ) − (δi1 γi2 + δi2 γi1 )

j1 j2

j1 j2

j2 j1

j1

j2

j2

j1

+(δi1 γi2 + δi2 γi1 ) − 2Γi1i2

j2

j1

j1

j2

j1 j2

G-condition: G-matrix is positive semidefinite

Gi1i2

= (δi2 γi1 − 2Γi1 j2) ≥ 0,

j1 j2

j2

j1

j1i2

P-condition: Γi1i2 is positive semidefinite

j1 j2

N-representability conditions

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - G-condition: G-matrix is positive semidefinite

Gi1i2

= (δi2 γi1 − 2Γi1 j2) ≥ 0,

j1 j2

j2

j1

j1i2

Q-condition: Q-matrix is positive semidefinite

Qi1i2

= (δi1 δi2 − δi1 δi2 ) − (δi1 γi2 + δi2 γi1 )

j1 j2

j1 j2

j2 j1

j1

j2

j2

j1

+(δi1 γi2 + δi2 γi1 ) − 2Γi1i2

j2

j1

j1

j2

j1 j2

N-representability conditions

P-condition: Γi1i2 is positive semidefinite

j1 j2

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - G-condition: G-matrix is positive semidefinite

Gi1i2

= (δi2 γi1 − 2Γi1 j2) ≥ 0,

j1 j2

j2

j1

j1i2

N-representability conditions

P-condition: Γi1i2 is positive semidefinite

j1 j2

Q-condition: Q-matrix is positive semidefinite

Qi1i2

= (δi1 δi2 − δi1 δi2 ) − (δi1 γi2 + δi2 γi1 )

j1 j2

j1 j2

j2 j1

j1

j2

j2

j1

+(δi1 γi2 + δi2 γi1 ) − 2Γi1i2

j2

j1

j1

j2

j1 j2

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - N-representability conditions

P-condition: Γi1i2 is positive semidefinite

j1 j2

Q-condition: Q-matrix is positive semidefinite

Qi1i2

= (δi1 δi2 − δi1 δi2 ) − (δi1 γi2 + δi2 γi1 )

j1 j2

j1 j2

j2 j1

j1

j2

j2

j1

+(δi1 γi2 + δi2 γi1 ) − 2Γi1i2

j2

j1

j1

j2

j1 j2

G-condition: G-matrix is positive semidefinite

Gi1i2

= (δi2 γi1 − 2Γi1 j2) ≥ 0,

j1 j2

j2

j1

j1i2

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - N-representability conditions

∑

For any operator A such that A =

ci ja†a†,

i

j

expectation value of A A† should be non negative.

Here are explict representations mentioned before.

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model - Application to potential energy curve

Dissociation curve of N2 (triple bond) the world first result.

[Nakata-Nakatsuji-Ehara 2002]

Potential curve for N2 (STO-6G)

Hartree-Fock

-108.5

PQG

FullCI

MP2

CCSD(T)

-108.55

-108.6

-108.65

-108.7

Total energy(atomic unit)

-108.75

1

1.5

2

2.5

3

distance(Angstrom)

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application

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Hubbard model