direct variational calculation of second-order reduced density matrix : application to the...

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Direct variational calculation ofsecond-order reduced density matrix :

application to the two-dimensionalHubbard model

中田「アングリーバード」真秀[email protected]

http://nakatamaho.riken.jp/

RIKEN, Advanced Center for Computing and Communication

GCOE interdisciplinary workshop on numerical methods for many-body correlations, Faculity ofScience Building 4, Room 1320, Hongo Campus, Univ of Tokyo, 14:10-14:50

NAKATA “AngryBirds” Maho (RIKEN, ACCC) Direct variational calculation of second-order reduced density matrix : application to the two-dimensional Hubbard modelGCOE CMSI 1 / 45

Today’s Angry Birds Score

Overview

Introduction of reduced density matrix methodIntroduction of semidefinite programmingApplication to two dimensional Hubbard modelThe fundamental theoretical limitations ofcalculating the ground and excited state oncomputers

Introduction of reduced density matrix method

Problem statement

98% of chemistry is explained by obtaining the ground statewavefunction and the total energy by solving Schrodingerequation with Born-Oppenheimer approximation (+ basis set expansion).

Hamiltonian H

H =N∑

− ~2

2me∇2

j −K∑A

4πε0rA j

+∑i> j

4πε0ri j

Schrodinger equation

HΨ(1, 2, · · · N) = EΨ(1, 2, · · · N)

Pauli exclusion principle

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

Problem statement

Hamiltonian H

H =N∑

− ~2

2me∇2

j −K∑A

4πε0rA j

+∑i> j

4πε0ri j

HΨ(1, 2, · · · N) = EΨ(1, 2, · · · N)

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

Problem statement

Hamiltonian H

H =N∑

− ~2

2me∇2

j −K∑A

4πε0rA j

+∑i> j

4πε0ri j

HΨ(1, 2, · · · N) = EΨ(1, 2, · · · N)

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

Problem statement

Hamiltonian H

H =N∑

− ~2

2me∇2

j −K∑A

4πε0rA j

+∑i> j

4πε0ri j

HΨ(1, 2, · · · N) = EΨ(1, 2, · · · N)

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

Problem statement

Hamiltonian H

H =N∑

− ~2

2me∇2

j −K∑A

4πε0rA j

+∑i> j

4πε0ri j

HΨ(1, 2, · · · N) = EΨ(1, 2, · · · N)

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

Problem statement

Hamiltonian H

H =N∑

− ~2

2me∇2

j −K∑A

4πε0rA j

+∑i> j

4πε0ri j

HΨ(1, 2, · · · N) = EΨ(1, 2, · · · N)

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

Problem statement

Hamiltonian H

H =N∑

− ~2

2me∇2

j −K∑A

4πε0rA j

+∑i> j

4πε0ri j

HΨ(1, 2, · · · N) = EΨ(1, 2, · · · N)

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

“The Schrodinger equation is much too complected to be soluble.”

The general theory of quantummechanics is now almost com-plete. · · · the whole of chemistryare thus completely known, andthe difficultly is only that the exactapplication of these laws leads toequations much too complectedto be soluble.

[Dirac 1929] “Quantum Mechanics of Many-Electron Systems.”�� However, Dirac did’t say how difficult it is!

[Dirac 1929] “Quantum Mechanics of Many-Electron Systems.”

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

[Dirac 1929] “Quantum Mechanics of Many-Electron Systems.”�� However, Dirac did’t say how difficult it is!

Classical answers to sigh of Dirac (I)�� Density Functional Theory (DFT)

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

ρ(r) = N∫

∫d3r3 · · ·

∫d3rNΨ

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

Hohenberg-Kohn theorem states that there exists universal functionalF[ρ]

E[ρ] = F[ρ] +∫

V(r)ρ(r)d3r

Kohn-Sham equation.(− ~

2m∇2 + veff(r)

)φi(r) = εiφi(r)

v-representability : simply we ignore.

Classical answers to sigh of Dirac (I)“Electron” density can be used fundamental variable, instead of wavefunction:

ρ(r) = N∫

∫d3r3 · · ·

∫d3rNΨ

∗(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 formolecules!

systematic improvement is not possible.�� So what? Is it an answer to Dirac?

�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∫

∫d3r3 · · ·

∫d3rNΨ

∗(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 formolecules!

systematic improvement is not possible.�� So what? Is it an answer to Dirac?�

�No, I don’t think so.

Classical answers to sigh of Dirac (II)

[Coulson 1960] “wave functions tell us more than we need toknow...All the necessary information required for energy andcalculating properties of molecules is embodied in the first andsecond order density matrices.”

[Husimi 1940], [Lowdin 1954], [Mayer 1955], [Coulson 1960], [Coleman 1963], [Rosina 1968]

Γ(12|1′2′) =(N2

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

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 Kodi was a leading physist who first defined the RDM of general order.

Γ(12|1′2′) =(N2

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

γ(1|1′) = N∫Ψ(12 · · · N)Ψ∗(1′2 · · · N)d2d3 · · · dN

Γ(12|1′2′) =(N2

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

γ(1|1′) = N∫Ψ(12 · · · N)Ψ∗(1′2 · · · N)d2d3 · · · dN

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

Γ(12|1′2′) =(N2

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

γ(1|1′) = N∫Ψ(12 · · · N)Ψ∗(1′2 · · · N)d2d3 · · · dN.

Then, minimizeEg = Min TrHΓ

2-RDM has only four variables.

Equivalent to solve Schrodinger equation.

Minimization of linear functional.

Γ(12|1′2′) =(N2

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

γ(1|1′) = N∫Ψ(12 · · · N)Ψ∗(1′2 · · · N)d2d3 · · · dN.

Γ(12|1′2′) =(N2

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

γ(1|1′) = N∫Ψ(12 · · · N)Ψ∗(1′2 · · · N)d2d3 · · · dN.

Γ(12|1′2′) =(N2

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

γ(1|1′) = N∫Ψ(12 · · · N)Ψ∗(1′2 · · · N)d2d3 · · · dN.

Classical answers to sigh of Dirac (II)[Husimi 1940], [Lowdin 1954], [Mayer 1955], [Coulson 1960], [Coleman 1963], [Rosina 1968]

Γ(12|1′2′) =(N2

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

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 berecalculated when two-particle interaction has changed.

All observables up to two-particles including kinetic energy can beevaluated exactly, while it’s impossible in DFT.

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

The number of variables becomes four instead of one.

Just minimize linear functional.

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

Γ(12|1′2′) =(N2

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

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.

Γ(12|1′2′) =(N2

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

γ(1|1′) = N∫Ψ(12 · · · N)Ψ∗(1′2 · · · N)d2d3 · · · dN.

Eg = Min TrHΓ

�� Is it an answer to Dirac?

�Yes, partially.

Γ(12|1′2′) =(N2

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

γ(1|1′) = N∫Ψ(12 · · · N)Ψ∗(1′2 · · · N)d2d3 · · · dN.

Eg = Min TrHΓ

�� Is it an answer to Dirac?�

�Yes, partially.

The RDM method : ground state calculation using 2-RDM

H =∑

vija†ia j +

∑i1 i2 j1 j2

wi1 i2j1 j2

a†i1

a†i2

a j2 a j1

The ground stat energy becomes:

Eg = min〈Ψ|H|Ψ〉

= min∑

vij〈Ψ|a

†ia j|Ψ〉 +

∑i1 i2 j1 j2

wi1 i2j1 j2〈Ψ|a†

i1a†

i2a j2 a j1 |Ψ〉

= min{∑

∑i1 i2 j1 j2

wi1 i2j1 j2Γ

i1 i2j1 j2}

Second quantized version of 1, 2-RDM.

Γi1 i2j1 j2=

12〈Ψ|a†

i1a†

i2a j2 a j1 |Ψ〉, γi

j = 〈Ψ|a†ia j|Ψ〉.

H =∑

vija†ia j +

∑i1 i2 j1 j2

wi1 i2j1 j2

a†i1

a†i2

a j2 a j1

= min∑

vij〈Ψ|a

†ia j|Ψ〉 +

∑i1 i2 j1 j2

i1a†

i2a j2 a j1 |Ψ〉

= min{∑

∑i1 i2 j1 j2

wi1 i2j1 j2Γ

i1 i2j1 j2}

Γi1 i2j1 j2=

12〈Ψ|a†

i1a†

H =∑

vija†ia j +

∑i1 i2 j1 j2

wi1 i2j1 j2

a†i1

a†i2

a j2 a j1

= min∑

vij〈Ψ|a

†ia j|Ψ〉 +

∑i1 i2 j1 j2

i1a†

i2a j2 a j1 |Ψ〉

= min{∑

∑i1 i2 j1 j2

wi1 i2j1 j2Γ

i1 i2j1 j2}

Γi1 i2j1 j2=

12〈Ψ|a†

i1a†

H =∑

vija†ia j +

∑i1 i2 j1 j2

wi1 i2j1 j2

a†i1

a†i2

a j2 a j1

= min∑

vij〈Ψ|a

†ia j|Ψ〉 +

∑i1 i2 j1 j2

i1a†

i2a j2 a j1 |Ψ〉

= min{∑

∑i1 i2 j1 j2

wi1 i2j1 j2Γ

i1 i2j1 j2}

Γi1 i2j1 j2=

12〈Ψ|a†

i1a†

Eg = min〈Ψ|H|Ψ〉= min{

∑i j

∑i1i2 j1 j2

wi1i2j1 j2Γ

i1i2j1 j2}

��

��Looks too simple, but what is the barter?

Eg = min〈Ψ|H|Ψ〉= min{

∑i j

∑i1i2 j1 j2

wi1i2j1 j2Γ

i1i2j1 j2}

��

��Looks too simple, but what is the barter?

The N-representability conditions

[Mayers 1955], [Tredgold 1957]: early attempts failed with too lowenergies from the exact value.N-representability condition coined by [Coleman 1963].

Eg = minP{∑

∑i1i2 j1 j2

wi1i2j1 j2Γ

i1i2j1 j2}

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

Γ(12|1′2′) → Ψ(123 · · · N)

γ(1|1′) → Ψ(123 · · · N).

[Mayers 1955], [Tredgold 1957]: early attempts failed with too lowenergies from the exact value.

N-representability condition coined by [Coleman 1963].

Eg = minP{∑

∑i1i2 j1 j2

wi1i2j1 j2Γ

i1i2j1 j2}

Γ(12|1′2′) → Ψ(123 · · · N)

γ(1|1′) → Ψ(123 · · · N).

[Mayers 1955], [Tredgold 1957]: early attempts failed with too lowenergies from the exact value.N-representability condition coined by [Coleman 1963].

Eg = minP{∑

∑i1i2 j1 j2

wi1i2j1 j2Γ

i1i2j1 j2}

Γ(12|1′2′) → Ψ(123 · · · N)

γ(1|1′) → Ψ(123 · · · N).

The N-representability conditionSome explict forms of N-representability conditions:

1, 2-RDM should be Hermitian

γij = (γ j

i)∗, Γi1 i2

j1 j2= (Γ j2 j1

i1 i2)∗, Γi1 i2

j1 j2= −Γi2 i1

j1 j2= −Γi1 i2

j2 j1...

Trace condition∑i=1

γii = N,

∑i, j=1

Γi ji j=

N(N − 1)2

Trace condition (higher to lower)

N − 12γi

j =∑k=1

Γikjk

γij = (γ j

i)∗, Γi1 i2

j1 j2= (Γ j2 j1

i1 i2)∗, Γi1 i2

j1 j2= −Γi2 i1

j1 j2= −Γi1 i2

j2 j1...

γii = N,

∑i, j=1

Γi ji j=

N(N − 1)2

N − 12γi

j =∑k=1

Γikjk

γij = (γ j

i)∗, Γi1 i2

j1 j2= (Γ j2 j1

i1 i2)∗, Γi1 i2

j1 j2= −Γi2 i1

j1 j2= −Γi1 i2

j2 j1...

γii = N,

∑i, j=1

Γi ji j=

N(N − 1)2

N − 12γi

j =∑k=1

Γikjk

γij = (γ j

i)∗, Γi1 i2

j1 j2= (Γ j2 j1

i1 i2)∗, Γi1 i2

j1 j2= −Γi2 i1

j1 j2= −Γi1 i2

j2 j1...

γii = N,

∑i, j=1

Γi ji j=

N(N − 1)2

N − 12γi

j =∑k=1

Γikjk

Some known facts about N-representability conditions

Approximate (necessity) condtion : some well known conditios.

P, Q-condition, complete condition for 1-RDM [Coleman1963]

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-Percus1964].

Computational complexity of complete N-rep. condition :QMA-complete [Liu 2007].

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

Computational complexity of complete N-rep. condition :QMA-complete [Liu 2007].�� How effective and how to calculate 2-RDM systematically had been not known

Positive semidefinite type of N-representability conditions

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

〈Ψ|a†ia†

jal ak|Ψ〉 � 0

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

〈Ψ|aia ja†l a†k|Ψ〉 � 0

G-condition: eigenvalues of Γ should be non-negative [Garrod-Perucs1964]

〈Ψ|a†ia ja†l ak|Ψ〉 � 0

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

〈Ψ|(a†ia†

kanama` + anama`a†i a†

k|Ψ〉 � 0

〈Ψ|(a†ia†

jaka†nama` + a†nama`a†i a†

jak|Ψ〉 � 0

〈Ψ|a†ia†

jal ak|Ψ〉 � 0

〈Ψ|(a†ia†

k|Ψ〉 � 0

〈Ψ|(a†ia†

jak|Ψ〉 � 0

〈Ψ|a†ia†

jal ak|Ψ〉 � 0

〈Ψ|(a†ia†

k|Ψ〉 � 0

〈Ψ|(a†ia†

jak|Ψ〉 � 0

〈Ψ|a†ia†

jal ak|Ψ〉 � 0

〈Ψ|(a†ia†

k|Ψ〉 � 0

〈Ψ|(a†ia†

jak|Ψ〉 � 0

〈Ψ|a†ia†

jal ak|Ψ〉 � 0

〈Ψ|(a†ia†

k|Ψ〉 � 0

〈Ψ|(a†ia†

jak|Ψ〉 � 0

〈Ψ|a†ia†

jal ak|Ψ〉 � 0

〈Ψ|(a†ia†

k|Ψ〉 � 0

〈Ψ|(a†ia†

jak|Ψ〉 � 0

Positive semidefinite N-representability conditions

P,Q,G,T1,T2-matrices are all positive semidefinite↔eigenvalues are all non-negatieve.

U†ΓU =

λ1 0λ2. . .

Positive semidefinite N-representability conditions

P,Q,G,T1,T2-matrices are all positive semidefinite↔eigenvalues are all non-negatieve.

U†ΓU =

λ1 0λ2. . .

Geometical interpretation of approximateN-representability condition

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, poorresults on nucleon systems, etc.

Revival: introduction of the semidefinite programming

Eg = MinΓ∈P

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

Positive semidefinite programmingThe 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

Eg = MinΓ∈P

[Nakata-Nakatsuji-Ehara 2002]

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

Eg = MinΓ∈P

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

System State N r ∆ EGT1T2 ∆ EGT1T2′ ∆ ECCSD(T) ∆ EHF EFCIC 3 P 6 20 −0.0004 −0.0001 +0.00016 +0.05202 −37.73653O 1 D 8 20 −0.0013 −0.0012 +0.00279 +0.10878 −74.78733Ne 1S 10 20 −0.0002 −0.0001 −0.00005 +0.11645 −128.63881O+

22Πg 15 20 −0.0022 −0.0020 +0.00325 +0.17074 −148.79339

BH 1Σ+ 6 24 −0.0001 −0.0001 +0.00030 +0.07398 −25.18766CH 2Πr 7 24 −0.0008 −0.0003 +0.00031 +0.07895 −38.33735NH 1∆ 8 24 −0.0005 −0.0004 +0.00437 +0.11495 −54.96440HF 1Σ+ 14 24 −0.0003 −0.0003 +0.00032 +0.13834 −100.16031SiH4

1 A1 18 26 −0.0002 −0.0002 +0.00018 +0.07311 −290.28490F− 1S 10 26 −0.0003 −0.0003 +0.00067 +0.15427 −99.59712P 4S 15 26 −0.0001 −0.0000 +0.00003 +0.01908 −340.70802H2O 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 triplesHF : Hartree-FockFCI : FullCI

Introduction of semidefinite programming

What is the Semidefinite programming (SDP)?

primal minimize A0 • Xs.t.: Ai • X = bi (i = 1, 2, · · · , m)

X � 0

dual maximizem∑

s.t.:m∑

Ai zi + Y = A0

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.

Primal Dual Interior Point MethodEstablished algorithm and there are many good implementations

Step 0: Choose initial points:x0, X0, Y0, X0 � 0, Y0 � 0. Set h = 0 andchoose γ ∈ (0, 1)

Step 1: Calculate Shur complementary matrix B ∈ Sn:

Bi j = ((Xh)−1FiYh) • F j

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 positivesemidefinitenessα = 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 + 1and return to step 1.

Semidefinite programmingGood news

We never stucked to local minima: convex optimization.

Quantum many body problems can be casted exactly to semidefiniteprogramming.

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 andCPU [Fujisawa-Endo-Sato-Yamashita-Matsuoka-Nakata Supercomputing2012]

Bad news

Still not so large calculation is possible.

Attaining very accurate solution is theoretically difficult therefore multipleprecision calculation is needed [Nakata 2008].

Application to two dimensional Hubbard model

Application to the two-dimensional Hubbard model

The Hamiltonian of the Hubbard model:

H = −tN∑〈i, j〉

∑σ=↑,↓

a†i,σ

a j,σ + UL∑

a†j,↑a j,↑a†j,↓a j,↓

We choose 4 × 4 lattice with 16 electrons and S = 0(largest configuration with this lattice), and calculationsare done with various U/t, and N-representabilityconditions.

Why and what is the two-dimensional Hubbard model?

It is believed that two-dimensional Hubbard model is the simplest modelthat exhibits the high-Tc superconductivity of copper oxide.

The simplest and a theoretically important model for strongly correlatedelectronic systems is the Hubbard model; Hartree-Fock nor CCSD, don’twork.

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

Large size calculation on the 2D hubbard model is challenging : DMRGfails as the number of basis required exponentailly grows!

Quantum monte carlo method doesn’t give wavefunction (but the RDMmethod gives 2-RDM).

The FullCI (aka ED; exact diagonalization) is not feasible. The largestone is 20 lattices [Tohyama 2007]

Application to the two-dimensional Hubbard modelWe choose 4 × 4 lattice with 16 electrons and S = 0 (largest configuration withthis lattice), and calculations are done with various U/t, and N-representabilityconditions.[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.

SummaryGood news

The RDM method with positive semidefinite type N-representabilityconditions is quantum mechanical method which scales polynomially.

The RDM method doesn’t have empirical parameters, and we do nothave local minima in the energy.

The RDM method doesn’t depend on Hartree-Fock solutions unlike postHartree-Fock methods.

The PQGT1T2′ N-representability conditions are quite good even forhigh correlation system like two-dimensional Hubbard model: the largestdeviation was 0.03 per site.

The PQGT1T2′ N-representability conditions gives good energies foratoms and molecules :100.1% of correation energy.

Bad newsThis method is size-consistent nor extensive.

Complete N-representability conditions may not scale like polynomially.

Still computational cost is somewhat disappointing.

The fundamental theoretical limitations of calculatingthe ground and excited state on computers.

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 Schrodingerequation in general?

What if when we use a quantum computer?

What is computational complexity?From wikipedia:

Computational complexity theory is a branch of the theory ofcomputation in theoretical computer science and mathematics thatfocuses on classifying computational problems according to theirinherent difficulty, and relating those classes to each other. In thiscontext, a computational problem is understood to be a task that isin principle amenable to being solved by a computer (i.e. theproblem can be stated by a set of mathematical instructions).Informally, a computational problem consists of problem instancesand solutions to these problem instances. For example, primalitytesting is the problem of determining whether a given number isprime or not. The instances of this problem are natural numbers,and the solution to an instance is yes or no based on whether thenumber is prime or not.

Restate: Dirac’s question:�� What is the computational complexity of solving Schodinger equation?

Some known NP-complete problems

NP-complete is a subset of NP, the set of all decisionproblems whose solutions can be verified in polynomialtime; NP may be equivalently defined as the set ofdecision problems that can be solved in polynomialtime 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 = −∑〈i j〉

J i jSiS j.

Max-cut problem

H = −∑〈i j〉

J i jSiS j.

Max-cut problem

H = −∑〈i j〉

J i jSiS j.

Max-cut problem

H = −∑〈i j〉

J i jSiS j.

Problem: given the costs and a number x, decidewhether there is a round-trip route cheaper than x.

Max-cut problem

A decision problem

Given a graph G and an integer k, determine whether there is acut of size at least k

The polynomial-time approximation algorithm for max-cut with the best knownapproximation ratio is a method by Goemans and Williamson using semidefiniteprogramming and randomized rounding that achieves an approximation ratioα ≈ 0.878.

What if when we use a quantum computer?From wikipedia:

A quantum computer is a computation device that makes directuse of quantum mechanical phenomena, such as superpositionand entanglement, to perform operations on data. Quantumcomputers are different from digital computers based ontransistors. Whereas digital computers require data to be encodedinto binary digits (bits), quantum computation uses quantumproperties 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.

Decision problem for N-representability of 2-RDM isQMA-complete [http://arxiv.org/pdf/quant-ph/0609125.pdf]

Decision problem for universal density functional isQMA-complete [Nature Physics, 5, pp. 732-735, 2009]

QMA-complete even for interacting fermions under two-bodyinteractions [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 Schrodinger equation in general seems to be very difficult.

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 Schrodinger equation in general seems to be very difficult.

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 Schrodinger equation in general seems to be very difficult.

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 Schrodinger equation in general seems to be very difficult.

Complementary slides

Computational cost and complexityApproximate number of floating-point operations per iteration (FPOI) andtheoretical 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 orP, Q, G, T1, and T2

algorithm FPOI # iterations FPOI # iterationsPrimal SDP Formulation

PDIPM r12 r ln ε−1 r18 r3/2 ln ε−1

RRSDP at least r10? none at least r15? noneDual 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 isNP-complete.

The RDM method is a method which scales polynomially.

N-representability conditions

P-condition: Γi1i2j1 j2

is positive semidefinite

Q-condition: Q-matrix is positive semidefinite

Qi1i2j1 j2= (δi1

j1δi2

j2− δi1

j2δi2

j1) − (δi1

j1γi2

j2+ δi2

j2γi1

+(δi1j2γi2

j1+ δi2

j1γi1

j2) − 2Γi1i2

G-condition: G-matrix is positive semidefinite

Gi1i2j1 j2= (δi2

j2γi1

j1− 2Γi1 j2

j1i2) ≥ 0,

Qi1i2j1 j2= (δi1

j1δi2

j2− δi1

j2δi2

j1) − (δi1

j1γi2

j2+ δi2

j2γi1

+(δi1j2γi2

j1+ δi2

j1γi1

j2) − 2Γi1i2

Gi1i2j1 j2= (δi2

j2γi1

j1− 2Γi1 j2

j1i2) ≥ 0,

Qi1i2j1 j2= (δi1

j1δi2

j2− δi1

j2δi2

j1) − (δi1

j1γi2

j2+ δi2

j2γi1

+(δi1j2γi2

j1+ δi2

j1γi1

j2) − 2Γi1i2

Gi1i2j1 j2= (δi2

j2γi1

j1− 2Γi1 j2

j1i2) ≥ 0,

Qi1i2j1 j2= (δi1

j1δi2

j2− δi1

j2δi2

j1) − (δi1

j1γi2

j2+ δi2

j2γi1

+(δi1j2γi2

j1+ δi2

j1γi1

j2) − 2Γi1i2

Gi1i2j1 j2= (δi2

j2γi1

j1− 2Γi1 j2

j1i2) ≥ 0,

For any operator A such that A =∑

ci ja†i a†j,

expectation value of 〈AA†〉 should be non negative.Here are explict representations mentioned before.

Application to potential energy curve

Dissociation curve of N2 (triple bond) the world first result.[Nakata-Nakatsuji-Ehara 2002]

-108.75

-108.7

-108.65

-108.6

-108.55

-108.5

1 1.5 2 2.5 3

distance(Angstrom)

Potential curve for N2 (STO-6G)Hartree-Fock

PQGFullCIMP2

CCSD(T)

direct variational calculation of second-order reduced density matrix : application to the...

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