The document describes the Hamiltonian operator (H) and its application to the Hartree-Fock wavefunction (|ΦHF⟩) to obtain energy eigenvalues (E0, E1, etc.). The Hartree-Fock wavefunction can be expressed as a linear combination of Slater determinants (|Ψ0⟩, |Ψ1⟩, etc.). Applying the exponential of the Hamiltonian operator over time (eiHt) to |ΦHF⟩ yields the time-dependent Hartree-Fock wavefunction.

4. ̂HΨ = EΨ

5. ̂HΨ(r1, ⋯, rN) = EΨ(r1, ⋯, rN)
̂H
Ψ E
̂H = −
ℏ2
2me
∇2
1 −
ℏ2
2me
∇2
2 −
e2
4πϵ0 (
1
ra1
+
1
rb1
+
1
ra2
+
1
rb2 )
+
e2
4πϵ0
1
r12
+
e2
4πϵ0R
Ψ(x1, y1, z1, x2, y2, z2)

6. ̂Hx = λx

13. N 1 N − 1 N − 1
3N 3 N
|⟨ΨfullCI |ΨHartree−Fock⟩|2
≈ 1
1
3N
N

14. |Ψ0⟩ {χa}i=1,⋯,r
Ψ0⟩ = χ1χ2⋯χa χb⋯χN⟩
E0 = ⟨Ψ0 | ̂ℋ|Ψ0⟩ =
∑
a
⟨a|h|a⟩ +
1
2 ∑
ab
⟨ab||ab⟩
⟨χa |χb⟩ = δab

15. [
h(1) +
N
∑
b=1
𝒥b(1) − 𝒦b(1)
]
χa(1) =
N
∑
b=1
εba χb(1)
f χa⟩ =
N
∑
b=1
εba xb⟩
f χa⟩ = εa χa⟩

16. {ϕμ}
ψi =
K
∑
μ=1
Cμiϕμ
Sμv =
∫
dr1ϕ*μ (1)ϕv(1)
Fμv =
∫
dr1ϕ*μ (1)f(1)ϕv(1)
∑
v
FμvCvi = εi ∑
v
SμvCvi FC = SCε

17. FC = SCε
Fμv = Hcore
μv +
∑
a
∑
λσ
CλaC*σa[2(μv|σλ) − (μλ|σv)]

22. $ sudo apt install psi4 python3-numpy libopenblas-dev
…

24. #! STO-3G H2 Hartree-Fock Energy Point
# see also: http://pubchemqc.riken.jp/cgi-bin/molecularquery.py?name=hydrogen
molecule h2 {
H 0.0 0.0 0.0
H 0.0 0.0 1.4
unit bohr
}
set {
basis STO-3G
}
thisenergy = energy('hf')

27. #! STO-3G H2 Hartree-Fock Energy Point
# see also: http://pubchemqc.riken.jp/cgi-bin/molecularquery.py?name=hydrogen
molecule h2 {
H 0.0 0.0 0.0
H 0.0 0.0 1.4
unit bohr
}
set {
basis STO-3G
}
thisenergy = energy('hf')

28. ****
H 0
S 3 1.00
3.42525091 0.15432897
0.62391373 0.53532814
0.16885540 0.44463454

31. en, wfn = energy('hf', return_wfn=True)
nbf = wfn.nso()
mints = psi4.core.MintsHelper(wfn.basisset())
S = np.asarray(mints.ao_overlap())
for i in range(nbf):
for j in range(nbf):
print (i+1,j+1, S[i,j])

32. (μv|λσ)

38. |Ψ0⟩
{χa}i=1,⋯,r
|Ψ0⟩ = |χ1χ2⋯χa χb⋯χN⟩
{χa}i=1,⋯,r

39. |Ψ0⟩
|Ψ⟩ = c0 |Φ0⟩ +
occ
∑
i
vir
∑
a
ca
i |Φa
i ⟩ +
occ
∑
i<j
vir
∑
a<b
cab
ij |Φab
ij ⟩ +
occ
∑
i<j<k
vir
∑
a<b<c
cabc
ijk |Φabc
ijk ⟩ + ⋯
|Φ0⟩
|Φ0⟩ = |χ1χ2⋯χa χb⋯χN⟩
|Φa
i ⟩
|Φa
i ⟩ = |χ1χ2⋯χi−1χa χi+1⋯χN⟩
|Φab
ij ⟩
|Φab
ij ⟩ = |χ1χ2⋯χi−1χa χi+1⋯χj−1χb χj+1⋯χN⟩
ca
i , cab
ij

40. |Ψ⟩ = c0 |Φ0⟩ +
occ
∑
i
vir
∑
a
ca
i |Φa
i ⟩ +
occ
∑
i<j
vir
∑
a<b
cab
ij |Φab
ij ⟩ +
occ
∑
i<j<k
vir
∑
a<b<c
cabc
ijk |Φabc
ijk ⟩ + ⋯
min
⟨Ψ|Ψ⟩=1
⟨Ψ| ̂H|Ψ⟩

41. rCN

43. #! STO-3G H2 FullCI Energy Point
# see also: http://pubchemqc.riken.jp/cgi-bin/molecularquery.py?name=hydrogen
molecule h2 {
H 0.0 0.0 0.0
H 0.0 0.0 1.4
unit bohr
}
set {
basis STO-3G
}
thisenergy = energy(‘fci')

44. |⟨ΨFullCI |ΦHF⟩|2
= − 0.9872

46. |Ψ⟩ = c0 |Φ0⟩ +
occ
∑
i
vir
∑
a
ca
i |Φa
i ⟩ +
occ
∑
i<j
vir
∑
a<b
cab
ij |Φab
ij ⟩ +
occ
∑
i<j<k
vir
∑
a<b<c
cabc
ijk |Φabc
ijk ⟩ + ⋯
|ΨfullCI⟩ = − 0.993627|ΦHF⟩ + 0.112716|Φ2¯2
1¯1
⟩
|⟨ΨFullCI |ΦHF⟩|2
= 0.9872

49. |⟨ΨFullCI |ΦHF⟩|2
r(H − H)
r(H − H)

52. |⟨ΨFullCI |ΦHF⟩|2

53. (7C5)2
= 21 * 21 = 441

59. : CCSD(T), MP2 14_n2_631g_cc_mp_pes.dat
N-N FullCI Hartree-Fock CCSD(T) MP2
fullCI (1.5A ?)

61. |⟨ΨfullCI |ΨHF⟩|2

64. |11⟩, |01⟩, |10⟩, |00⟩
|111⟩, |101⟩, |110⟩, |100⟩
|011⟩, |001⟩, |010⟩, |000⟩
|1⟩, |0⟩
|00000000⟩ + |00000001⟩ + . . . + |11111111⟩

68. ei ̂Ht
|ΦHF⟩ = c0eiE0t
|Ψ0⟩ + c1eiE1t
|Ψ1⟩ + c2eiE2t
|Ψ2⟩⋯
ei ̂Hτ
≈
∏
i
eHΔτi ≈
∏
i
e
∑j
HjΔτi
̂H|Ψ⟩ = E|Ψ⟩ → ei ̂Hτ
|Ψ⟩ = ̂U|Ψ⟩ = ei2πϕ
|Ψ⟩
ei2πϕ
|Ψ⟩ ⊗ |R⟩ → ei2πϕ
|Ψ⟩ ⊗ ei2πϕ
|R⟩ → measure|R⟩ → E = 2πϕ/τ

69. |⟨ΦFullCI |ΦHF⟩|2
≈ 1
E0 ≤ E1 ≤ E2 ≤ ⋯

75. $ dotnet new -i “Microsoft.Quantum.ProjectTemplates::0.7-*”
…

76. $ mkdir -p qsharp
$ cd qsharp
$ sudo apt install git
$ git clone https://github.com/Microsoft/Quantum.git
$ git clone https://github.com/microsoft/QuantumLibraries

79. |ψ⟩
CNOT(H ⊗ 1)|00⟩ ⟶
|00⟩ + |11⟩
2

80. |ψ⟩ = α|0⟩ + β|1⟩ |ψ⟩ ⊗
|00⟩ + |11⟩
2
(H ⊗ 1 ⊗ 1)[CNOT ⊗ 1]|ψ⟩ ⊗
|00⟩ + |11⟩
2
=
1
2
[α|000⟩ + α|100⟩ + α|011⟩ + α|111⟩ + β|010⟩ − β|110⟩ + β|001⟩ − β|101⟩]
1
2
[|00⟩(α|0⟩ + β|1⟩) + |01⟩(α|1⟩ + β|0⟩) + |10⟩(α|0⟩ − β|1⟩) + |11⟩(α|1⟩ − β|0⟩)]

82. $ cd Quantum/Samples/src/Teleportation
$ dotnet run
Round 0: Sent True, got True.
Teleportation successful!!
Round 1: Sent False, got False.
Teleportation successful!!
Round 2: Sent False, got False.
Teleportation successful!!
Round 3: Sent True, got True.
Teleportation successful!!
Round 4: Sent False, got False.
Teleportation successful!!
Round 5: Sent True, got True.
Teleportation successful!!
Round 6: Sent True, got True.
Teleportation successful!!
Round 7: Sent False, got False.
Teleportation successful!!

83. operation Teleport (msg : Qubit, target : Qubit) : Unit {
using (register = Qubit()) {
// Create some entanglement that we can use to send our message.
H(register);
CNOT(register, target);
// Encode the message into the entangled pair,
// and measure the qubits to extract the classical data
// we need to correctly decode the message into the target qubit:
CNOT(msg, register);
H(msg);
let data1 = M(msg);
let data2 = M(register);
// decode the message by applying the corrections on
// the target qubit accordingly:
if (data1 == One) { Z(target); }
if (data2 == One) { X(target); }
// Reset our "register" qubit before releasing it.
Reset(register);
}
}

84. // Copyright (c) Microsoft Corporation. All rights reserved.
// Licensed under the MIT License.
using Microsoft.Quantum.Simulation.Simulators;
using System.Linq;
namespace Microsoft.Quantum.Samples.Teleportation {
class Program
{
static void Main(string[] args)
{
using (var sim = new QuantumSimulator())
{
var rand = new System.Random();
foreach (var idxRun in Enumerable.Range(0, 8))
{
var sent = rand.Next(2) == 0;
var received = TeleportClassicalMessage.Run(sim, sent).Result;
System.Console.WriteLine($"Round {idxRun}:tSent {sent},tgot
{received}.");
System.Console.WriteLine(sent == received ? "Teleportation
successful!!n" : "n");
}
}
}
}
}

87. ̂H =
∑
i,j
vija†
i
aj +
1
2 ∑
ijkl
wijkla†
i
a†
j
alak
|Ψ⟩
̂H|Ψ⟩ = E|Ψ⟩

88. |⟨ΦFullCI |ΦHF⟩|2
≈ 1
E0 ≤ E1 ≤ E2 ≤ ⋯

89. |Ψ⟩ = c0 |Φ0⟩ +
occ
∑
i
vir
∑
a
ca
i |Φa
i ⟩ +
occ
∑
i<j
vir
∑
a<b
cab
ij |Φab
ij ⟩ +
occ
∑
i<j<k
vir
∑
a<b<c
cabc
ijk |Φabc
ijk ⟩ + ⋯
|Φ0⟩ = |χ1χ2⋯χa χb⋯χN⟩
|Φa
i ⟩ = |χ1χ2⋯χi−1χa χi+1⋯χN⟩
|Φab
ij ⟩ = |χ1χ2⋯χi−1χa χi+1⋯χj−1χb χj+1⋯χN⟩
= |111111000000⟩
= |111110001000⟩
= |110110001010⟩
2r
r ≃ log(2r
)

90. 1/ N!
̂P12Ψ(x1, x2) = ± Ψ(x1, x2)

91. ̂H|Ψ⟩ = E|Ψ⟩ → ei ̂Hτ
|Ψ⟩ = ̂U|Ψ⟩ = ei2πϕ
|Ψ⟩
ei2πϕ
|Ψ⟩ ⊗ |R⟩ → ei2πϕ
|Ψ⟩ ⊗ ei2πϕ
|R⟩ → measure|R⟩ → E = 2πϕ/τ

92. ̂H =
∑
i,j
vija†
i
aj +
1
2 ∑
ijkl
wijkla†
i
a†
j
alak
̂H =
∑
ijkl
Hijkla†
i
a†
j
alak
ei ̂Ht
|ΦHF⟩ = c0eiE0t
|Ψ0⟩ + c1eiE1t
|Ψ1⟩ + c2eiE2t
|Ψ2⟩⋯
exp(i ̂Ht)
Hijkla†
i
a†
j
alak
ei ̂Ht
≈
∏
Δt
eHΔt
≈
∏
Δt
e
∑ijkl
HijklΔt
≈
∏
Δt,ijkl
eHijklΔt

93. |000000111111 >

94. |⟨ΦFullCI |ΦHF⟩|2
≈ 1
E0 ≤ E1 ≤ E2 ≤ ⋯
ei ̂Ht
|ΦHF⟩ = c0eiE0t
|Ψ0⟩ + c1eiE1t
|Ψ1⟩ + c2eiE2t
|Ψ2⟩⋯

97. $ cd Quantum/Chemistry/MolecularHydrogen
$ dotnet run
----- Print Hamiltonian
PP has 4 entries).
[1 * 0u 0d, -1.252477495]
[1 * 1u 1d, -1.252477495]
[1 * 2u 2d, -0.475934275]
[1 * 3u 3d, -0.475934275]
PQQP has 6 entries).
[1 * 0u 1u 1d 0d, 0.674493166]
[1 * 0u 2u 2d 0d, 0.482184583]
[1 * 1u 3u 3d 1d, 0.482184583]
[1 * 1u 2u 2d 1d, 0.663472101]
[1 * 0u 3u 3d 0d, 0.663472101]
[1 * 2u 3u 3d 2d, 0.69739801]
PQRS has 2 entries).
[1 * 0u 3u 2d 1d, -0.362575036]
[1 * 0u 1u 3d 2d, 0.362575036]
Identity has 1 entries).
[1 * , 0.713776188]
----- End Print Hamiltonian
----- Creating Jordan–Wigner encoding
----- End Creating Jordan–Wigner encoding
----- Print Hamiltonian
Identity has 1 entries).
[Identity: [ ], -0.098834446]
Z has 4 entries).
[Z: [ 0 ], 0.171201285]
[Z: [ 1 ], 0.171201285]
[Z: [ 2 ], -0.222796536]
[Z: [ 3 ], -0.222796536]
ZZ has 6 entries).
[ZZ: [ 0 1 ], 0.1686232915]
[ZZ: [ 0 2 ], 0.12054614575]
[ZZ: [ 1 3 ], 0.12054614575]
[ZZ: [ 1 2 ], 0.16586802525]
[ZZ: [ 0 3 ], 0.16586802525]
[ZZ: [ 2 3 ], 0.1743495025]
v01234 has 1 entries).
[v01234: [ 0 1 2 3 ], 0, -0.0453218795, 0, 0.0453218795]
----- End Print Hamiltonian
$ psi4 07_h2_sto3g_hf_dumpint.dat
1 electron integral
1 1 -1.25247730398
2 2 -0.475934461144
2 electron integral
1 1 1 1 0.674493103326
1 2 1 2 0.181287535812
2 2 1 1 0.663472044861
2 2 2 2 0.69739794982

98. Exact molecular Hydrogen ground state energy: -1.137260278.
----- Performing quantum energy estimation by Trotter simulation algorithm
Rep #1/5: Energy estimate: -1.16005261648883; Phase estimate: -0.424487268195532
Rep #2/5: Energy estimate: -1.11126176600452; Phase estimate: -0.404970928001809
Rep #3/5: Energy estimate: -1.14194138176224; Phase estimate: -0.417242774304894
Rep #4/5: Energy estimate: -1.12383014703564; Phase estimate: -0.409998280414257
Rep #5/5: Energy estimate: -1.11126176600452; Phase estimate: -0.404970928001809
----- End Performing quantum energy estimation by Trotter simulation algorithm
----- Performing quantum energy estimation by Qubitization simulation algorithm
Rep #1/1: Energy estimate: -1.1347488350418; Phase estimate: -0.581804128657449
----- End Performing quantum energy estimation by Qubitization simulation algorithm

99. namespace Microsoft.Quantum.Chemistry.Samples.Hydrogen
{
class Program
{
static void Main(string[] args)
{
//////////////////////////////////////////////////////////////////////////
// Introduction //////////////////////////////////////////////////////////
//////////////////////////////////////////////////////////////////////////
// In this example, we will create a spin-orbital representation of the molecular
// Hydrogen Hamiltonian `H`, given ovelap coefficients for its one- and
// two - electron integrals.
// We when perform quantum phase estimation to obtain an estimate of
// the molecular Hydrogen ground state energy.
#region Building the Hydrogen Hamiltonian through orbital integrals
// One of the simplest representations of Hydrogen uses only two
// molecular orbitals indexed by `0` and `1`.
var nOrbitals = 2;
// This representation also has two occupied spin-orbitals.
var nElectrons = 2;
// The Coulomb repulsion energy between nuclei is
var energyOffset = 0.713776188;
// One-electron integrals are listed below
// <0|H|0> = -1.252477495
// <1|H|1> = -0.475934275
// Two-electron integrals are listed below
// <00|H|00> = 0.674493166
// <01|H|01> = 0.181287518
// <01|H|10> = 0.663472101
// <11|H|11> = 0.697398010
•
$ psi4 07_h2_sto3g_hf_dumpint
1 electron integral
1 1 -1.25247730398
2 2 -0.475934461144
2 electron integral
1 1 1 1 0.674493103326
1 2 1 2 0.181287535812
2 2 1 1 0.663472044861
2 2 2 2 0.69739794982

100. // We initialize a fermion Hamiltonian data structure and add terms to it
var fermionHamiltonian = new
OrbitalIntegralHamiltonian(orbitalIntegrals).ToFermionHamiltonian();
// These orbital integral terms are automatically expanded into
// spin-orbitals. We may print the Hamiltonian to see verify what it contains.
Console.WriteLine("----- Print Hamiltonian");
Console.Write(fermionHamiltonian);
Console.WriteLine("----- End Print Hamiltonian n");
// We also need to create an input quantum state to this Hamiltonian.
// Let us use the Hartree–Fock state.
var fermionWavefunction = fermionHamiltonian.CreateHartreeFockState(nElectrons);
#endregion
#region Jordan–Wigner representation
// The Jordan–Wigner encoding converts the fermion Hamiltonian,
// expressed in terms of Fermionic operators, to a qubit Hamiltonian,
// expressed in terms of Pauli matrices. This is an essential step
// for simulating our constructed Hamiltonians on a qubit quantum
// computer.
Console.WriteLine("----- Creating Jordan–Wigner encoding");
var jordanWignerEncoding =
fermionHamiltonian.ToPauliHamiltonian(Paulis.QubitEncoding.JordanWigner);
Console.WriteLine("----- End Creating Jordan–Wigner encoding n");
// Print the Jordan–Wigner encoded Hamiltonian to see verify what it contains.
Console.WriteLine("----- Print Hamiltonian");
Console.Write(jordanWignerEncoding);
Console.WriteLine("----- End Print Hamiltonian n");
#endregion