static void Main(string[] args)
{
//////////////////////////////////////////////////////////////////////////
// Introduction //////////////////////////////////////////////////////////
//////////////////////////////////////////////////////////////////////////
// In this example, we will estimate the ground state energy of
// 1D Hubbard Hamiltonian using the quantum chemistry library.
// The 1D Hubbard model has `n` sites. Let `i` be the site index,
// `s` = 1,0 be the spin index, where 0 is up and 1 is down, `t` be the
// hopping coefficient, `u` the repulsion coefficient, and aᵢₛ the fermionic
// annihilation operator on the fermion indexed by `(i,s)`. The Hamiltonian
// of this model is
//
// H ≔ - t Σᵢ (a†ᵢₛ aᵢ₊₁ₛ + a†ᵢ₊₁ₛ aᵢₛ) + u Σᵢ a†ᵢ₀ a†ᵢ₁ aᵢ₁ aᵢ₀
//
// Note that we use closed boundary conditions.
#region Building the Hubbard Hamiltonian through orbital integrals
var t = 0.2; // hopping coefficient
var u = 1.0; // repulsion coefficient
var nSites = 6; // number of sites;
// Construct Hubbard Hamiltonian
var hubbardOrbitalIntegralHamiltonian = new OrbitalIntegralHamiltonian();
foreach (var i in Enumerable.Range(0, nSites))
{
hubbardOrbitalIntegralHamiltonian.Add(new OrbitalIntegral(new[] { i, (i + 1) % nSites }, -t));
hubbardOrbitalIntegralHamiltonian.Add(new OrbitalIntegral(new[] { i, i, i, i }, u));
}
// Create fermion representation of Hamiltonian
// In this case, we use the spin-orbital to integer
// indexing convention `x = orbitalIdx + spin * nSites`; as it
// minimizes the length of Jordan–Wigner strings
var hubbardFermionHamiltonian = hubbardOrbitalIntegralHamiltonian.ToFermionHamiltonian(IndexConvention.HalfUp);
#endregion
#region Estimating energies by simulating quantum phase estimation
// Create Jordan–Wigner representation of Hamiltonian
var jordanWignerEncoding = hubbardFermionHamiltonian.ToPauliHamiltonian();
// Create data structure to pass to QSharp.
var qSharpData = jordanWignerEncoding.ToQSharpFormat().Pad();
Console.WriteLine($"Estimate Hubbard Hamiltonian energy:");
// Bits of precision in phase estimation.
var bits = 7;
// Repetitions to find minimum energy.
var reps = 5;
// Trotter step size
var trotterStep = 0.5;
using (var qsim = new QuantumSimulator())
{
for (int i = 0; i < reps; i++)
{
// EstimateEnergyByTrotterization
// Name should make clear that it does it by trotterized
var(phaseEst, energyEst) = GetEnergy.Run(qsim, qSharpData, bits, trotterStep).Result;
Console.WriteLine($"Rep #{i}: Energy estimate: {energyEst}; Phase estimate: {phaseEst}");
}
}
Console.WriteLine("Press Enter to continue...");
if (System.Diagnostics.Debugger.IsAttached)
{
Console.ReadLine();
}
#endregion
}
static void Main(string[] args)
{
//////////////////////////////////////////////////////////////////////////
// Introduction //////////////////////////////////////////////////////////
//////////////////////////////////////////////////////////////////////////
// In this example, we will estimate the ground state energy of
// 1D Hubbard Hamiltonian using the quantum chemistry library.
// The 1D Hubbard model has `n` sites. Let `i` be the site index,
// `s` = 1,0 be the spin index, where 0 is up and 1 is down, `t` be the
// hopping coefficient, `u` the repulsion coefficient, and aᵢₛ the fermionic
// annihilation operator on the fermion indexed by `(i,s)`. The Hamiltonian
// of this model is
//
// H ≔ - t Σᵢ (a†ᵢₛ aᵢ₊₁ₛ + a†ᵢ₊₁ₛ aᵢₛ) + u Σᵢ a†ᵢ₀ a†ᵢ₁ aᵢ₁ aᵢ₀
//
// Note that we use closed boundary conditions.
#region Building the Hubbard Hamiltonian through orbital integrals
var t = 0.2; // hopping coefficient
var u = 1.0; // repulsion coefficient
var nSites = 6; // number of sites;
// Construct Hubbard Hamiltonian
var hubbardHamiltonian = new FermionHamiltonian(nOrbitals: nSites, nElectrons: nSites);
foreach (var i in Enumerable.Range(0, nSites))
{
hubbardHamiltonian.AddFermionTerm(new OrbitalIntegral(new[] { i, (i + 1) % nSites }, -t));
hubbardHamiltonian.AddFermionTerm(new OrbitalIntegral(new[] { i, i, i, i }, u));
}
// Let us verify that both Hamiltonians are identical
hubbardHamiltonian.SortAndAccumulate();
Console.WriteLine($"Hubbard Hamiltonian:");
Console.WriteLine(hubbardHamiltonian + "\n");
#endregion
#region Estimating energies by simulating quantum phase estimation
var jordanWignerEncoding = JordanWignerEncoding.Create(hubbardHamiltonian);
var qSharpData = jordanWignerEncoding.QSharpData();
Console.WriteLine($"Estimate Hubbard Hamiltonian energy:");
// Bits of precision in phase estimation.
var bits = 7;
// Repetitions to find minimum energy.
var reps = 5;
// Trotter step size
var trotterStep = 0.5;
using (var qsim = new QuantumSimulator())
{
for (int i = 0; i < reps; i++)
{
// EstimateEnergyByTrotterization
// Name shold make clear that it does it by trotterized
var(phaseEst, energyEst) = GetEnergy.Run(qsim, qSharpData, bits, trotterStep).Result;
Console.WriteLine($"Rep #{i}: Energy estimate: {energyEst}; Phase estimate: {phaseEst}");
}
}
Console.WriteLine("Press Enter to continue...");
if (System.Diagnostics.Debugger.IsAttached)
{
Console.ReadLine();
}
#endregion
}