Creating Hamiltonian Simulation Operator in Q#

Question

I am trying to create a unitary operator $U = \sum^{T - 1}_{k=0}$ $|k\rangle$ $\langle k |$ $ \otimes$ $e^{i A k}$ in Q#, where A is a Hermitian matrix. For the beginning, I just want A to be a combination of 2 Pauli matrices, say $A = X + 2Z$. This is the code that I have, where input is of type Qubit[3] and register of Qubit[2]:

   ...
   let SIZE_OF_MATRIX = 2; 
   let unitaryGenerator = (2 ^ SIZE_OF_MATRIX, ConstructU);
   let registerLE = LittleEndian(register);

   MultiplexOperationsFromGenerator(unitaryGenerator, registerLE, input);
   ...

function ConstructU (j : Int) : (Qubit[] => Unit is Adj + Ctl) {
    let generatorSystem = GeneratorSystem(2, MapToGeneratorIndex);

    let evolutionGenerator = EvolutionGenerator(PauliEvolutionSet(), generatorSystem);

    let unitaryOperator = TrotterStep(evolutionGenerator, 1, - IntAsDouble(j));

    return unitaryOperator;
}

// The purpose of this function is to map each part of the Hamiltonian generator
// to a generator index.
// Initially, we want to test the matrix A = X + 2Z.
function MapToGeneratorIndex (index : Int) : GeneratorIndex {
    // We only have 2 terms, hence index can only be 0 or 1

    if (index == 0) {
        // Here we just want X
        return GeneratorIndex(([1], [1.0]), [0]);
    }

    elif (index == 1) {
        // Here we want 2Z
        return GeneratorIndex(([3], [2.0]), [0]);
    }

    // TODO: throw an error
    return GeneratorIndex( ([1000], [1000.0]), [0]);
}

Does anyone know what am I doing incorrectly? I am not getting the result that I am expecting. I know the code is messy but I am just trying to make it work for a basic 2x2 matrix first.

Thanks for the help!

Answer 1

Check your arguments to TrotterStep(). You are passing -IntAsDouble(j) as the trotterStepSize argument, but I believe this should just be a constant, since it indicates the simulation time for each Trotter step.

If I change that line of your code to do something like:

let unitaryOperator = TrotterStep(evolutionGenerator, 1, 0.1);

where I arbitrarily chose a value of 0.1 for trotterStepSize, I get some behavior that I believe is more like what you are expecting.