The qINTERCONNECT solver can be used to calculate the evolution of a quantum state in a photonic integrated circuit (PIC), which is represented classically in the frequency domain by its scattering matrix. The quantum state in the Fock (number) basis is the input to the solver, which returns the output as a density matrix. Additionally, the solver can report the fidelity of the final state with respect to an expected final state and the probability of success for measuring a given output.

## qINTERCONNECT and INTERCONNECT

The S-Parameter Simulator in INTERCONNECT is used to calculate the frequency domain response for a photonic integrated circuit. qINTERCONNECT relies on these s-parameters to determine the response in the Fock basis, represented by a quantum S-matrix.

## Solver Physics

The evolution of a single photon in a photonic circuit is completely described by the classical S-parameters given by INTERCONNECT. For more than one photon, non-classical interference between photons must be accounted for, which is done by summing the complex amplitude coefficients of the final states.

A general photonic circuit with K channels has an S-matrix with components \(S_{ij}\) describing the amplitude of an incoming state in channel \(j\) being scattered to channel \(i\), where \(i\) and \(j\) range from \(1\) to \(K\).

$$ S = \begin{bmatrix} S_{11} & \cdots & S_{1K} \\ \vdots & \ddots & \vdots \\ S_{K1} & \cdots & S_{KK} \end{bmatrix} $$

A general input to such a circuit, in the Fock basis, can be written as:

$$ \vert \psi_{in} \rangle =\vert n_1, n_2, \dots, n_K \rangle = \frac{ \left( {a_1}^{\dagger} \right)^{n_1} }{ \sqrt{n_1 !} } \frac{ \left( {a_2}^{\dagger} \right)^{n_2} }{ \sqrt{n_2 !} } \dots \frac{ \left( {a_K}^{\dagger} \right)^{n_K} }{ \sqrt{n_K !} } \vert vac \rangle = \prod_i^K \frac{\left( {a_i}^{\dagger} \right)^n_i }{ \sqrt{n_i !} } \vert vac \rangle $$

where \( {a_i}^{\dagger} \) are creation operators. The quantum S matrix is defined by applying the classical S matrix to each of the input modes.

$$ \vert \psi_{out} \rangle = QS \vert n_1, n_2, \dots, n_K \rangle = \sum_j^K \prod_i^K \frac{\left( S_{ji}{a_i}^{\dagger} \right)^{n_i}}{\sqrt{n_i !}} \vert vac \rangle $$

In terms of a density matrix, the evolution of the system can be described as

$$ \rho_1 = \left( QS \right) \rho_{in} \left( QS \right)^{\dagger} $$

Finally, the output state is changed by any measurements that are performed on any of the output channels. In the Fock basis, a measurement consists of detecting a certain number of photons at the output of a given channel. This has the effect of reducing the dimension of the final density matrix and renormalizing it by the probability of a successful measurement [1]

$$ \rho_{final} = Tr \left[ \frac{A \rho_1 A^{\dagger}}{p_{success}} \right] = \frac{A \rho_1 A^{\dagger}}{Tr \left[ \rho_1 A^{\dagger}A \right] } $$

Where \(M = A^{\dagger}A \) is a projective measurement operator and \(p_{success}\) is the probability of success for a measurement outcome.

Optionally, the fidelity of the final density matrix can be calculated with respect to an expected density matrix:

$$ F = Tr \sqrt{{\rho_{final}}^{\frac{1}{2}} \rho_{expected} {\rho_{final}}^{\frac{1}{2}} } $$

## Units and Normalization

Unless otherwise specified, all quantities are returned in SI units. Please see Units and Normalization for more information.

## Examples

## Reference

- M. Nielsen & I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press (2010)