nonunitary_quantum_computing

Linear Combination of Unitaries

An example LCU implementation of a $2 \times 2$ non-unitary matrix.

Implementation

Coefficients $\alpha_i$ are determined performing a PauliTerms decomposition.

Consequently, SELECT and PREPARE operators are prepared so that $\langle 0| \text{PREPARE}^{\dagger}.\text{SELECT}.\text{PREPARE} |0\rangle = \hat{U} |\psi\rangle$. Here, $\hat{U}$ is the quantum circuit. For our problem $\hat{U} = 0.5\hat{I} + 0.5\hat{Z}$. This means applying $\hat{I}$ when controller qubit is in state $|0\rangle$ and applying $\hat{Z}$ when controller qubit is in state $|1\rangle$. The required operations are defined in lcu_controllers(controller: QNum, psi: QNum).

The rest is simply preparing the state $|\psi\rangle$ which is to be operated by our non-unitary operator and applying the lcu_controllers on it.

Software Requirements

The following Python packages are required to run this notebook:

• numpy
• classiq

Note: These packages are included in the top-level requirements.txt. If you have set up the base environment as described in the main README, no additional installation is needed.

Outputs

The notebook may generate the following file when executed:

• lcu-2x2.qmod - Quantum model file (generated if the write_qmod cell is executed)

This file is saved in the nonunitary_quantum_computing/ directory. The notebook also displays measurement results showing the application of the non-unitary operator, but these are displayed inline and not saved to files by default.

GNU GPL v3+

Copyright (C) 2024 Mohammadreza Khellat GNU GPL v3+

This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.

See also https://www.gnu.org/licenses/gpl.html