Pejman Jouzdani

Quantum error correction (QEC) is a way to protect quantum information against noise. It consists of encoding input information into entangled quantum states known as the code space. Furthermore, to classify if the encoded information is corrupted or intact, a step known as syndrome detection is performed. For stabilizer codes, this step consists of measuring a set of stabilizer operators. In this paper, inspired by the QEC approach, and specifically stabilizer codes, we propose a prototype quantum circuit model for classification of classical data. The core quantum circuit can be considered as a quantum perceptron where the classification is based on syndrome detection. In this proposal, a quantum perceptron is realized by one stabilizer as part of a stabilizer code, while a quantum neural network (QNN) layer is realized by a stabilizer code which consists of many stabilizers. The concatenation of stabilizer codes results in complex QNNs. The QNN is trained by performing measurements and optimization of a set of parametrized stabilizers. We demonstrate the concept numerically. In this paper we also consider the first challenge to most applications of quantum computers, including data classification, which is to load data into the memory of the quantum computer. This loading amounts to representing the data as a quantum state, i.e., quantum state preparation. An exact amplitude encoding algorithm requires a circuit of exponential depth. We introduce an alternative recursive algorithm which approximates amplitude encoding with only a polynomial number of elementary gates. We name it the recursive approximate-scheme algorithm.

Quantum error correction (QEC) is a way to protect quantum information against noise. It consists of encoding input information into entangled quantum states known as the code space. Furthermore, to classify if the encoded information is corrupted or intact, a step known as syndrome detection is performed. For stabilizer codes, this step consists of measuring a set of stabilizer operators. In this paper, inspired by the QEC approach, and specifically stabilizer codes, we propose a prototype quantum circuit model for classification of classical data. The core quantum circuit can be considered as a \emph{quantum perceptron} where the classification is based on syndrome detection. In this proposal, a quantum perceptron is realized by one stabilizer as part of a stabilizer code, while a quantum neural network (QNN) layer is realized by a stabilizer code which consists of many stabilizers. The concatenation of stabilizer codes results in complex QNNs. The QNN is trained by performing measurements and optimization of a set of parameterized stabilizers. We demonstrate the concept numerically. In this paper we also consider the first challenge to most applications of quantum computers, including data classification, which is to load data into the memory of the quantum computer. This loading amounts to representing the data as a quantum state, i.e., quantum state preparation. An exact amplitude encoding algorithm requires a circuit of exponential depth. We introduce an alternative recursive algorithm which approximates amplitude encoding with only a polynomial number of elementary gates. We name it recursive approximate-scheme algorithm (RASA).

The use of near-term quantum devices that lack quantum error correction, for addressing quantum chemistry and physics problems, requires hybrid quantum-classical algorithms and techniques. Here, we present a process for obtaining the eigenenergy spectrum of electronic quantum systems. This is achieved by projecting the Hamiltonian of a quantum system onto a limited effective Hilbert space specified by a set of computational bases. From this projection, an effective Hamiltonian is obtained. Furthermore, a process for preparing short depth quantum circuits to measure the corresponding diagonal and off-diagonal terms of the effective Hamiltonian is given, whereby quantum entanglement and ancilla qubits are used. The effective Hamiltonian is then diagonalized on a classical computer using numerical algorithms to obtain the eigenvalues. The use case of this approach is demonstrated for ground state and excited states of BeH₂ and LiH molecules, and the density of states, which agrees well with exact solutions. Additionally, hardware demonstration is presented using IBM quantum devices for H₂ molecule.

The use of near-term quantum devices that lack quantum error correction, for addressing quantum chemistry and physics problems, requires hybrid quantum-classical algorithms and techniques. Here we present a process for obtaining the eigenenergy spectrum of electronic quantum systems. This is achieved by projecting the Hamiltonian of a quantum system onto a limited effective Hilbert space specified by a set of computational bases. From this projection an effective Hamiltonian is obtained. Furthermore, a process for preparing short depth quantum circuits to measure the corresponding diagonal and off-diagonal terms of the effective Hamiltonian is given, whereby quantum entanglement and ancilla qubits are used. The effective Hamiltonian is then diagonalized on a classical computer using numerical algorithms to obtain the eigenvalues. The use case of this approach is demonstrated for ground sate and excited states of BeH$_2$ and LiH molecules, and the density of states, which agrees well with exact solutions. Additionally, hardware demonstration is presented using IBM quantum devices for H$_2$ molecule.