#
Algorithms

Ansatz

A parametrized quantum circuit $U(\vec{\theta})$ and an ansatz quantum state $|\psi(\vec{\theta})\rangle = U(\vec{\theta})|0\rangle$ are often involved in quantum algorithms for NISQ devices. Here we introduce several quantum circuits and ansatze.

## Hardware efficient ansatz

The parametrized quantum circuit of the hardware efficient ansatz consists of a sequence of single qubit rotation gates and "entangling" 2-qubit gates. One example of the circuits for the ansatz is shown in Fig. 1 (the case of 4-qubit systems). Note that each single qubit rotation gate, $R_{Y(Z)} = e^{i\theta Y(Z)}, $ has an angle $\theta$ as a parameter. The number of repetitions of the single qubit rotations and 2-qubit gates (here we choose controlled Z gates) are called depth, denoted as $D$ in the figure. The structure of the circuit is easy to implement in real NISQ devices (especially ones composed of superconducting qubits) because the 2-qubit gates are applied only to adjacent qubits. More details are found in the reference below.

### Reference

- "Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets", A. Kandala
*et al.*, Nature**549**, 242–246 (2017).

## Unitary coupled cluster (UCC) ansatz

UCC ansatz is a cousin of the coupled-cluster method in quantum chemistry and often applied to calculation of a molecular system by quantum computers. The most common choice is the UCC singles and doubles ansatz (unitary CCSD), defined as $$ |\psi_{\rm unitary-CCSD}\rangle = U(\vec{\theta}) |HF\rangle, \ U(\vec{\theta}) = \exp(T(\vec{\theta}) - T^\dagger(\vec{\theta})), \ T( \vec{\theta}) = \sum_{ij} \theta_{ij} a_i^\dagger a_j + \frac{1}{2}\sum_{ijkl} \theta_{ijkl} a_i^\dagger a_j^\dagger a_k a_l, $$ where $|HF\rangle$ is the Hartree-Fock state of the system. This ansatz often gives a good approximation of the ground state of a given molecular Hamiltonian because it nicely captures the essence of the electron correlations in the molecule.

In actual implementation of this ansatz on NISQ devices, one has to decompose $U(\vec{\theta})$ into a set of small gates available on NISQ devices.

### Reference

- "A variational eigenvalue solver on a photonic quantum processor" A. Peruzzo
*et al.*, Nature Communications**5**, 4213 (2014).

## Adaptive Derivative-Assembled Pseudo-Trotter ansatz Variational Quantum Eigensolver (ADAPT-VQE)

ADAPT-VQE is an algorithm where the structure of the ansatz is determined in an adaptive manner. Please refer the reference for details.

### Reference

"An adaptive variational algorithm for exact molecular simulations on a quantum computer", H. R. Grimsley *et al.*, Nature Communications **10**, 3007 (2019).

## Quantum circuit structure learning

"Quantum circuit structure learning" proposed by Ostaszewski *et al.* is an algorithm similar to the ADAPT-VQE.
It optimizes operators of single qubit rotation gates in the ansatz as well as the parameters (rotation angle) of them.

### Reference

"Quantum circuit structure learning", M. Ostaszewski, E. Grant, and M. Benedetti, https://arxiv.org/abs/1905.09692