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