Gate Selection

Overview

In adaptive Variational Quantum Eigensolver (adaptive-VQE) algorithms, gate selection refers to the process of deciding which gate to add next from a predefined gate pool to the variational circuit. It is therefore the central component that leads to short circuits with few parameters while achieving high accuracy, which is ideal for near-term quantum hardware.

(See Quantum Circuits for Computational Chemists — for an explanation of the quantum computing nomenclature.)

The key steps in this central part of adaptive-VQE algorithms, such as ADAPT-VQE and FAST-VQE, are as follows:

  1. Evaluate the importance of each candidate gate in the pool.

  2. Select the gate with the highest potential to lower the energy.

  3. Append this gate to the variational circuit and re-optimize parameter(s).

Constructing the Gate Pool

The gate pool contains all candidate gates the algorithm is allowed to choose from.

For quantum chemistry problems, it may for example be built from single and double excitation operators :

  • Single excitations: promote one electron from an occupied orbital \(i\) to a virtual orbital \(a\)

  • Double excitations: promote two electrons simultaneously, \(i \to a\) and \(j \to b\)

For the reasons explanationed in Quantum Circuits for Computational Chemists, these excitation operators are expressed as anti-Hermitian fermionic operators which induce unitary transformations of the form:

\[U(\theta) = e^{-\theta \hat{\kappa_I}}\]

Here, \(\hat{\kappa_I}\) is an anti-Hermitian operator corresponding to excitation \(I\), where \(I\) is a composite index representing the specific orbitals involved in the excitation. In FAST-VQE, we approximate these unitaries using QEB (Qubit Excitation-Based) gates, which are hardware-efficient implementations suitable for near-term quantum devices. In this language, the operators \(\hat{\kappa}_I\) will adopt another meaning, representing the qubit-based analogs of the fermionic excitation operators. Otherwise the equations hold.

\[U(\theta) = e^{-i \theta \hat{T}}\]

The mapping from fermionic to qubit operators can be done via:

  • Jordan–Wigner transformation — preserves fermionic anticommutation via \(Z\)-strings

  • Qubit-Excitation-Based (QEB) transformation — more hardware efficient, drops \(Z\)-strings at the cost of losing exact fermionic algebra

In FAST-VQE, we use QEB for efficiency.

Selection Criteria

Different algorithms define importance differently:

  • ADAPT-VQE — selects the gate with the largest energy gradient:

    \[g_I = \langle \Psi \,|\, [\hat{\kappa}_I, \hat{H}] \,|\, \Psi \rangle\]

  • FAST-VQE — uses the heuristic gradient:

    \[\mu_I = \sum_{D_i \in S} \Re \big( \langle D_i | \hat{\kappa}_I^\dagger \hat{H} | D_i \rangle \big)\]

    Here, \(S\) is the set of sampled determinants from the last energy evaluation. The heuristic gradient can be computed classically once \(S\) is known, reducing quantum hardware calls. In contrast the ADAPT-VQE gradient requires all the expectation values to be measured on quantum hardware.

  • Heuristic Selected CI — a classical approach introduced alongside the heuristic gradient in arXiv:2303.07417, which selects gates based on Configuration Interaction (CI) coefficients. However, the heuristic gradient has generally been found to give better performance and is the recommended choice in FAST-VQE.

Advantages of the Heuristic Gradient

The heuristic gradient in FAST-VQE:

  • Avoids extra quantum evaluations after each iteration

  • Requires fewer circuit executions, which is cost-effective on paid quantum backends

  • Retains high selection accuracy in practice

Applying the Selected Gate

Assume we have a current wave function \(|\Psi_i\rangle\) in iteration \(i\) of the adaptive algorithm. Using this wave function, we compute the importance scores for all candidate gates in the pool, and select the gate \(\kappa_{i+1}\) with the best score.

Once a gate \(\kappa_{i+1}\) is selected:

  1. It is appended to the circuit as a parameterized unitary \(U_{i+1}(\theta_{i+1}) = e^{-\theta_{i+1} \kappa_{i+1}}\)

  2. The parameter \(\theta_{i+1}\) is variationally optimized alongside existing parameters \(\theta_1, \ldots, \theta_i\) (VQE) to yield the updated wave function:

    \[|\Psi(\boldsymbol{\theta})\rangle = U_{i+1}(\theta_{i+1}) U_{i}(\theta_{i}) \cdots U_1(\theta_1) |\text{HF}\rangle\]

This incremental, on-demand ansatz growth ensures computational resources are focused on the most impactful operators.

Summary

Gate selection is the decision-making core of adaptive VQE methods, which determine how efficiently and accurately the ansatz approaches the true ground state.

See Also