What Is an Adaptive-VQE and FAST-VQE

Background: The Challenge with Static VQE

Standard VQE uses a fixed ansatz e.g. unitary coupled cluster ansatz (e.g. UCCSD), meaning the structure of the quantum circuit is predetermined. While often effective, this approach may:

  • Require deep circuits, which is prohibitive on noisy, near-term hardware.

  • Be suboptimal for strongly correlated systems because the ansatz may not align well with the true ground state.

  • Suffer from barren plateaus and local minima that make training difficult.

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

Adaptive-VQE: A Smarter, Adaptive Strategy

One way to overcome these challenges is to use an adaptive variational quantum eigensolver (adaptive-VQE), which builds the ansatz dynamically during the optimization process. Instead of starting with a fixed circuit, adaptive-VQE iteratively adds gates from a predefined pool based on their expected contribution to lowering the energy (see Gate Selection). The original adaptive-VQE algorithm was introduced by Grimsley et al. in 2019 and has since inspired several variants.

  • ADAPT-VQE - The original adaptive-VQE algorithm is the Adaptive Derivative-Assembled Pseudo-Trotter Variational Quantum Eigensolver. Reference: An adaptive variational algorithm for exact molecular simulations on a quantum computer

  • Qubit-excitation-based adaptive variational quantum eigensolver - A variant that focuses on qubit excitations to enhance performance. Reference: arXiv:2011.10540

  • Overlap-ADAPT-VQE — Selects operators based on maximizing overlap with an intermediate target state, helping to avoid local minima and compress the ansatz. Reference: arXiv:2301.10196

  • Qubit-ADAPT-VQE — Uses a hardware-efficient operator pool optimized for near-term devices, often producing shallower circuits while retaining accuracy. Reference: arXiv:1911.10205

  • TETRIS-ADAPT-VQE — Adds multiple disjoint operators per iteration, reducing the number of adaptation steps and final circuit depth. Reference: arXiv:2209.10562

  • CEO-ADAPT-VQE — Introduces a coupled exchange operator pool for dramatically reduced measurement costs and resource usage. Reference: arXiv:2407.08696

Key Idea:

  1. Start with a minimal quantum circuit — often the Quantum circuit corresponding to the Hartree-Fock reference state.

  2. Maintain an operator pool (e.g., single and double excitation operators relevant to chemistry, hardware-efficient gates, etc.).

  3. At each iteration:

    • select the next gate or gates (Gate Selection: See Gate Selection)

    • add gate(s) to circuit

    • Optimize the gate parameter(s) of the expanding circuit. (Parameter optimization with VQE)

  4. Repeat until convergence.

This yields a problem-tailored and compact ansatz.

The Advantages of ADAPT-VQE:

  1. Circuit depth minimized Each added gate must earn its place by maximizing energy improvement—so that the ansatz remains shallow which is suitable for NISQ devices. This also reduces the number of parameters to optimize.

  2. Improved convergence and accuracy Works particularly well for strongly-correlated small molecules (stretched versions of LiH, BeH₂, H₆) where it can achieve chemical accuracy.

  3. Avoiding barren plateaus By gradually growing the ansatz, ADAPT-VQE “burrows” out of poor initializations and avoids large flat regions in the parameter landscape.

FAST-VQE: A Resource Efficient Adaptive Strategy

One major bottleneck with ADAPT-VQE is that significant measurement resources are required for estimating the importance of operators in the wave function. Instead, we implement a more resource-friendly and patented version:

Fermionic Adaptive Sampling Theory for Variational Quantum Eigensolvers (FAST-VQE),

which uses heuristic scoring instead of true gradients to reduce measurement cost while maintaining convergence. Reference: arXiv:2303.07417

The FAST-VQE algorithm is the recommended and default adaptive VQE in Kvantify Qrunch, and can be easily configured (see Calculate the Ground State Energy Using the FAST-VQE).

Summary

FAST-VQE:

  • Builds minimal, and accurate ansatz.

  • Offers hardware-friendly circuit depth and measurement efficiency.

See Also: