Source-linked AI summary
An adaptive variational algorithm for exact molecular simulations on a quantum computer
Harper R. Grimsley, Sophia E. Economou, Edwin Barnes, Nicholas J. Mayhall
TL;DR
VQE’s reliance on a pre-selected ansatz limits molecular simulations to approximate wavefunctions and energies. ADAPT-VQE grows a molecule-specific ansatz one operator at a time, producing compact circuits and improved performance over UCCSD in accuracy and operator count. The paper also notes that local operator selection can slow convergence for strongly correlated systems and does not guarantee the truly optimal compact ansatz.
Problem
VQE ansätze generally produce approximate ground states, while strongly correlated systems can require higher-rank excitations that are prohibitively expensive for classical subroutines and NISQ devices.
Method
ADAPT-VQE discovers an ansatz by iteratively selecting fermionic operators that recover correlation energy, using a gradient-norm convergence threshold.
Results
ADAPT-VQE creates more compact and accurate ansätze than UCCSD, routinely outperforming it while keeping errors controllable.
Takeaways & Limitations
ADAPT-VQE is an operator- and parameter-efficient approach with potential for high-accuracy quantum chemistry simulations on NISQ devices.
Takeaways & Limitations
Local operator selection may slow convergence in strongly correlated systems, and the method does not guarantee the truly optimal compact ansatz.
Abstract
from arXiv · showhide
Quantum simulation of chemical systems is one of the most promising near-term applications of quantum computers. The variational quantum eigensolver, a leading algorithm for molecular simulations on quantum hardware, has a serious limitation in that it typically relies on a pre-selected wavefunction ansatz that results in approximate wavefunctions and energies. Here we present an arbitrarily accurate variational algorithm that instead of fixing an ansatz upfront, this algorithm grows it systematically one operator at a time in a way dictated by the molecule being simulated. This generates an ansatz with a small number of parameters, leading to shallow-depth circuits. We present numerical simulations, including for a prototypical strongly correlated molecule, which show that our algorithm performs much better than a unitary coupled cluster approach, in terms of both circuit depth and chemical accuracy. Our results highlight the potential of our adaptive algorithm for exact simulations with present-day and near-term quantum hardware.
I. INTRODUCTION
Quantum computers may benefit molecular simulation, but VQE accuracy is constrained by its pre-selected ansatz. ADAPT-VQE instead discovers a compact ansatz by adding operators iteratively, improving accuracy and reducing operator counts relative to UCCSD in molecular simulations.
- Motivation: Quantum computers could help overcome the exponential Hilbert-space growth that limits classical molecular simulation, provided quantum algorithms are efficient.The paper identifies computational chemistry as a domain likely to benefit from quantum technologies.
- VQE background: VQE shares computational work between classical and quantum hardware while optimizing ansatz parameters to minimize the molecular Hamiltonian expectation value.Classical hardware constructs Hamiltonian terms and updates parameters; quantum hardware prepares states and measures interaction terms.
- VQE limitation: VQE uses shorter circuits than phase estimation but generally obtains only approximate ground states because the ansatz limits variational flexibility.This trades reduced circuit depth for a higher number of measurements.
- Problem: Strongly correlated systems expose UCCSD's limitations, while adding higher-rank excitations would be prohibitively expensive for classical subroutines and NISQ devices.The paper motivates avoiding an ad hoc, fixed ansatz for these systems.
- Contribution: ADAPT-VQE grows a molecule-specific ansatz one fermionic operator at a time, selecting operators to recover maximal correlation energy with minimal operators for a target accuracy.The ansatz is discovered by the algorithm rather than predicted in advance by an excitation-based scheme such as UCCSD.
- Results: Numerical simulations of LiH, BeH2, and H6 report vastly improved performance over UCCSD in operator count and chemical accuracy.The paper evaluates molecules of increasing complexity.
A. Specification of the adopted notation
The notation defines orbital-index conventions and generalized excitation operators used to formulate unitary coupled-cluster ansätze. It also explains their unitary, Trotterized implementation and the approximation introduced by truncating the Trotter expansion.
- A. Specification of the adopted notation: Indices i and j denote occupied orbitals, a and b virtual orbitals, while p, q, r, and s denote arbitrary molecular orbitals.These conventions establish the orbital labels used in subsequent excitation operators.
- A. Specification of the adopted notation: UCCSD replaces coupled-cluster excitation operators with an anti-Hermitian sum of excitation and de-excitation operators.The resulting operator is unitary and supports variational Hamiltonian optimization.
- A. Specification of the adopted notation: The UCCSD expectation value can be written as a normalized Hamiltonian expectation using the Baker-Campbell-Hausdorff formula, but the BCH expansion does not truncate at finite order.Its unitary structure remains useful for quantum algorithms because it corresponds to coherent time evolution.
- A. Specification of the adopted notation: Generalized excitations remove the Hartree–Fock-based subspace restriction by allowing orbital indices to refer to arbitrary orbitals.This includes excitation operators that immediately annihilate the Hartree–Fock state.
- A. Specification of the adopted notation: Because excitation operators generally do not commute, gate-model implementation uses a Trotter expansion to break the matrix exponential into a time-ordered sequence of few-particle operators.A truncated Trotter expansion approximates the underlying UCCSD ansatz; prior work reported that n = 1 can reproduce UCCSD results.
B. ADAPT-VQE algorithm
ADAPT-VQE constructs an ansatz adaptively by selecting operators from a pool according to measured gradients, then re-optimizing its parameters. The procedure continues until a convergence criterion is met, trading additional measurements and repeated VQE optimizations for controllable ansatz accuracy and potentially shallow circuits.
- Algorithm overview: ADAPT-VQE aims to approximate FCI with arbitrary accuracy using a maximally compact sequence of unitary operators.The identity of each operator in the converged ansatz is determined by the algorithm.
- Algorithm overview: The algorithm defines an operator pool, initializes an appropriate reference state and identity ansatz, then prepares the current trial state on quantum hardware.The pool can contain spin-complemented one- and two-body operators, with possible extensions to higher-body terms.
- Adaptive operator selection: At each iteration, commutator measurements estimate every pool operator's gradient, potentially in parallel across uncoupled quantum computers.The gradient identifies which operator is likely to recover the most correlation energy in the subsequent VQE minimization.
- Convergence: The procedure exits when the gradient norm falls below threshold ϵ, although alternative convergence indicators could replace this criterion.Len(⃗θ(n)) equals the number of ansatz operators, and each iteration adds one nonzero parameter.
- Adaptive operator selection: The operator with the largest gradient is added to the ansatz with a new parameter, without removing it from the pool, and all parameters are re-optimized.Because the pool is not drained, an operator may be selected multiple times.
- Resource considerations: ADAPT-VQE reduces circuit depth at the expense of more measurements and repeated VQE re-optimizations, while retaining controllability over ansatz accuracy.For larger systems, discovering the ansatz requires a corresponding number of VQE re-optimizations; freezing early parameters is proposed as one possible mitigation.
C. Molecular dissociation simulation results
ADAPT-VQE was evaluated on LiH, BeH2, and strongly correlated H6 dissociation curves using gradient-norm convergence thresholds. Across these systems, adaptive ansätze achieved high accuracy with fewer operators than UCCSD, though lenient convergence could produce discontinuous curves.
- Simulation setup: ADAPT-VQE simulations evaluated LiH, BeH2, and H6 dissociation using thresholds ϵ1, ϵ2, and ϵ3.The threshold is based on the norm of the operator-pool gradient; ADAPT(ϵ3) denotes convergence below 0.001.
- LiH: ADAPT(ϵ3) outperformed UCCSD across the LiH curve by at least one order of magnitude and sometimes up to four orders of magnitude in error.ADAPT(ϵ2) was comparable to UCCSD at short distances and better at longer distances.
- LiH: Fewer than half as many parameters as UCCSD were needed in all three ADAPT calculations for LiH.UCCSD used 92 parameters, or 64 after combining spin complements; ADAPT(ϵ1) used fewer than 10 across the curve.
- BeH2: For BeH2, ADAPT(ϵ2) and ADAPT(ϵ3) produced nearly exact results with a small fraction of UCCSD’s parameters, while UCCSD failed chemical accuracy beyond about 3 Å.ADAPT(ϵ1) and UCCSD were comparable at smaller bond distances but both exceeded 1 kcal/mol error beyond approximately 3 Å.
- H6: For strongly correlated H6, ADAPT(ϵ2) and ADAPT(ϵ3) remained accurate using only one- and two-body operators, with ADAPT(ϵ2) using fewer operators than UCCSD at most distances.ADAPT(ϵ3) also used fewer parameters than UCCSD up to the distance where UCCSD lost chemical accuracy.
- Convergence behavior: Adaptive parameter counts could change abruptly during bond breaking, causing discontinuous potential-energy curves when optimization stopped at false gradient troughs.Using a tighter threshold such as 0.001 avoided premature termination and yielded high-accuracy results for H6.
D. Dependence of convergence on operator ordering
ADAPT-VQE converges dramatically faster than random and lexical operator-growth schemes in BeH2, demonstrating that gradient-based ordering yields a compact ansatz for a given state.
- D. Dependence of convergence on operator ordering: ADAPT-VQE converges dramatically faster than the four random and lexical operator-ordering schemes compared in BeH2.The comparison includes restricted and unrestricted index pools for both random and lexical growth.
- D. Dependence of convergence on operator ordering: Random-growth ansätze converge relatively similarly whether restricted indices are used or not.
- D. Dependence of convergence on operator ordering: Lexical ordering differs between restricted and unrestricted indices because early operators involving occupied-orbital creation do not contribute until the wavefunction is entangled.
- D. Dependence of convergence on operator ordering: The comparison marks the un-Trotterized UCCSD result for reference and shows that iterative gradient minimization produces a highly compact ansatz for a given state.
III. DISCUSSION
ADAPT-VQE targets shallow, accurate molecular simulations by adapting the ansatz to each system and selecting operators iteratively. The discussion reports strong accuracy and compactness, while noting increased gradient-measurement costs and possible slow convergence for strongly correlated systems.
- III. DISCUSSION: ADAPT-VQE seeks to minimize circuit depth while accepting an increased number of measurements.
- III. DISCUSSION: ADAPT-VQE creates a more compact and accurate ansatz than UCCSD by identifying the operator set and ordering for each problem.
- III. DISCUSSION: ADAPT-VQE likely requires more measurements than UCCSD-based VQE because of necessary gradient measurements.
- III. DISCUSSION: The algorithm is presented as versatile, with multiple implementation choices for gradient updates, convergence determination, and related steps.
- III. DISCUSSION: For strongly correlated systems, convergence will likely be slower because updates use only local information from two-body operators, and a truly optimal compact ansatz is not guaranteed.
- III. DISCUSSION: Adding three- or four-body interactions or multiple operators per iteration are proposed strategies for addressing possible slow convergence.
- III. DISCUSSION: ADAPT-VQE is operator- and parameter-efficient, achieves high accuracy with controllable errors, and routinely outperforms UCCSD.
- III. DISCUSSION: Its compatibility with classical circuit compilation and quantum parallelism is described as useful for molecular simulations on current and future quantum computers.