6x More Accurate Quantum Algorithm Initialization with CAFQA Method

Quantum computing is a rapidly growing field that offers the potential to solve some of the most challenging computational problems. However, quantum machines are noisy and expensive resources, which can lead to inaccurate and slow computations. In this article, we discuss the paper titled “CAFQA: A classical simulation bootstrap for variational quantum algorithms,” which proposes a novel method to improve the accuracy and speed of variational quantum algorithms (VQAs) by finding better initializations for the ansatz, a parameterized unitary circuit.

Quantum Algorithm

Variational Quantum Algorithms

VQAs are a class of quantum algorithms that can be used to solve optimization problems. They are hybrid algorithms that combine classical optimization with quantum circuits to solve problems that are intractable for classical computers. VQAs have been shown to be effective in solving problems such as molecular energy estimation and optimization, and have been implemented on near-term quantum devices.

However, VQAs face many challenges due to the noisy and error-prone nature of quantum machines. One of the main challenges is finding good initializations for the ansatz, a parameterized unitary circuit that is used to prepare the quantum state. The ansatz is typically initialized randomly, which can lead to poor performance and slow convergence.

CAFQA addresses this challenge by finding better initializations for the ansatz using a classical simulation bootstrap. CAFQA uses a Clifford-only circuit as the ansatz and applies a classical optimization algorithm to find the optimal parameters for the circuit. The authors show that CAFQA can improve the performance of VQAs on the Variational Quantum Eigensolver (VQE) task of estimating molecular ground state energy.

CAFQA Method

The CAFQA (Clifford Ansatz For Quantum Accuracy) method presented in the paper offers a solution to the problem of improving initialization of VQA circuits by leveraging classical simulation to explore the Clifford space of VQA problems. CAFQA achieves this by using a hardware-efficient circuit built with only Clifford gates, which can be simulated classically in polynomial time. The parameters for tunable gates are found by efficiently searching through the Clifford parameter space using classical simulation.

The authors demonstrate that CAFQA outperforms traditional classical initialization methods and enables higher-accuracy VQA estimations. The advantages of using Clifford-only circuits are that they are highly scalable and computationally efficient, and the discrete Clifford space can be searched efficiently via Bayesian Optimization. CAFQA allows for preliminary ground state energy estimation for the challenging chromium dimer (Cr2) molecule.

The CAFQA algorithm starts with a parameterized circuit with all fixed gates being Clifford, typically from a hardware-efficient ansatz. It performs a discrete search over tunable circuit parameters limited to angles that make the tunable gates Clifford. Since both fixed and tunable gates are Clifford, the resulting circuit can be simulated classically, even as the size of the circuits grows. Simulating ansatz circuits and measuring the expectation produce the objective function value for the iterative tuning process.

Electron and spin preservation constraints can be added to the Hamiltonian or directly to the objective function. Simulations performed classically are free of noise, potentially eliminating noise-induced barren plateaus in variational tuning. One-shot simulation is required for each Pauli term, as expectation values are strictly +1, -1, or 0 for stabilizer states in Clifford circuits.

The search continues until convergence of the minimum value is obtained or for a specific tuning budget. The resulting circuit with Clifford parameters corresponding to the minimum objective function value is the Clifford ansatz, ready for traditional VQA optimization.

Subsequent VQA tuning on the quantum device is noisy but allows for exploring the entire quantum space, with CAFQA’s initial state potentially enabling faster and more accurate convergence. The CAFQA algorithm can be implemented in Python using Qiskit for evaluations, and classical computations can be carried out predominantly on the Google Compute Cloud.

Comparison of Different Methods of Ansatz Tuning

In the paper, the authors present a comparison of different methods of ansatz tuning for a 2-qubit XX Hamiltonian using a hardware-efficient ansatz with one tuning parameter. The methods compared are Hartree-Fock initialization, CAFQA, and a noisy quantum device exploration of the entire tuning space.

The results show that while noisy quantum devices are able to explore the entire tuning space, the effect of noise limits their performance. In contrast, the Clifford-based approach (CAFQA) is able to achieve the global minimum even though the search space is limited. The Hartree-Fock initialization method is unable to produce useful results in this case.

The comparison shows that CAFQA offers advantages over the other methods by providing a noise-free, efficient search through the discrete Clifford parameter space. Additionally, CAFQA enables high accuracy even prior to execution on a quantum device, which can accelerate the VQA convergence by up to 2.5x, even for small molecules.

Estimation of Ground State Energies of Several Molecules

The paper evaluates the performance of CAFQA in estimating the ground state energies of several molecules, including H2, LiH, H2O, H6, Cr2, N2, NaH, H2-S1, and BeH2. They construct Hamiltonians in the STO-3G basis with parity mapping and Z2 symmetry/two-qubit reduction and use a SU2 parameterized circuit with one layer of linear entanglement as an ansatz.

The authors compare the performance of CAFQA with the exact energy estimations computed classically and the Hartree-Fock (HF) computational basis state. The evaluation is based on four metrics: ground state energy, energy estimation accuracy, recovered correlation energy, and relative accuracy.

The results show that CAFQA outperforms the Hartree-Fock method, achieving more accurate energy estimates and recovering up to 99.998% of the correlation energy over HF as bond lengths increase. The relative accuracy of CAFQA compared to the state-of-the-art Hartree-Fock approach is also presented, showing that CAFQA achieves significant average relative accuracy improvements over all applications, with a mean of 6.4x.

The authors also explore post-CAFQA VQE exploration, where CAFQA initialization allows for more focused tuning on the machine, resulting in faster and more accurate convergence. Additionally, the number of iterations consumed by CAFQA’s discrete search to converge to a minimum energy estimate is shown to be reasonable. The authors conclude by discussing the future exploration of allowing a few T gates within the CAFQA framework to extend simulation beyond Cliffords.


The CAFQA algorithm proposes a novel approach to improve the performance of variational quantum algorithms by finding better initializations for the ansatz, a parameterized unitary circuit. CAFQA leverages classical simulation to explore the Clifford space of VQA problems, providing high accuracy VQA ansatz initialization and outperforming state-of-the-art methods. It emphasizes the potential for quantum-inspired classical techniques and a synergistic quantum-classical paradigm to boost NISQ-era quantum computing.

In conclusion, the CAFQA algorithm presents a promising solution to improve the performance of variational quantum algorithms by providing better initialization states. The algorithm offers a significant improvement in the accuracy of the ground state energy estimation of several molecules and demonstrates the potential for quantum-inspired classical techniques to boost NISQ-era quantum computing towards real-world applicability.


Hello! I'm a Quantum Computing Scientist based in Silicon Valley with a strong background in software engineering. My blog is dedicated to sharing the tools and trends I come across in my research and development work, as well as fun everyday anecdotes.

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