Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2312.01036 · IEEE International Conference on Rebooting Computing (ICRC)

Optimal Clifford Initial States for Ising Hamiltonians

Bikrant Bhattacharyya, Gokul Subramanian Ravi

Published Dec 5, 2023• 3 citations
arXivHTMLPDF

Abstract

Modern day quantum devices are in the NISQ era, meaning that the effects of size restrictions and noise are essential considerations when developing practical quantum algorithms. Recent developments have demonstrated that Variational Quantum Algorithms (VQAs) are an appropriate choice for current era quantum devices. VQAs make use of classical computation to iteratively optimize parameters of a quantum circuit, referred to as the ansatz. These parameters are usually chosen such that a given objective function is minimized. Generally, the cost function to be minimized is computed on a quantum device, and then the parameters are updated using a classical optimizer. One of the most promising VQAs for practical hardware is the Variational Quantum Eigensolver (VQE). VQE uses the expectation value of a given Hamiltonian as the cost function, and the minimal value of this particular cost function corresponds to the minimal eigenvalue of the Hamiltonian.Because evaluating quantum circuits is currently very noisy, developing classical bootstraps that help minimize the number of times a given circuit has to be evaluated is a powerful technique for improving the practicality of VQE. One possible such bootstrapping method is creating an ansatz which can be efficiently simulated on classical computers for restricted parameter values. Once the optimal set of restricted parameters is determined, they can be used as initial parameters for a VQE optimization which has access to the full parameter space. Stabilizer states are states which are generated by a particular group of operators called the Clifford Group. Because of the underlying structure of these operators, circuits consisting of only Clifford operators can be simulated efficiently on classical computers. Clifford Ansatz For Quantum Algorithms (CAFQA) is a proposed classical bootstrap for VQAs that uses ansatzes which reduce to clifford operators for restricted parameter values [1].CAFQA has been shown to produce fairly accurate initialization for VQE applied to molecular Hamiltonians. Motivated by this result, in this paper we seek to analyze the Clifford states that optimize the cost function for a new type of Hamiltonian, namely Transverse Field Ising Hamiltonians. Our primary result is contained in theorem IV.1 which connects the problem of finding the optimal CAFQA initialization to a submodular minimization problem which in turn can be solved in polynomial time.

DOI
10.1109/icrc60800.2023.10386971

Related papers

CAFQA: A classical simulation bootstrap for variational quantum algorithms

Gokul Subramanian Ravi, Pranav Gokhale, Yi Ding +7 more

This work tackles the problem of finding a good ansatz initialization for Variational Quantum Algorithms (VQAs), by proposing CAFQA, a Clifford Ansatz For Quantum Accuracy. The CAFQA ansatz is a hardware-efficient circuit built with only Clifford gates. In this ansatz, the parameters for the tunable gates are chosen by searching efficiently through the Clifford parameter space via classical simulation. The resulting initial states always equal or outperform traditional classical initialization (e.g., Hartree-Fock), and enable high-accuracy VQA estimations. CAFQA is well-suited to classical computation because: a) Clifford-only quantum circuits can be exactly simulated classically in polynomial time, and b) the discrete Clifford space is searched efficiently via Bayesian Optimization. For the Variational Quantum Eigensolver (VQE) task of molecular ground state energy estimation (up to 18 qubits), CAFQA's Clifford Ansatz achieves a mean accuracy of nearly 99% and recovers as much as 99.99% of the molecular correlation energy that is lost in Hartree-Fock initialization. CAFQA achieves mean accuracy improvements of 6.4x and 56.8x, over the state-of-the-art, on different metrics. The scalability of the approach allows for preliminary ground state energy estimation of the challenging chromium dimer (Cr$_2$) molecule. With CAFQA's high-accuracy initialization, the convergence of VQAs is shown to accelerate by 2.5x, even for small molecules. Furthermore, preliminary exploration of allowing a limited number of non-Clifford (T) gates in the CAFQA framework, shows that as much as 99.9% of the correlation energy can be recovered at bond lengths for which Clifford-only CAFQA accuracy is relatively limited, while remaining classically simulable.

arXiv:2202.12924•Feb 25, 2022•9 citations

CAFQA: A Classical Simulation Bootstrap for Variational Quantum Algorithms

Gokul Subramanian Ravi, Pranav Gokhale, Yi Ding +7 more

Classical computing plays a critical role in the advancement of quantum frontiers in the NISQ era. In this spirit, this work uses classical simulation to bootstrap Variational Quantum Algorithms (VQAs). VQAs rely upon the iterative optimization of a parameterized unitary circuit (ansatz) with respect to an objective function. Since quantum machines are noisy and expensive resources, it is imperative to classically choose the VQA ansatz initial parameters to be as close to optimal as possible to improve VQA accuracy and accelerate their convergence on today’s devices.

International Conference on Architectural Support for Programming Languages and Operating Systems (ASPLOS)•Dec 19, 2022•39 citations

Excited-CAFQA: A classical simulation bootstrap for the variational estimation of molecular excited states

Bikrant Bhattacharyya, Gokul Subramanian Ravi

Variational Quantum Algorithms (VQAs) are iterative algorithms suited to implementation on current-era quantum devices. VQAs employ classical optimization to minimize cost functions evaluated on quantum circuits. However, the extent to which VQAs manage noise is often insufficient for quantum chemistry applications. One method of improving VQAs is through accurate ansatz initialization. The CAFQA (Clifford Ansatz For Quantum Accuracy) protocol runs a discrete search through a classically simulatable subset of the entire state space to find a desirable initialization. Prior work has evaluated CAFQA applied to the Variational Quantum Eigensolver (VQE), a VQA that computes grounds states of a Hamiltonian. Motivated by CAFQA's success, we propose Excited-CAFQA initialization for Variational Quantum Deflation (VQD), a quantum algorithm that extends VQE by allowing the computation of excited states. VQD recursively computes excited states, by constraining the kth state to be orthogonal to the previous k-1 computed energy states via a penalty term appended to the standard VQE cost function. Just as with VQE, the VQD cost function can be efficiently computed classically for the states considered in the discrete CAFQA search, allowing for the discrete CAFQA optimizer to find good initial parameters for each energy level computation. Preliminary evaluation shows that Excited-CAFQA achieves 90 to 99+% accuracy across a variety of bond lengths and excited states for H2 and HeH+ molecular systems.

arXiv:2509.20588•Sep 24, 2025•0 citations

Classical Regularization in Variational Quantum Eigensolvers

Yury Chernyak, Ijaz Ahamed Mohammad, Martin Plesch

While quantum computers are a very promising tool for the far future, in their current state of the art they remain limited both in size and quality. This has given rise to hybrid quantum-classical algorithms, where the quantum device performs only a small but vital part of the overall computation. Among these, variational quantum algorithms (VQAs), which combine a classical optimization procedure with quantum evaluation of a cost function, have emerged as particularly promising. However, barren plateaus and ill-conditioned optimization landscapes remain among the primary obstacles faced by VQAs, often leading to unstable convergence and high sensitivity to initialization. Motivated by this challenge, we investigate whether a purely classical remedy, standard L2 squared-norm regularization, can systematically stabilize hybrid quantum-classical optimization. Specifically, we augment the Variational Quantum Eigensolver (VQE) objective with a quadratic penalty proportional to the squared norm of the parameters, without modifying the quantum circuit or measurement process. Across all tested Hamiltonians, H2, LiH, and the Random Field Ising Model (RFIM), we observe improved performance over a broad window of the regularization strength. Our large-scale numerical results demonstrate that classical regularization provides a robust, system-independent mechanism for mitigating VQE instability, enhancing the reliability and reproducibility of variational quantum optimization without altering the underlying quantum circuit.

quant-pharXiv:2601.16679•Jan 26, 2026•0 citations

Quadratic Clifford expansion for efficient benchmarking and initialization of variational quantum algorithms

Kosuke Mitarai, Yasunari Suzuki, Wataru Mizukami +2 more

Variational quantum algorithms are considered to be appealing applications of near-term quantum computers. However, it has been unclear whether they can outperform classical algorithms or not. To reveal their limitations, we must seek a technique to benchmark them on large-scale problems. Here we propose a perturbative approach for efficient benchmarking of variational quantum algorithms. The proposed technique performs perturbative expansion of a circuit consisting of Clifford and Pauli rotation gates, which is enabled by exploiting the classical simulatability of Clifford circuits. Our method can be applied to a wide family of parameterized quantum circuits consisting of Clifford gates and single-qubit rotation gates. The approximate optimal parameter obtained by the method can also serve as an initial guess for further optimizations on a quantum device. As the first application of the method, we perform a benchmark of so-called hardware-efficient-type ansatzes when they are applied to the variational quantum eigensolver (VQE) of one-dimensional hydrogen chains up to ${\mathrm{H}}_{24}$, which corresponds to a 48-qubit system using a standard workstation. This is the largest scale benchmark of the VQE to the best of our knowledge and reveals the limitation of hardware-efficient-type ansatzes.

arXiv:2011.09927•Physical Review Research•Jul 6, 2022•32 citations

Classical Regularization in Variational Quantum Eigensolvers

Yury Chernyak

Abstract While quantum computers are a very promising tool for the far future, in their current state of the art they remain limited both in size and quality. This has given rise to hybrid quantum--classical algorithms, where the quantum device performs only a small but vital part of the overall computation. Among these, variational quantum algorithms (VQAs), which combine a classical optimization procedure with quantum evaluation of a cost function, have emerged as particularly promising. However, barren plateaus and ill-conditioned optimization landscapes remain among the primary obstacles faced by VQAs, often leading to unstable convergence and high sensitivity to initialization. Motivated by this challenge, we investigate whether a purely classical remedy---standard L2 squared-norm regularization---can systematically stabilize hybrid quantum--classical optimization. Specifically, we augment the Variational Quantum Eigensolver (VQE) objective with a quadratic penalty \((R(\boldsymbol{\theta}) = \lambda\|\boldsymbol{\theta}\|^2)\), without modifying the quantum circuit or measurement process. Across all tested Hamiltonians---H\((_2)\), LiH, and the Random Field Ising Model (RFIM)---we observe improved performance over a broad window of the regularization strength \((\lambda)\). Our large-scale numerical results demonstrate that classical regularization provides a robust, system-independent mechanism for mitigating VQE instability, enhancing the reliability and reproducibility of variational quantum optimization without altering the underlying quantum circuit.

•Apr 13, 2026•0 citations
Modern day quantum devices are in the NISQ era, meaning that the effects of size restrictions and noise are essential considerations when developing practical quantum algorithms. Recent developments have demonstrated that Variational Quantum Algorithms (VQAs) are an appropriate choice for current era quantum devices. VQAs make use of classical computation to iteratively optimize parameters of a quantum circuit, referred to as the ansatz. These parameters are usually chosen such that a given objective function is minimized. Generally, the cost function to be minimized is computed on a quantum device, and then the parameters are updated using a classical optimizer. One of the most promising VQAs for practical hardware is the Variational Quantum Eigensolver (VQE). VQE uses the expectation value of a given Hamiltonian as the cost function, and the minimal value of this particular cost function corresponds to the minimal eigenvalue of the Hamiltonian.Because evaluating quantum circuits is currently very noisy, developing classical bootstraps that help minimize the number of times a given circuit has to be evaluated is a powerful technique for improving the practicality of VQE. One possible such bootstrapping method is creating an ansatz which can be efficiently simulated on classical computers for restricted parameter values. Once the optimal set of restricted parameters is determined, they can be used as initial parameters for a VQE optimization which has access to the full parameter space. Stabilizer states are states which are generated by a particular group of operators called the Clifford Group. Because of the underlying structure of these operators, circuits consisting of only Clifford operators can be simulated efficiently on classical computers. Clifford Ansatz For Quantum Algorithms (CAFQA) is a proposed classical bootstrap for VQAs that uses ansatzes which reduce to clifford operators for restricted parameter values [1].CAFQA has been shown to produce fairly accurate initialization for VQE applied to molecular Hamiltonians. Motivated by this result, in this paper we seek to analyze the Clifford states that optimize the cost function for a new type of Hamiltonian, namely Transverse Field Ising Hamiltonians. Our primary result is contained in theorem IV.1 which connects the problem of finding the optimal CAFQA initialization to a submodular minimization problem which in turn can be solved in polynomial time.
This work tackles the problem of finding a good ansatz initialization for Variational Quantum Algorithms (VQAs), by proposing CAFQA, a Clifford Ansatz For Quantum Accuracy. The CAFQA ansatz is a hardware-efficient circuit built with only Clifford gates. In this ansatz, the parameters for the tunable gates are chosen by searching efficiently through the Clifford parameter space via classical simulation. The resulting initial states always equal or outperform traditional classical initialization (e.g., Hartree-Fock), and enable high-accuracy VQA estimations. CAFQA is well-suited to classical computation because: a) Clifford-only quantum circuits can be exactly simulated classically in polynomial time, and b) the discrete Clifford space is searched efficiently via Bayesian Optimization. For the Variational Quantum Eigensolver (VQE) task of molecular ground state energy estimation (up to 18 qubits), CAFQA's Clifford Ansatz achieves a mean accuracy of nearly 99% and recovers as much as 99.99% of the molecular correlation energy that is lost in Hartree-Fock initialization. CAFQA achieves mean accuracy improvements of 6.4x and 56.8x, over the state-of-the-art, on different metrics. The scalability of the approach allows for preliminary ground state energy estimation of the challenging chromium dimer (Cr2_22​) molecule. With CAFQA's high-accuracy initialization, the convergence of VQAs is shown to accelerate by 2.5x, even for small molecules. Furthermore, preliminary exploration of allowing a limited number of non-Clifford (T) gates in the CAFQA framework, shows that as much as 99.9% of the correlation energy can be recovered at bond lengths for which Clifford-only CAFQA accuracy is relatively limited, while remaining classically simulable.
Classical computing plays a critical role in the advancement of quantum frontiers in the NISQ era. In this spirit, this work uses classical simulation to bootstrap Variational Quantum Algorithms (VQAs). VQAs rely upon the iterative optimization of a parameterized unitary circuit (ansatz) with respect to an objective function. Since quantum machines are noisy and expensive resources, it is imperative to classically choose the VQA ansatz initial parameters to be as close to optimal as possible to improve VQA accuracy and accelerate their convergence on today’s devices.
Variational Quantum Algorithms (VQAs) are iterative algorithms suited to implementation on current-era quantum devices. VQAs employ classical optimization to minimize cost functions evaluated on quantum circuits. However, the extent to which VQAs manage noise is often insufficient for quantum chemistry applications. One method of improving VQAs is through accurate ansatz initialization. The CAFQA (Clifford Ansatz For Quantum Accuracy) protocol runs a discrete search through a classically simulatable subset of the entire state space to find a desirable initialization. Prior work has evaluated CAFQA applied to the Variational Quantum Eigensolver (VQE), a VQA that computes grounds states of a Hamiltonian. Motivated by CAFQA's success, we propose Excited-CAFQA initialization for Variational Quantum Deflation (VQD), a quantum algorithm that extends VQE by allowing the computation of excited states. VQD recursively computes excited states, by constraining the kth state to be orthogonal to the previous k-1 computed energy states via a penalty term appended to the standard VQE cost function. Just as with VQE, the VQD cost function can be efficiently computed classically for the states considered in the discrete CAFQA search, allowing for the discrete CAFQA optimizer to find good initial parameters for each energy level computation. Preliminary evaluation shows that Excited-CAFQA achieves 90 to 99+% accuracy across a variety of bond lengths and excited states for H2 and HeH+ molecular systems.
While quantum computers are a very promising tool for the far future, in their current state of the art they remain limited both in size and quality. This has given rise to hybrid quantum-classical algorithms, where the quantum device performs only a small but vital part of the overall computation. Among these, variational quantum algorithms (VQAs), which combine a classical optimization procedure with quantum evaluation of a cost function, have emerged as particularly promising. However, barren plateaus and ill-conditioned optimization landscapes remain among the primary obstacles faced by VQAs, often leading to unstable convergence and high sensitivity to initialization. Motivated by this challenge, we investigate whether a purely classical remedy, standard L2 squared-norm regularization, can systematically stabilize hybrid quantum-classical optimization. Specifically, we augment the Variational Quantum Eigensolver (VQE) objective with a quadratic penalty proportional to the squared norm of the parameters, without modifying the quantum circuit or measurement process. Across all tested Hamiltonians, H2, LiH, and the Random Field Ising Model (RFIM), we observe improved performance over a broad window of the regularization strength. Our large-scale numerical results demonstrate that classical regularization provides a robust, system-independent mechanism for mitigating VQE instability, enhancing the reliability and reproducibility of variational quantum optimization without altering the underlying quantum circuit.
Variational quantum algorithms are considered to be appealing applications of near-term quantum computers. However, it has been unclear whether they can outperform classical algorithms or not. To reveal their limitations, we must seek a technique to benchmark them on large-scale problems. Here we propose a perturbative approach for efficient benchmarking of variational quantum algorithms. The proposed technique performs perturbative expansion of a circuit consisting of Clifford and Pauli rotation gates, which is enabled by exploiting the classical simulatability of Clifford circuits. Our method can be applied to a wide family of parameterized quantum circuits consisting of Clifford gates and single-qubit rotation gates. The approximate optimal parameter obtained by the method can also serve as an initial guess for further optimizations on a quantum device. As the first application of the method, we perform a benchmark of so-called hardware-efficient-type ansatzes when they are applied to the variational quantum eigensolver (VQE) of one-dimensional hydrogen chains up to H24{\mathrm{H}}_{24}H24​, which corresponds to a 48-qubit system using a standard workstation. This is the largest scale benchmark of the VQE to the best of our knowledge and reveals the limitation of hardware-efficient-type ansatzes.
Abstract While quantum computers are a very promising tool for the far future, in their current state of the art they remain limited both in size and quality. This has given rise to hybrid quantum--classical algorithms, where the quantum device performs only a small but vital part of the overall computation. Among these, variational quantum algorithms (VQAs), which combine a classical optimization procedure with quantum evaluation of a cost function, have emerged as particularly promising. However, barren plateaus and ill-conditioned optimization landscapes remain among the primary obstacles faced by VQAs, often leading to unstable convergence and high sensitivity to initialization. Motivated by this challenge, we investigate whether a purely classical remedy---standard L2 squared-norm regularization---can systematically stabilize hybrid quantum--classical optimization. Specifically, we augment the Variational Quantum Eigensolver (VQE) objective with a quadratic penalty (R(θ)=λ∥θ∥2)(R(\boldsymbol{\theta}) = \lambda\|\boldsymbol{\theta}\|^2)(R(θ)=λ∥θ∥2), without modifying the quantum circuit or measurement process. Across all tested Hamiltonians---H(2)(_2)(2​), LiH, and the Random Field Ising Model (RFIM)---we observe improved performance over a broad window of the regularization strength (λ)(\lambda)(λ). Our large-scale numerical results demonstrate that classical regularization provides a robust, system-independent mechanism for mitigating VQE instability, enhancing the reliability and reproducibility of variational quantum optimization without altering the underlying quantum circuit.