Abstract In variational quantum algorithms (VQAs), parameterization is typically applied to single-qubit gates. In this study, we instead parameterize a generalized controlled gate and propose an algorithm to locally minimize the cost function by maximally optimizing these parameters. This method extends the free quaternion selection technique, which was originally developed for single-qubit gate optimization. To evaluate its performance, we apply the proposed method to a variety of quantum optimization tasks, including the variational quantum eigensolver for both Ising and molecular Hamiltonians, fidelity maximization in general VQAs, and unitary compilation of time evolution operators. Across these applications, our method demonstrates efficient optimization, enhanced expressibility, and the ability to construct shallower circuits compared to existing techniques. Moreover, the method can be generalized to optimize particle-number-conserving gates, which are particularly relevant for quantum chemistry. Leveraging this capability, we further demonstrate that the method achieves superior quantum compilation of molecular time-evolution operators by approximating them with shallower circuits than standard Trotter decomposition.
Abstract In variational quantum algorithms (VQAs), parameterization is typically applied to single-qubit gates. In this study, we instead parameterize a generalized controlled gate and propose an algorithm to locally minimize the cost function by maximally optimizing these parameters. This method extends the free quaternion selection technique, which was originally developed for single-qubit gate optimization. To evaluate its performance, we apply the proposed method to a variety of quantum optimization tasks, including the variational quantum eigensolver for both Ising and molecular Hamiltonians, fidelity maximization in general VQAs, and unitary compilation of time evolution operators. Across these applications, our method demonstrates efficient optimization, enhanced expressibility, and the ability to construct shallower circuits compared to existing techniques. Moreover, the method can be generalized to optimize particle-number-conserving gates, which are particularly relevant for quantum chemistry. Leveraging this capability, we further demonstrate that the method achieves superior quantum compilation of molecular time-evolution operators by approximating them with shallower circuits than standard Trotter decomposition.
We propose two-gate extensions of the sequential single-qubit optimizers, Free Axis Selection (Fraxis) and Free Quaternion Selection (FQS), termed Two-Gate Fraxis (TGF) and Two-Gate FQS (TGFQS), respectively. In contrast to Fraxis and FQS, which update one single-qubit gate at a time via a quadratic local cost function and matrix diagonalization, TGF and TGFQS optimize two parameterized single-qubit gates simultaneously by constructing an exact quartic local cost function and optimizing it using classical optimizers. We further investigate how different gate pairing strategies affect optimization performance. Using numerical experiments on lattice Hamiltonians, molecular Hamiltonians, and quantum state preparation tasks, we find that TGF and TGFQS frequently achieve a lower final relative error to the ground state energy or infidelity than their single-gate counterparts. We observe that the random and half-shifted gate pairing strategies for TGF and TGFQS perform best across many of the tested settings. In the additional finite-shot experiments on Fermi–Hubbard and transverse-field Ising model Hamiltonians, the best gate pairing strategies retain their advantage across the tested shot counts in shallow circuits. These improvements come at the cost of increased circuit evaluations per gate update, highlighting a trade-off between the power of local optimization and measurement overhead.
We propose two-gate extensions of the sequential single-qubit optimizers, Free Axis Selection (Fraxis) and Free Quaternion Selection (FQS), termed Two-Gate Fraxis (TGF) and Two-Gate FQS (TGFQS), respectively. In contrast to Fraxis and FQS, which update one single-qubit gate at a time via quadratic local cost function and matrix diagonalization, TGF and TGFQS optimize two parameterized single-qubit gates simultaneously by constructing an exact quartic local cost function and optimizing it using classical optimizers. We further investigate how different gate pairing strategies affect optimization performance. Using numerical experiments on spin Hamiltonians, molecular Hamiltonians, and quantum state preparation tasks, we find that TGF and TGFQS frequently achieve a lower final relative error to the ground state energy or infidelity than their single gate counterparts. We observe that the random and half-shifted gate pairing strategies for TGF and TGFQS perform best in many of the tested settings. In the additional finite-shot experiments on Fermi-Hubbard and transverse-field Ising model Hamiltonians, the best gate pairing strategies retain their advantage across the tested shot counts in shallow circuits. These improvements come at the cost of increased circuit evaluations per gate update, highlighting a trade-off between the power of local optimization and measurement overhead.
Parameterized quantum circuits (PQCs) play an essential role in the application of variational quantum algorithms (VQAs) in noisy intermediate-scale quantum (NISQ) devices. The PQCs are a leading candidate to achieve a quantum advantage in NISQ devices and have already been applied in various domains such as quantum chemistry, quantum machine learning, combinatorial optimization, and many others. There is no single definitive way to optimize PQCs. The most commonly used methods are based on computing the gradient via the parameter-shift rule to use classical gradient descent (GD) optimizers like Adam, stochastic GD, and others. In addition, sequential single-qubit optimizers have been proposed, such as Rotosolve, Free-Axis Selection (Fraxis), Free-Quaternion Selection (FQS), and hybrid algorithms from the aforementioned optimizers. We further develop hybrid algorithms than those represented in the previous work by drawing inspiration from the early stopping method used in classical machine learning. The switch between the optimizers depends on the previous cost function values compared to the previous ones. We introduce two new hybrid algorithms that are more robust and scalable, and they outperform previous hybrid methods in terms of convergence towards the global minima across various cost functions. In addition, we find that they are feasible for NISQ devices with different noise profiles.
Variational quantum circuits (VQCs) are an essential tool in applying noisy intermediate-scale quantum computers to practical problems. VQCs are used as a central component in many algorithms, for example, in quantum machine learning, optimization, and quantum chemistry. Several methods have been developed to optimize VQCs. In this work, we enhance the performance of the well-known Rotosolve method, a gradient-free optimization algorithm specifically designed for VQCs. We develop two hybrid algorithms that combine an improved version of Rotosolve with the free quaternion selection (FQS) algorithm, which is the main focus of this study. Through numerical simulations, we observe that these hybrid algorithms achieve higher accuracy and better average performance across different ansatz circuit sizes and cost functions. For shallow variational circuits, we identify a trade-off between the expressivity of the variational ansatz and the speed of convergence to the optimum: a more expressive ansatz ultimately reaches a closer approximation to the true minimum, but at the cost of requiring more circuit evaluations for convergence. By combining the less expressive but fast-converging Rotosolve with the more expressive FQS, we construct hybrid algorithms that benefit from the rapid initial convergence of Rotosolve while leveraging the superior expressivity of FQS. As a result, these hybrid approaches outperform either method used independently.
Variational quantum algorithms (VQAs) were once believed to be the fastest route to demonstrating practical quantum advantage. VQAs use a parameterized quantum circuit to perform a machine-learning task. For example, VQAs can employ the Rayleigh–Ritz method to estimate a molecule's eigenenergies. While VQAs are already used to bootstrap quantum processors, their utility for quantum chemistry tasks has been questioned. The three main concerns are runtime, noise on near-term devices, and optimizability. This thesis presents data to corroborate these concerns before making progress on addressing all three. First, I present numerical runtime estimates and quantify the noise-induced errors for a class of VQAs that is believed to be readily optimizable: adaptive derivative-assembled pseudo-Trotter variational quantum eigensolvers (ADAPT-VQEs). These initial results demonstrate that VQAs will only become feasible when device noise levels fall below typical fault-tolerance thresholds. Second, to overcome these concerns, I replace the gate-based quantum circuit approach with a Hamiltonian-control approach. This necessitates a device model. Herein, I focus on quantum-dot spin-qubit devices with electron and hole spins. To perform a study of Hamiltonian control on these processors, I developed a performant software suite that includes an integrator and automatic differentiation. Using this software, I estimate the minimal evolution times for state preparation. This demonstrates Hamiltonian-level VQAs can run up to 100 times faster than the fastest gate-based VQAs. Next, I numerically demonstrate that Hamiltonian-level VQAs are robust to device imperfections and leakage, and up to 10⁵ times less susceptible to decoherence than the state-of-the-art VQAs. Finally, I consider the scalability of Hamiltonian-level VQAs in four ways: (i) Increasing the expressibility of the device Hamiltonian by developing a technique for adding terms virtually. (ii) Demonstrating, via Lie algebraic methods, that the microwave antenna, used for single-qubit control via electron spin resonance, is a redundant component. (Similarly, electric-dipole–spin-resonance controls can be removed.) (iii) Calibrating modules with Hamiltonian-control techniques to enhance scalability. (iv) Designing an adaptive control technique to limit the ansatz parameter count. This thesis restores much of the lost belief in variational quantum algorithms.
We propose an algorithm for variational quantum algorithms (VQAs) to optimize the structure of parameterized quantum circuits (PQCs) efficiently. The algorithm optimizes the PQC structure on-the-fly in VQA by sequentially replacing a single-qubit gate with the optimal one to minimize the objective function. To directly find the optimal gate, our method uses the factorization of matrices whose elements are evaluated on a set of the slightly-modified circuits. The matrix factorization enables us to not only unify the existing sequential methods for further extension but also provide rigorous proofs of their limitation and potential in comparison with conventional gradient-based optimizers. Firstly, when the circuits are sufficiently deep, the sequential methods encounter a barren plateau that the spectrum of the matrix concentrates on a single value exponentially fast with respect to the number of qubits. Secondly, if the objective functions are local observables, they can avoid barren plateaus as long as the depth of the n-qubit PQCs is O(logn). Although the family of these optimizers does not directly employ gradients of the objective function, our results establish their connection with conventional optimizations providing a consistent picture of the barren plateau. We also perform numerical experiments showing the advantages over conventional VQAs and confirm the successful optimization getting over the barren plateau in the ground state problem of the mixed field Ising model up to 12 qubits.