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.
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.
Fault-tolerant quantum computers, when realized, guarantee significant speedups for particular computational tasks. Quantum hardware in the near term, however, is limited to low hundreds of noisy qubits that are not error-corrected. To leverage such devices as well as accommodate rapid developments in both hardware and algorithms, variational quantum algorithms (VQAs) were introduced. These algorithms have the potential to be useful in areas such as chemistry, optimization, machine learning, among others. Despite this potential, the largest experiment to date has employed just twelve qubits to implement the Variational Quantum Eigensolver (VQE), a VQA for simulating ground state energies of fermionic systems. In this dissertation, we present several methods for extending and further developing VQAs to step closer towards realizing applications in which quantum computers can provide meaningful advantages over classical computers. In the first part, we present methods for analyzing and optimizing parameterized quantum circuits (PQCs), a core component in VQAs for approximating target quantum states, e.g. ground state wavefunctions in VQE. To compare among PQCs and guide their design and selection, we introduce a set of algorithm-independent descriptors that can be estimated from simulations. Using these descriptors, we rule out circuits with limited capabilities in expressive and entangling powers. Next, we present a strategy for optimizing PQCs with large parameter counts and high redundancy by removing circuit operations that are less important for optimizing the algorithm objective, effectively reducing the search space. We demonstrate that this method enables larger VQE simulations by converging previously difficult optimizations while additionally reducing circuit resources. In the second part, we present methods for developing VQAs that are related to the quantum hardware. We introduce a new algorithm for simulating components of a quantum processor using an existing quantum computer. The goal of this application is to leverage near-term devices and algorithms to aid in the design of more powerful next-generation quantum hardware. Lastly, we present a guide for algorithm prototyping by deploying instances of VQAs on quantum computers via the cloud. We demonstrate instances of two algorithms, the quantum autoencoder and classifier, on an eight-qubit device. While further advancements in hardware and algorithms lie ahead, our work, along with developments from previous studies, represents significant progress towards realizing practical quantum advantage.
Variational quantum algorithms (VQAs) are a modern family of quantum algorithms designed to solve optimization problems using a quantum computer. Typically VQAs rely on a feedback loop between the quantum device and a classical optimization algorithm. The appeal of VQAs lies in their versatility, resistance to noise, and ability to demonstrate some results even with circuits of small depth. We primarily focus on the algorithm called variational quantum eigensolver (VQE), which takes a qubit Hamiltonian and returns its approximate ground state. We first present our numerical findings regarding VQE applied to two spin models and a variant of the Hubbard model. Next, we briefly touch the topic of quantum machine learning by developing a quantum classifier to partition quantum data. We further study the phenomenon of vanishing derivatives in VQAs, also known as barren plateaus phenomenon. We derive a new lower bound on variance of the derivatives, which depends on the causal structure of the ansatz circuit and the individual terms entering the Pauli decomposition of the problem Hamiltonian. In the final chapter of the thesis, we present our results on bounding the fidelity of experimentally prepared Clifford states using their parent Hamiltonians.
Variational quantum algorithms (VQAs) have emerged as the leading strategy for using these noisy devices to achieve an advantage over their classical counterparts.VQAs use a hybrid quantum-classical framework, in which one employs the quantum computer to prepare a parameterized quantum state and outsource the optimization to a classical subroutine. VQAs have been proposed for applications in various fields, including physics, quantum chemistry, machine learning, and optimization. Nevertheless, many challenges must be addressed before we can use them for practical tasks. In this dissertation, we propose strategies to overcome some challenges in implementing VQAs. We first look at the issue of trainability for VQAs and propose three methods for optimizing them.The first method is based on natural evolutionary strategies (NES) and can alleviate the issue of vanishing gradients for randomly initialized parameterized quantum circuits (PQCs). The second method is based on the notion of information flow in PQCs and uses a mutual information-based distance metric to define paths for stochastic optimization of VQAs. Finally, we develop gradient evaluation procedures for optimizing objectives with the unitary coupled-cluster ansatz. Next, we study the issue of expressibility of PQCs used in variational algorithms.First, using the theory of quantum optimal control we define a notion of reachability of general ans\"atze used in VQAs. Next, using a variational autoencoder (VAE) coupled with a regression model, we try to gain a better understanding of the parameterization of PQCs. Finally, we attempt to remedy the inefficient usage of quantum resources for storing quantum data using evolutionary strategies to design quantum circuits for compressing quantum data. We present numerical evidence that the work presented here can increase trainability, efficiency, and understanding of VQAs.We believe our work is of interest to the quantum computing community as we continue to push the limits of NISQ devices for solving practical problems.
Currently developed quantum devices have the potential to show a computational advantage if quantum applications exploit the devices efficiently. Variational quantum algorithms (VQAs) combine a quantum processor with a classical computer to minimize the expectation value of a problem Hamiltonian. An important element of VQAs is the state-preparation circuit, referred to as variational quantum circuit. To improve the ability for solving combinatorial optimizations with VQAs, we show that the variational quantum circuit with single-qubit gates only can find the ground state more quickly. We demonstrate our VQA by using real quantum processors through IBM Quantum. As a result, the variational circuit with single-qubit gates takes four times smaller numbers of function evaluations than that with entangling gates, such as controlled-X gates.
Variational quantum algorithms (VQAs) offer the most promising path to obtaining quantum advantages via noisy intermediate-scale quantum (NISQ) processors. Such systems leverage classical optimization to tune the parameters of a parameterized quantum circuit (PQC). The goal is minimizing a cost function that depends on measurement outputs obtained from the PQC. Optimization is typically implemented via stochastic gradient descent (SGD). On NISQ computers, gate noise due to imperfections and decoherence affects the stochastic gradient estimates by introducing a bias. Quantum error mitigation (QEM) techniques can reduce the estimation bias without requiring any increase in the number of qubits, but they in turn cause an increase in the variance of the gradient estimates. This work studies the impact of quantum gate noise on the convergence of SGD for the variational eigensolver (VQE), a fundamental instance of VQAs. The main goal is ascertaining conditions under which QEM can enhance the performance of SGD for VQEs. It is shown that quantum gate noise induces a nonzero error-floor on the convergence error of SGD (evaluated with respect to a reference noiseless PQC), which depends on the number of noisy gates, the strength of the noise, as well as the eigenspectrum of the observable being measured and minimized. In contrast, with QEM, any arbitrarily small error can be obtained. Furthermore, for error levels attainable with or without QEM, QEM can reduce the number of required iterations, but only as long as the quantum noise level is sufficiently large, and a sufficiently large number of measurements is allowed at each SGD iteration. Numerical examples for a max-cut problem corroborate the main theoretical findings.
In recent years, Variational Quantum Algorithms (VQAs) have emerged as a promising approach for solving optimization problems on quantum computers in the NISQ era. However, one limitation of VQAs is their reliance on fixed-structure circuits, which may not be taylored for specific problems or hardware configurations. A leading strategy to address this issue are Adaptative VQAs, which dynamically modify the circuit structure by adding and removing gates, and optimize their parameters during the training. Several Adaptative VQAs, based on heuristics such as circuit shallowness, entanglement capability and hardware compatibility, have already been proposed in the literature, but there is still lack of a systematic comparison between the different methods. In this paper, we aim to fill this gap by analyzing three Adaptative VQAs: Evolutionary Variational Quantum Eigensolver (EVQE), Variable Ansatz (VAns), already proposed in the literature, and Random Adapt-VQE (RA-VQE), a random approach we introduce as a baseline. In order to compare these algorithms to traditional VQAs, we also include the Quantum Approximate Optimization Algorithm (QAOA) in our analysis. We apply these algorithms to QUBO problems and study their performance by examining the quality of the solutions found and the computational times required. Additionally, we investigate how the choice of the hyperparameters can impact the overall performance of the algorithms, highlighting the importance of selecting an appropriate methodology for hyperparameter tuning. Our analysis sets benchmarks for Adaptative VQAs designed for near-term quantum devices and provides valuable insights to guide future research in this area.