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.
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 eigensolver (VQE), a kind of quantum-classical hybrid method, has latent powers to leverage noisy quantum devices by effectively managing a limited number of qubits with finite coherent times. We implemented VQE with real quantum processors through IBM Quantum Experience and applied it to combinatorial optimizations. Notably, a short-depth quantum circuit consisting of a lower number of quantum gate than that in the quantum approximate optimization algorithm can search the ground states of diagonal Hamiltonians encoding the problems. Since the simplest case of the quantum circuit for VQE requires only single-qubit gates, it can be realized using current quantum computers.
The variational quantum eigensolver (VQE), which is a quantum–classical hybrid approach, has latent powers to leverage near-term quantum devices by effectively managing a limited number of qubits with finite coherent lifetimes. While it is generally argued that the quantum approximate optimization algorithm (QAOA), which is a special case of VQE with a variational Ansatz based on the adiabatic theorem, may enable practical applications of noisy quantum devices for classical combinatorial optimizations, the strategy to improve the performance of this algorithm by increasing the circuit depth conflicts with the limited coherence time of near-term quantum devices. Here, we introduce strategies involving the VQE to reduce the circuit resources required for solving combinatorial optimizations. Our concept of a parameterized quantum circuit allows the Ansatz preparation to be achieved by only single-qubit operation. We find that the variational Ansatz without controlled X-gates leads to quick convergence in a classical subroutine used to determine the variational parameters. In addition, the variational Ansatz with optimized parameters maintains performance over the problem sizes both on the numerical simulation and IBM 27-qubit processor “ibm_kawasaki.” Therefore, the variational Ansatz introduced in this study has several advantages considering the total calculation time and performance scaling over the problem sizes. We also show that the variational Ansatz consisting of a lower number of gate operations than that of QAOA can approximate the eigenstates of diagonal Hamiltonians with high accuracy. We illustrate our ideas with a maximum-cut problem and show that near-term quantum applications may be feasible using short-depth circuits.
In recent years, as datasets have grown larger, quantum computing has shown potential in addressing the rapidly increasing costs of data processing due to its superposition and entanglement capabilities. Limited by quantum hardware and the number of controllable qubits, variational quantum algorithms (VQAs) are algorithms suitable for NISQ devices, capable of computation on medium-scale noisy devices. VQAs are hybrid classical-quantum algorithms that use parameterized quantum circuits to prepare parameterized quantum states and employ classical optimizers to minimize objective functions, enabling efficient solutions for complex problems. Classical optimizers reduce the stringent requirements on quantum hardware and can achieve quantum advantage on current devices. This paper systematically elaborates on the core foundational theories of quantum computing and neural networks, with a focus on reviewing the research and application status of VQAs in three main scenarios: machine learning (supervised, unsupervised, and reinforcement learning), solving systems of equations (linear and nonlinear), and network attack detection. It analyzes the current technical bottlenecks in this field and provides prospects for future development, offering a reference for further research on 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.
The promise of quantum computing to address complex problems requiring high computational resources has long been hindered by the intrinsic and demanding requirements of quantum hardware development. Nonetheless, the current state of quantum computing, denominated Noisy Intermediate-Scale Quantum (NISQ) era, has introduced algorithms and methods that are able to harness the computational power of current quantum computers with advantages over classical computers (referred to as quantum advantage). Achieving quantum advantage is of particular relevance for the combinatorial optimization domain, since it often implies solving an NP-Hard optimization problem. Moreover, combinatorial problems are highly relevant for practical application areas, such as operations research, or resource allocation problems. Among quantum computing methods, Variational Quantum Algorithms (VQA) have emerged as one of the strongest candidates towards reaching practical applicability of NISQ systems. This paper explores the current state and recent developments of VQAs, emphasizing their applicability to combinatorial optimization. We identify the Quantum Approximate Optimization Algorithm (QAOA) as the leading candidate for these problems. Furthermore, we implement QAOA circuits with varying depths to solve the MaxCut problem on graphs with 10 and 20 nodes, demonstrating the potential and challenges of using VQAs in practical optimization tasks. We release our code, dataset and optimized circuit parameters under https://github.com/DanielFPerez/VQA-for-MaxCut.