Variational Quantum Algorithms (VQAs) are expected to be promising algorithms with quantum advantages that can be run at quantum computers in the close future. In this work, we review simple rules in basic quantum circuits, and propose a simplification method, Measurement Simplification, that simplifies the expression for the measurement of quantum circuit. By the Measurement Simplification, we simplified the specific result expression of VQAs and obtained large improvements in calculation time and required memory size. Here we applied Measurement Simplification to Variational Quantum Linear Solver (VQLS), Variational Quantum Eigensolver (VQE) and other Quantum Machine Learning Algorithms to show an example of speedup in the calculation time and required memory size.
Variational Quantum Algorithms (VQAs) are expected to be promising algorithms with quantum advantages that can be run at quantum computers in the close future. In this work, we review simple rules in basic quantum circuits, and propose a simplification method, Measurement Simplification, that simplifies the expression for the measurement of quantum circuit. By the Measurement Simplification, we simplified the specific result expression of VQAs and obtained large improvements in calculation time and required memory size. Here we applied Measurement Simplification to Variational Quantum Linear Solver (VQLS), Variational Quantum Eigensolver (VQE) and other Quantum Machine Learning Algorithms to show an example of speedup in the calculation time and required memory size.
Variational quantum algorithms (VQAs) provide a promising approach to achieve quantum advantage in the noisy intermediate-scale quantum era. In this era, quantum computers experience high error rates and quantum error detection and correction is not feasible. VQAs can utilize noisy qubits in tandem with classical optimization algorithms to solve hard problems. However, VQAs are still slow relative to their classical counterparts. Hence, improving the performance of VQAs will be necessary to make them competitive. While VQAs are expected perform better as the problem sizes increase, increasing their performance will make them a viable option sooner. In this work we show that circuit-level concurrency provides a means to increase the performance of variational quantum algorithms on noisy quantum computers. This involves mapping multiple instances of the same circuit (program) onto the quantum computer at the same time, which allows multiple samples in a variational quantum algorithm to be gathered in parallel for each training iteration. We demonstrate that this technique provides a linear increase in training speed when increasing the number of concurrently running quantum circuits. Furthermore, even with pessimistic error rates concurrent quantum circuit sampling can speed up the quantum approximate optimization algorithm by up to 20x with low mapping and run time overhead.
The variational quantum algorithm (VQA) is a hybrid classical-quantum algorithm. It can actually run in an intermediate-scale quantum device where the number of available qubits is too limited to perform quantum error correction, so it is one of the most promising quantum algorithms in the noisy intermediate-scale quantum era. In this paper, two ideas for solving the learning with errors problem (LWE) using VQA are proposed. First, after reducing the LWE problem into the bounded distance decoding problem, the quantum approximation optimization algorithm (QAOA) is introduced to improve classical methods. Second, after the LWE problem is reduced into the unique shortest vector problem, the variational quantum eigensolver (VQE) is used to solve it, and the number of qubits required is calculated in detail. Small-scale experiments are carried out for the two LWE variational quantum algorithms, and the experiments show that VQA improves the quality of the classical solutions.
The superiority of variational quantum algorithms (VQAs) such as quantum neural networks (QNNs) and variational quantum eigensolvers (VQEs) heavily depends on the expressivity of the employed Ansätze. Namely, a simple Ansatz is insufficient to capture the optimal solution, while an intricate Ansatz leads to the hardness of trainability. Despite its fundamental importance, an effective strategy of measuring the expressivity of VQAs remains largely unknown. Here, we exploit an advanced tool in statistical learning theory, i.e., covering number, to study the expressivity of VQAs. Particularly, we first exhibit how the expressivity of VQAs with an arbitrary Ansätze is upper bounded by the number of quantum gates and the measurement observable. We next explore the expressivity of VQAs on near-term quantum chips, where the system noise is considered. We observe an exponential decay of the expressivity with increasing circuit depth. We also utilize the achieved expressivity to analyze the generalization of QNNs and the accuracy of VQE. We numerically verify our theory employing VQAs with different levels of expressivity. Our Letter opens the avenue for quantitative understanding of the expressivity of VQAs.
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 (VQAs) have emerged as promising methods for tackling complex problems on near-term quantum devices. Among these algorithms, the Variational Quantum Linear Solver (VQLS) addresses linear systems of the form Ax=b, aiming to prepare a quantum state ∣x⟩ such that A∣x⟩ is proportional to the quantum state corresponding to b. A key advantage of VQLS is its use of amplitude encoding, which requires a number of qubits that scales logarithmically with the linear system size. However, the existing literature has primarily focused on linear systems of limited size or with a specific structure. In this study, we extend the application of VQLS to more general and larger problem instances, including problems where state preparation is non-trivial and problems within the real domain of fluid dynamics. Our investigation reveals some critical challenges inherent to VQLS, including the need for a sufficiently expressive ansatz, the large number of circuit executions required to estimate the cost function, and the high gate count in the circuits in the most general setting. Our analysis highlights the obstacles that need to be addressed for a broader application of VQLS and concludes that further research is necessary to fully leverage the algorithm's capabilities in addressing real-world problems.
The variational quantum eigensolver (or VQE), first developed by Peruzzo et al. (2014), has received significant attention from the research community in recent years. It uses the variational principle to compute the ground state energy of a Hamiltonian, a problem that is central to quantum chemistry and condensed matter physics. Conventional computing methods are constrained in their accuracy due to the computational limits facing exact modeling of the exponentially growing electronic wavefunction for these many-electron systems. The VQE may be used to model these complex wavefunctions in polynomial time, making it one of the most promising near-term applications for quantum computing. One important advantage is that variational algorithms have been shown to present some degree of resilience to the noise in the quantum hardware. Finding a path to navigate the relevant literature has rapidly become an overwhelming task, with many methods promising to improve different parts of the algorithm, but without clear descriptions of how the diverse parts fit together. The potential practical advantages of the algorithm are also widely discussed in the literature, but with varying conclusions. Despite strong theoretical underpinnings suggesting excellent scaling of individual VQE components, studies have pointed out that their various pre-factors could be too large to reach a quantum computing advantage over conventional methods. This review aims at disentangling the relevant literature to provide a comprehensive overview of the progress that has been made on the different parts of the algorithm, and to discuss future areas of research that are fundamental for the VQE to deliver on its promises. All the different components of the algorithm are reviewed in detail. These include the representation of Hamiltonians and wavefunctions on a quantum computer, the optimization process to find ground state energies, the post processing mitigation of quantum errors, and suggested best practices. We identify four main areas of future research: (1) optimal measurement schemes for reduction of circuit repetitions required; (2) large scale parallelization across many quantum computers; (3) ways to overcome the potential appearance of vanishing gradients in the optimization process for large systems, and how the number of iterations required for the optimization scales with system size; (4) the extent to which VQE suffers for quantum noise, and whether this noise can be mitigated in a tractable manner. The answers to these open research questions will determine the routes for the VQE to achieve quantum advantage as the quantum computing hardware scales up and as the noise levels are reduced.