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The variational quantum eigensolver (VQE) is a promising algorithm to compute eigenstates and eigenenergies of a given quantum system that can be performed on a near-term quantum computer. Obtaining eigenstates and eigenenergies in a specific symmetry sector of the system is often necessary for practical applications of the VQE in various fields ranging from high energy physics to quantum chemistry. It is common to add a penalty term in the cost function of the VQE to calculate such a symmetry-resolving energy spectrum, but systematic analysis on the effect of the penalty term has been lacking, and the use of the penalty term in the VQE has not been justified rigorously. In this work, we investigate two major types of penalty terms for the VQE that were proposed in the previous studies. We show a penalty term in one of the two types works properly in that eigenstates obtained by the VQE with the penalty term reside in the desired symmetry sector. We further give a convenient formula to determine the magnitude of the penalty term, which may lead to the faster convergence of the VQE. Meanwhile, we prove that the other type of penalty terms does not work for obtaining the target state with the desired symmetry in a rigorous sense and even gives completely wrong results in some cases. We finally provide numerical simulations to validate our analysis. Our results apply to general quantum systems and lay the theoretical foundation for the use of the VQE with the penalty terms to obtain the symmetry-resolving energy spectrum of the system, which fuels the application of a near-term quantum computer.
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Recent practical approaches for the use of current generation noisy quantum devices in the simulation of quantum many-body problems have been dominated by the use of a variational quantum eigensolver (VQE). These coupled quantum-classical algorithms leverage the ability to perform many repeated measurements to avoid the currently prohibitive gate depths often required for exact quantum algorithms, with the restriction of a parameterized circuit to describe the states of interest. In this work, we show how the calculation of zero-temperature dynamic correlation functions defining the linear response characteristics of quantum systems can also be recast into a modified VQE algorithm, which can be incorporated into the current variational quantum infrastructure. This allows for these important physical expectation values describing the dynamics of the system to be directly converged on the frequency axis, and they approach exactness over all frequencies as the flexibility of the parameterization increases. The frequency resolution hence does not explicitly scale with gate depth, which is approximately twice as deep as a ground state VQE. We apply the method to compute the single-particle Greens function of ab initio dihydrogen and lithium hydride molecules, and demonstrate the use of a practical active space embedding approach to extend to larger systems. While currently limited by the fidelity of two-qubit gates, whose number is increased compared to the ground state algorithm on current devices, we believe the approach shows potential for the extraction of frequency dynamics of correlated systems on near-term quantum processors.
The variational quantum eigensolver (VQE) is one of the most representative quantum algorithms in the noisy intermediate-size quantum (NISQ) era, and is generally speculated to deliver one of the first quantum advantages for the ground-state simulations of some non-trivial Hamiltonians. However, short quantum coherence time and limited availability of quantum hardware resources in the NISQ hardware strongly restrain the capacity and expressiveness of VQEs. In this Letter, we introduce the variational quantum-neural hybrid eigensolver (VQNHE) in which the shallow-circuit quantum ansatz can be further enhanced by classical post-processing with neural networks. We show that VQNHE consistently and significantly outperforms VQE in simulating ground-state energies of quantum spins and molecules given the same amount of quantum resources. More importantly, we demonstrate that for arbitrary post-processing neural functions, VQNHE only incurs an polynomial overhead of processing time and represents the first scalable method to exponentially accelerate VQE with non-unitary post-processing that can be efficiently implemented in the NISQ era.
Establishing the nature of the ground state of the Heisenberg antiferromagnet (HAFM) on the kagome lattice is well known to be a prohibitively difficult problem for classical computers. Here, we give a detailed proposal for a Variational Quantum Eigensolver (VQE) with the aim of solving this physical problem on a quantum computer. At the same time, this VQE constitutes an explicit proposal for showing a useful quantum advantage on Noisy Intermediate-Scale Quantum (NISQ) devices because of its natural hardware compatibility. We classically emulate a noiseless quantum computer with the connectivity of a 2D square lattice and show how the ground state energy of a 20-site patch of the kagome HAFM, as found by the VQE, approaches the true ground state energy exponentially as a function of the circuit depth. Besides indicating the potential of quantum computers to solve for the ground state of the kagome HAFM, the classical emulation of the VQE serves as a benchmark for real quantum devices on the way towards a useful quantum advantage.
Hybrid quantum-classical algorithms have been proposed as a potentially viable application of quantum computers. A particular example - the variational quantum eigensolver, or VQE - is designed to determine a global minimum in an energy landscape specified by a quantum Hamiltonian, which makes it appealing for the needs of quantum chemistry. Experimental realizations have been reported in recent years and theoretical estimates of its efficiency are a subject of intense effort. Here we consider the performance of the VQE technique for a Hubbard-like model describing a one-dimensional chain of fermions with competing nearest- and next-nearest-neighbor interactions. We find that recovering the VQE solution allows one to obtain the correlation function of the ground state consistent with the exact result. We also study the barren plateau phenomenon for the Hamiltonian in question and find that the severity of this effect depends on the encoding of fermions to qubits. Our results are consistent with the current knowledge about the barren plateaus in quantum optimization.
Development of resource-friendly quantum algorithms remains highly desirable for noisy intermediate-scale quantum computing. Based on the variational quantum eigensolver (VQE) with unitary coupled cluster ansatz, we demonstrate that partitioning of the Hilbert space made possible by the point group symmetry of the molecular systems greatly reduces the number of variational operators by confining the variational search within a subspace. In addition, we found that instead of including all subterms for each excitation operator, a single-term representation suffices to reach required accuracy for various molecules tested, resulting in an additional shortening of the quantum circuit. With these strategies, VQE calculations on a noiseless quantum simulator achieve energies within a few meVs of those obtained with the full UCCSD ansatz for $mathrm{H}_4$ square, $mathrm{H}_4$ chain and $mathrm{H}_6$ hexagon molecules; while the number of controlled-NOT (CNOT) gates, a measure of the quantum-circuit depth, is reduced by a factor of as large as 35. Furthermore, we introduced an efficient score parameter to rank the excitation operators, so that the operators causing larger energy reduction can be applied first. Using $mathrm{H}_4$ square and $mathrm{H}_4$ chain as examples, We demonstrated on noisy quantum simulators that the first few variational operators can bring the energy within the chemical accuracy, while additional operators do not improve the energy since the accumulative noise outweighs the gain from the expansion of the variational ansatz.
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