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Register a new userAn efficient adaptive variational quantum solver of the Schrodinger equation based on reduced density matrices
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Recently, an adaptive variational algorithm termed Adaptive Derivative-Assembled Pseudo-Trotter ansatz Variational Quantum Eigensolver (ADAPT-VQE) has been proposed by Grimsley et al. (Nat. Commun. 10, 3007) while the number of measurements required to perform this algorithm scales O(N^8). In this work, we present an efficient adaptive variational quantum solver of the Schrodinger equation based on ADAPT-VQE together with the reduced density matrix reconstruction approach, which reduces the number of measurements from O(N^8) to O(N^4). This new algorithm is quite suitable for quantum simulations of chemical systems on near-term noisy intermediate-scale hardware due to low circuit complexity and reduced measurement. Numerical benchmark calculations for small molecules demonstrate that this new algorithm provides an accurate description of the ground-state potential energy curves. In addition, we generalize this new algorithm for excited states with the variational quantum deflation approach and achieve the same accuracy as ground-state simulations.
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By combining a parameterized Hermitian matrix, the realignment matrix of the bipartite density matrix $rho$ and the vectorization of its reduced density matrices, we present a family of separability criteria, which are stronger than the computable cross norm or realignment (CCNR) criterion. With linear contraction methods, the proposed criteria can be used to detect the multipartite entangled states that are biseparable under any bipartite partitions. Moreover, we show by examples that the presented multipartite separability criteria can be more efficient than the corresponding multipartite realignment criterion based on CCNR, multipartite correlation tensor criterion and multipartite covariance matrix criterion.
Previously proposed quantum algorithms for solving linear systems of equations cannot be implemented in the near term due to the required circuit depth. Here, we propose a hybrid quantum-classical algorithm, called Variational Quantum Linear Solver (VQLS), for solving linear systems on near-term quantum computers. VQLS seeks to variationally prepare $|xrangle$ such that $A|xranglepropto|brangle$. We derive an operationally meaningful termination condition for VQLS that allows one to guarantee that a desired solution precision $epsilon$ is achieved. Specifically, we prove that $C geq epsilon^2 / kappa^2$, where $C$ is the VQLS cost function and $kappa$ is the condition number of $A$. We present efficient quantum circuits to estimate $C$, while providing evidence for the classical hardness of its estimation. Using Rigettis quantum computer, we successfully implement VQLS up to a problem size of $1024times1024$. Finally, we numerically solve non-trivial problems of size up to $2^{50}times2^{50}$. For the specific examples that we consider, we heuristically find that the time complexity of VQLS scales efficiently in $epsilon$, $kappa$, and the system size $N$.
Recently a new class of quantum algorithms that are based on the quantum computation of the connected moment expansion has been reported to find the ground and excited state energies. In particular, the Peeters-Devreese-Soldatov (PDS) formulation is found variational and bearing the potential for further combining with the existing variational quantum infrastructure. Here we find that the PDS formulation can be considered as a new energy functional of which the PDS energy gradient can be employed in a conventional variational quantum solver. In comparison with the usual variational quantum eigensolver (VQE) and the original static PDS approach, this new variational quantum solver offers an effective approach to navigate the dynamics to be free from getting trapped in the local minima that refer to different states, and achieve high accuracy at finding the ground state and its energy through the rotation of the trial wave function of modest quality, thus improves the accuracy and efficiency of the quantum simulation. We demonstrate the performance of the proposed variational quantum solver for toy models, H$_2$ molecule, and strongly correlated planar H$_4$ system in some challenging situations. In all the case studies, the proposed variational quantum approach outperforms the usual VQE and static PDS calculations even at the lowest order. We also discuss the limitations of the proposed approach and its preliminary execution for model Hamiltonian on the NISQ device.
The superiority of variational quantum algorithms (VQAs) such as quantum neural networks (QNNs) and variational quantum eigen-solvers (VQEs) heavily depends on the expressivity of the employed ansatze. Namely, a simple ansatze is insufficient to capture the optimal solution, while an intricate ansatze leads to the hardness of the 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. In particular, we first exhibit how the expressivity of VQAs with an arbitrary ansatze 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 work opens the avenue for quantitative understanding of the expressivity of VQAs.
Variational quantum algorithms have found success in the NISQ era owing to their hybrid quantum-classical approach which mitigate the problems of noise in quantum computers. In our study we introduce the dynamic ansatz in the Variational Quantum Linear Solver for a system of linear algebraic equations. In this improved algorithm, the number of layers in the hardware efficient ansatz circuit is evolved, starting from a small and gradually increasing until convergence of the solution is reached. We demonstrate the algorithm advantage in comparison to the standard, static ansatz by utilizing fewer quantum resources and with a smaller quantum depth on average, in presence and absence of quantum noise, and in cases when the number of qubits or condition number of the system matrix are increased. The numbers of iterations and layers can be altered by a switching parameter. The performance of the algorithm in using quantum resources is quantified by a newly defined metric.
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