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Register a new userBenchmarking variational quantum eigensolvers for the square-octagon-lattice Kitaev model
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Quantum spin systems may offer the first opportunities for beyond-classical quantum computations of scientific interest. While general quantum simulation algorithms likely require error-corrected qubits, there may be applications of scientific interest prior to the practical implementation of quantum error correction. The variational quantum eigensolver (VQE) is a promising approach to find energy eigenvalues on noisy quantum computers. Lattice models are of broad interest for use on near-term quantum hardware due to the sparsity of the number of Hamiltonian terms and the possibility of matching the lattice geometry to the hardware geometry. Here, we consider the Kitaev spin model on a hardware-native square-octagon qubit connectivity map, and examine the possibility of efficiently probing its rich phase diagram with VQE approaches. By benchmarking different choices of variational ansatz states and classical optimizers, we illustrate the advantage of a mixed optimization approach using the Hamiltonian variational ansatz (HVA). We further demonstrate the implementation of an HVA circuit on Rigettis Aspen-9 chip with error mitigation.
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By design, the variational quantum eigensolver (VQE) strives to recover the lowest-energy eigenvalue of a given Hamiltonian by preparing quantum states guided by the variational principle. In practice, the prepared quantum state is indirectly assessed by the value of the associated energy. Novel adaptive derivative-assembled pseudo-trotter (ADAPT) ansatz approaches and recent formal advances now establish a clear connection between the theory of quantum chemistry and the quantum state ansatz used to solve the electronic structure problem. Here we benchmark the accuracy of VQE and ADAPT-VQE to calculate the electronic ground states and potential energy curves for a few selected diatomic molecules, namely H$_2$, NaH, and KH. Using numerical simulation, we find both methods provide good estimates of the energy and ground state, but only ADAPT-VQE proves to be robust to particularities in optimization methods. Another relevant finding is that gradient-based optimization is overall more economical and delivers superior performance than analogous simulations carried out with gradient-free optimizers. The results also identify small errors in the prepared state fidelity which show an increasing trend with molecular size.
Variational quantum eigensolver (VQE) optimizes parameterized eigenstates of a Hamiltonian on a quantum processor by updating parameters with a classical computer. Such a hybrid quantum-classical optimization serves as a practical way to leverage up classical algorithms to exploit the power of near-term quantum computing. Here, we develop a hybrid algorithm for VQE, emphasizing the classical side, that can solve a group of related Hamiltonians simultaneously. The algorithm incorporates a snake algorithm into many VQE tasks to collectively optimize variational parameters of different Hamiltonians. Such so-called collective VQEs~(cVQEs) is applied for solving molecules with varied bond length, which is a standard problem in quantum chemistry. Numeral simulations show that cVQE is not only more efficient in convergence, but also trends to avoid single VQE task to be trapped in local minimums. The collective optimization utilizes intrinsic relations between related tasks and may inspire advanced hybrid quantum-classical algorithms for solving practical problems.
Variational quantum eigensolver(VQE) typically minimizes energy with hybrid quantum-classical optimization, which aims to find the ground state. Here, we propose a VQE by minimizing energy variance, which is called as variance-VQE(VVQE). The VVQE can be viewed as an self-verifying eigensolver for arbitrary eigenstate by designing, since an eigenstate for a Hamiltonian should have zero energy variance. We demonstrate properties and advantages of VVQE for solving a set of excited states with quantum chemistry problems. Remarkably, we show that optimization of a combination of energy and variance may be more efficient to find low-energy excited states than those of minimizing energy or variance alone. We further reveal that the optimization can be boosted with stochastic gradient descent by Hamiltonian sampling, which uses only a few terms of the Hamiltonian and thus significantly reduces the quantum resource for evaluating variance and its gradients.
The discovery of superconductivity in twisted bilayer graphene has triggered a resurgence of interest in flat-band superconductivity. Here, we investigate the square-octagon lattice, which also exhibits two perfectly flat bands when next-nearest neighbour hopping or an external magnetic field are added to the system. We calculate the superconducting phase diagram in the presence of on-site attractive interactions and find two superconducting domes, as observed in several types of unconventional superconductors. The critical temperature shows a linear dependence on the coupling constant, suggesting that superconductivity might reach high temperatures in the square-octagon lattice. Our model could be experimentally realized using photonic or ultracold atoms lattices.
Quantum-enhanced computing methods are promising candidates to solve currently intractable problems. We consider here a variational quantum eigensolver (VQE), that delegates costly state preparations and measurements to quantum hardware, while classical optimization techniques guide the quantum hardware to create a desired target state. In this work, we propose a bosonic VQE using superconducting microwave cavities, overcoming the typical restriction of a small Hilbert space when the VQE is qubit based. The considered platform allows for strong nonlinearities between photon modes, which are highly customisable and can be tuned in situ, i.e. during running experiments. Our proposal hence allows for the realization of a wide range of bosonic ansatz states, and is therefore especially useful when simulating models involving degrees of freedom that cannot be simply mapped to qubits, such as gauge theories, that include components which require infinite-dimensional Hilbert spaces. We thus propose to experimentally apply this bosonic VQE to the U(1) Higgs model including a topological term, which in general introduces a sign problem in the model, making it intractable with conventional Monte Carlo methods.
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