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1. Quantum algorithms for escaping from saddle points

   https://doi.org/10.22331/q-2021-08-20-529  

   Zhang, Chenyi; Leng, Jiaqi; Li, Tongyang (August 2021, Quantum)

   null (Ed.)

   We initiate the study of quantum algorithms for escaping from saddle points with provable guarantee. Given a function f : R n → R , our quantum algorithm outputs an ϵ -approximate second-order stationary point using O ~ ( log 2 ⁡ ( n ) / ϵ 1.75 ) queries to the quantum evaluation oracle (i.e., the zeroth-order oracle). Compared to the classical state-of-the-art algorithm by Jin et al. with O ~ ( log 6 ⁡ ( n ) / ϵ 1.75 ) queries to the gradient oracle (i.e., the first-order oracle), our quantum algorithm is polynomially better in terms of log ⁡ n and matches its complexity in terms of 1 / ϵ . Technically, our main contribution is the idea of replacing the classical perturbations in gradient descent methods by simulating quantum wave equations, which constitutes the improvement in the quantum query complexity with log ⁡ n factors for escaping from saddle points. We also show how to use a quantum gradient computation algorithm due to Jordan to replace the classical gradient queries by quantum evaluation queries with the same complexity. Finally, we also perform numerical experiments that support our theoretical findings. 

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2. An adaptive variational algorithm for exact molecular simulations on a quantum computer

   https://doi.org/10.1038/s41467-019-10988-2  

   Grimsley, Harper R.; Economou, Sophia E.; Barnes, Edwin; Mayhall, Nicholas J. (July 2019, Nature Communications)

   Abstract Quantum simulation of chemical systems is one of the most promising near-term applications of quantum computers. The variational quantum eigensolver, a leading algorithm for molecular simulations on quantum hardware, has a serious limitation in that it typically relies on a pre-selected wavefunction ansatz that results in approximate wavefunctions and energies. Here we present an arbitrarily accurate variational algorithm that, instead of fixing an ansatz upfront, grows it systematically one operator at a time in a way dictated by the molecule being simulated. This generates an ansatz with a small number of parameters, leading to shallow-depth circuits. We present numerical simulations, including for a prototypical strongly correlated molecule, which show that our algorithm performs much better than a unitary coupled cluster approach, in terms of both circuit depth and chemical accuracy. Our results highlight the potential of our adaptive algorithm for exact simulations with present-day and near-term quantum hardware. 

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3. Emulating the Deutsch-Josza algorithm with an inverse-designed terahertz gradient-index lens

   https://doi.org/10.1364/OE.495919  

   Blackwell, Ashley N.; Yahiaoui, Riad; Chen, Yi-Huan; Chen, Pai-Yen; Searles, Thomas A.; Chase, Zizwe A. (January 2023, Optics Express)

   An all-dielectric photonic metastructure is investigated for application as a quantum algorithm emulator (QAE) in the terahertz frequency regime; specifically, we show implementation of the Deustsh-Josza algorithm. The design for the QAE consists of a gradient-index (GRIN) lens as the Fourier transform subblock and patterned silicon as the oracle subblock. First, we detail optimization of the GRIN lens through numerical analysis. Then, we employed inverse design through a machine learning approach to further optimize the structural geometry. Through this optimization, we enhance the interaction of the incident light with the metamaterial via spectral improvements of the outgoing wave. 

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4. Simultaneous estimation of multiple eigenvalues with short-depth quantum circuit on early fault-tolerant quantum computers

   https://doi.org/10.22331/q-2023-10-11-1136  

   Ding, Zhiyan; Lin, Lin (October 2023, Quantum)

   We introduce a multi-modal, multi-level quantum complex exponential least squares (MM-QCELS) method to simultaneously estimate multiple eigenvalues of a quantum Hamiltonian on early fault-tolerant quantum computers. Our theoretical analysis demonstrates that the algorithm exhibits Heisenberg-limited scaling in terms of circuit depth and total cost. Notably, the proposed quantum circuit utilizes just one ancilla qubit, and with appropriate initial state conditions, it achieves significantly shorter circuit depths compared to circuits based on quantum phase estimation (QPE). Numerical results suggest that compared to QPE, the circuit depth can be reduced by around two orders of magnitude under several settings for estimating ground-state and excited-state energies of certain quantum systems. 

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5. Efficient quantum algorithm for dissipative nonlinear differential equations

   https://doi.org/10.1073/pnas.2026805118  

   Liu, Jin-Peng; Kolden, Herman Øie; Krovi, Hari K.; Loureiro, Nuno F.; Trivisa, Konstantina; Childs, Andrew M. (August 2021, Proceedings of the National Academy of Sciences)

   null (Ed.)

   Nonlinear differential equations model diverse phenomena but are notoriously difficult to solve. While there has been extensive previous work on efficient quantum algorithms for linear differential equations, the linearity of quantum mechanics has limited analogous progress for the nonlinear case. Despite this obstacle, we develop a quantum algorithm for dissipative quadratic n-dimensional ordinary differential equations. Assuming R < 1 , where R is a parameter characterizing the ratio of the nonlinearity and forcing to the linear dissipation, this algorithm has complexity T 2 q   poly ( log ⁡ T , log ⁡ n , log ⁡ 1 / ϵ ) / ϵ , where T is the evolution time, ϵ is the allowed error, and q measures decay of the solution. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in T. While exponential decay precludes efficiency, driven equations can avoid this issue despite the presence of dissipation. Our algorithm uses the method of Carleman linearization, for which we give a convergence theorem. This method maps a system of nonlinear differential equations to an infinite-dimensional system of linear differential equations, which we discretize, truncate, and solve using the forward Euler method and the quantum linear system algorithm. We also provide a lower bound on the worst-case complexity of quantum algorithms for general quadratic differential equations, showing that the problem is intractable for R ≥ 2 . Finally, we discuss potential applications, showing that the R < 1 condition can be satisfied in realistic epidemiological models and giving numerical evidence that the method may describe a model of fluid dynamics even for larger values of R. 

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