01837nas a2200169 4500008004100000245009700041210006900138260001300207520131000220100001301530700001301543700002001556700001801576700002001594700001601614856003701630 2022 eng d00aEfficient quantum algorithm for nonlinear reaction-diffusion equations and energy estimation0 aEfficient quantum algorithm for nonlinear reactiondiffusion equa c5/2/20223 a

Nonlinear differential equations exhibit rich phenomena in many fields but are notoriously challenging to solve. Recently, Liu et al. [1] demonstrated the first efficient quantum algorithm for dissipative quadratic differential equations under the condition R<1, where R measures the ratio of nonlinearity to dissipation using the ℓ2 norm. Here we develop an efficient quantum algorithm based on [1] for reaction-diffusion equations, a class of nonlinear partial differential equations (PDEs). To achieve this, we improve upon the Carleman linearization approach introduced in [1] to obtain a faster convergence rate under the condition RD<1, where RD measures the ratio of nonlinearity to dissipation using the ℓ∞ norm. Since RD is independent of the number of spatial grid points n while R increases with n, the criterion RD<1 is significantly milder than R<1 for high-dimensional systems and can stay convergent under grid refinement for approximating PDEs. As applications of our quantum algorithm we consider the Fisher-KPP and Allen-Cahn equations, which have interpretations in classical physics. In particular, we show how to estimate the mean square kinetic energy in the solution by postprocessing the quantum state that encodes it to extract derivative information.

1 aAn, Dong1 aFang, Di1 aJordan, Stephen1 aLiu, Jin-Peng1 aLow, Guang, Hao1 aWang, Jiasu uhttps://arxiv.org/abs/2205.0114102587nas a2200157 4500008004100000245011000041210006900151260001500220300001200235520207500247100001902322700001302341700002002354700001802374856003702392 2018 eng d00aQuantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics0 aQuantum singular value transformation and beyond exponential imp c2018/06/05 a193-2043 a

Quantum computing is powerful because unitary operators describing the time-evolution of a quantum system have exponential size in terms of the number of qubits present in the system. We develop a new "Singular value transformation" algorithm capable of harnessing this exponential advantage, that can apply polynomial transformations to the singular values of a block of a unitary, generalizing the optimal Hamiltonian simulation results of Low and Chuang. The proposed quantum circuits have a very simple structure, often give rise to optimal algorithms and have appealing constant factors, while usually only use a constant number of ancilla qubits. We show that singular value transformation leads to novel algorithms. We give an efficient solution to a certain "non-commutative" measurement problem and propose a new method for singular value estimation. We also show how to exponentially improve the complexity of implementing fractional queries to unitaries with a gapped spectrum. Finally, as a quantum machine learning application we show how to efficiently implement principal component regression. "Singular value transformation" is conceptually simple and efficient, and leads to a unified framework of quantum algorithms incorporating a variety of quantum speed-ups. We illustrate this by showing how it generalizes a number of prominent quantum algorithms, including: optimal Hamiltonian simulation, implementing the Moore-Penrose pseudoinverse with exponential precision, fixed-point amplitude amplification, robust oblivious amplitude amplification, fast QMA amplification, fast quantum OR lemma, certain quantum walk results and several quantum machine learning algorithms. In order to exploit the strengths of the presented method it is useful to know its limitations too, therefore we also prove a lower bound on the efficiency of singular value transformation, which often gives optimal bounds.

1 aGilyen, Andras1 aSu, Yuan1 aLow, Guang, Hao1 aWiebe, Nathan uhttps://arxiv.org/abs/1806.0183801826nas a2200169 4500008004100000245005800041210005800099260001500157490000700172520131900179100001901498700002401517700002001541700001701561700002401578856005401602 2017 eng d00aHamiltonian Simulation with Optimal Sample Complexity0 aHamiltonian Simulation with Optimal Sample Complexity c2017/03/310 v133 a
We investigate the sample complexity of Hamiltonian simulation: how many copies of an unknown quantum state are required to simulate a Hamiltonian encoded by the density matrix of that state? We show that the procedure proposed by Lloyd, Mohseni, and Rebentrost [Nat. Phys., 10(9):631--633, 2014] is optimal for this task. We further extend their method to the case of multiple input states, showing how to simulate any Hermitian polynomial of the states provided. As applications, we derive optimal algorithms for commutator simulation and orthogonality testing, and we give a protocol for creating a coherent superposition of pure states, when given sample access to those states. We also show that this sample-based Hamiltonian simulation can be used as the basis of a universal model of quantum computation that requires only partial swap operations and simple single-qubit states.
 
 
1 aKimmel, Shelby1 aLin, Cedric, Yen-Yu1 aLow, Guang, Hao1 aOzols, Maris1 aYoder, Theodore, J. uhttps://www.nature.com/articles/s41534-017-0013-701534nas a2200157 4500008004100000245007200041210006900113260001500182300001100197490000700208520106200215100001901277700002001296700002401316856003601340 2015 eng d00aRobust Single-Qubit Process Calibration via Robust Phase Estimation0 aRobust SingleQubit Process Calibration via Robust Phase Estimati c2015/12/08 a0623150 v923 a An important step in building a quantum computer is calibrating experimentally implemented quantum gates to produce operations that are close to ideal unitaries. The calibration step involves estimating the error in gates and then using controls to correct the implementation. Quantum process tomography is a standard technique for estimating these errors, but is both time consuming, (when one only wants to learn a few key parameters), and requires resources, like perfect state preparation and measurement, that might not be available. With the goal of efficiently estimating specific errors using minimal resources, we develop a parameter estimation technique, which can gauge two key parameters (amplitude and off-resonance errors) in a single-qubit gate with provable robustness and efficiency. In particular, our estimates achieve the optimal efficiency, Heisenberg scaling. Our main theorem making this possible is a robust version of the phase estimation procedure of Higgins et al. [B. L. Higgins, New J. Phys. 11, 073023 (2009)]. 1 aKimmel, Shelby1 aLow, Guang, Hao1 aYoder, Theodore, J. uhttp://arxiv.org/abs/1502.02677