188 research outputs found
Efficient Quantum Algorithms for Quantum Optimal Control
In this paper, we present efficient quantum algorithms that are exponentially
faster than classical algorithms for solving the quantum optimal control
problem. This problem involves finding the control variable that maximizes a
physical quantity at time , where the system is governed by a time-dependent
Schr\"odinger equation. This type of control problem also has an intricate
relation with machine learning. Our algorithms are based on a time-dependent
Hamiltonian simulation method and a fast gradient-estimation algorithm. We also
provide a comprehensive error analysis to quantify the total error from various
steps, such as the finite-dimensional representation of the control function,
the discretization of the Schr\"odinger equation, the numerical quadrature, and
optimization. Our quantum algorithms require fault-tolerant quantum computers.Comment: 17 pages, 2 figure
Simulating Markovian open quantum systems using higher-order series expansion
We present an efficient quantum algorithm for simulating the dynamics of
Markovian open quantum systems. The performance of our algorithm is similar to
the previous state-of-the-art quantum algorithm, i.e., it scales linearly in
evolution time and poly-logarithmically in inverse precision. However, our
algorithm is conceptually cleaner, and it only uses simple quantum primitives
without compressed encoding. Our approach is based on a novel mathematical
treatment of the evolution map, which involves a higher-order series expansion
based on Duhamel's principle and approximating multiple integrals using scaled
Gaussian quadrature. Our method easily generalizes to simulating quantum
dynamics with time-dependent Lindbladians. Furthermore, our method of
approximating multiple integrals using scaled Gaussian quadrature could
potentially be used to produce a more efficient approximation of time-ordered
integrals, and therefore can simplify existing quantum algorithms for
simulating time-dependent Hamiltonians based on a truncated Dyson series.Comment: 28 pages, various minor changes. To appear in the 50th EATCS
International Colloquium on Automata, Languages and Programming (ICALP 2023
Implementation of the Density-functional Theory on Quantum Computers with Linear Scaling with respect to the Number of Atoms
Density-functional theory (DFT) has revolutionized computer simulations in
chemistry and material science. A faithful implementation of the theory
requires self-consistent calculations. However, this effort involves repeatedly
diagonalizing the Hamiltonian, for which a classical algorithm typically
requires a computational complexity that scales cubically with respect to the
number of electrons. This limits DFT's applicability to large-scale problems
with complex chemical environments and microstructures. This article presents a
quantum algorithm that has a linear scaling with respect to the number of
atoms, which is much smaller than the number of electrons. Our algorithm
leverages the quantum singular value transformation (QSVT) to generate a
quantum circuit to encode the density-matrix, and an estimation method for
computing the output electron density. In addition, we present a randomized
block coordinate fixed-point method to accelerate the self-consistent field
calculations by reducing the number of components of the electron density that
needs to be estimated.
The proposed framework is accompanied by a rigorous error analysis that
quantifies the function approximation error, the statistical fluctuation, and
the iteration complexity. In particular, the analysis of our self-consistent
iterations takes into account the measurement noise from the quantum circuit.
These advancements offer a promising avenue for tackling large-scale DFT
problems, enabling simulations of complex systems that were previously
computationally infeasible
Sublinear classical and quantum algorithms for general matrix games
We investigate sublinear classical and quantum algorithms for matrix games, a
fundamental problem in optimization and machine learning, with provable
guarantees. Given a matrix , sublinear algorithms
for the matrix game
were previously known only for two special cases: (1) being the
-norm unit ball, and (2) being either the -
or the -norm unit ball. We give a sublinear classical algorithm that
can interpolate smoothly between these two cases: for any fixed ,
we solve the matrix game where is a -norm unit ball
within additive error in time . We
also provide a corresponding sublinear quantum algorithm that solves the same
task in time with a
quadratic improvement in both and . Both our classical and quantum
algorithms are optimal in the dimension parameters and up to
poly-logarithmic factors. Finally, we propose sublinear classical and quantum
algorithms for the approximate Carath\'eodory problem and the -margin
support vector machines as applications.Comment: 16 pages, 2 figures. To appear in the Thirty-Fifth AAAI Conference on
Artificial Intelligence (AAAI 2021
Quantum-Inspired Sublinear Algorithm for Solving Low-Rank Semidefinite Programming
Semidefinite programming (SDP) is a central topic in mathematical
optimization with extensive studies on its efficient solvers. In this paper, we
present a proof-of-principle sublinear-time algorithm for solving SDPs with
low-rank constraints; specifically, given an SDP with constraint matrices,
each of dimension and rank , our algorithm can compute any entry and
efficient descriptions of the spectral decomposition of the solution matrix.
The algorithm runs in time
given access to a sampling-based low-overhead data structure for the constraint
matrices, where is the precision of the solution. In addition, we
apply our algorithm to a quantum state learning task as an application.
Technically, our approach aligns with 1) SDP solvers based on the matrix
multiplicative weight (MMW) framework by Arora and Kale [TOC '12]; 2)
sampling-based dequantizing framework pioneered by Tang [STOC '19]. In order to
compute the matrix exponential required in the MMW framework, we introduce two
new techniques that may be of independent interest:
Weighted sampling: assuming sampling access to each individual
constraint matrix , we propose a procedure that gives a
good approximation of .
Symmetric approximation: we propose a sampling procedure that gives
the \emph{spectral decomposition} of a low-rank Hermitian matrix . To the
best of our knowledge, this is the first sampling-based algorithm for spectral
decomposition, as previous works only give singular values and vectors.Comment: 37 pages, 1 figure. To appear in the Proceedings of the 45th
International Symposium on Mathematical Foundations of Computer Science (MFCS
2020
Well-tempered cosmology
We examine an approach to cosmology, known as Well-Tempering, that allows for a de Sitter phase whose expansion is independent of the cosmological constant. Starting from a generic scalar-tensor theory compatible with the recent gravitational wave observation, we impose the Well-Tempering conditions and derive a system that is capable of tuning away the cosmological constant within a sub-class of Horndeski theory, where the scalar has a canonical kinetic term and a general potential. This scenario improves upon the Fab-Four approach by allowing a standard fluid-cosmology before entering the de Sitter phase, and we present an explicit example of our general solution
Viia-hand: a Reach-and-grasp Restoration System Integrating Voice interaction, Computer vision and Auditory feedback for Blind Amputees
Visual feedback plays a crucial role in the process of amputation patients
completing grasping in the field of prosthesis control. However, for blind and
visually impaired (BVI) amputees, the loss of both visual and grasping
abilities makes the "easy" reach-and-grasp task a feasible challenge. In this
paper, we propose a novel multi-sensory prosthesis system helping BVI amputees
with sensing, navigation and grasp operations. It combines modules of voice
interaction, environmental perception, grasp guidance, collaborative control,
and auditory/tactile feedback. In particular, the voice interaction module
receives user instructions and invokes other functional modules according to
the instructions. The environmental perception and grasp guidance module
obtains environmental information through computer vision, and feedbacks the
information to the user through auditory feedback modules (voice prompts and
spatial sound sources) and tactile feedback modules (vibration stimulation).
The prosthesis collaborative control module obtains the context information of
the grasp guidance process and completes the collaborative control of grasp
gestures and wrist angles of prosthesis in conjunction with the user's control
intention in order to achieve stable grasp of various objects. This paper
details a prototyping design (named viia-hand) and presents its preliminary
experimental verification on healthy subjects completing specific
reach-and-grasp tasks. Our results showed that, with the help of our new
design, the subjects were able to achieve a precise reach and reliable grasp of
the target objects in a relatively cluttered environment. Additionally, the
system is extremely user-friendly, as users can quickly adapt to it with
minimal training
- …