14,476 research outputs found
Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics
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.Comment: 67 pages, 1 figur
Applicability of Measurement-based Quantum Computation towards Physically-driven Variational Quantum Eigensolver
Recently variational quantum algorithms have been considered promising
quantum computation methods, where the mainstream algorithms are based on the
conventional quantum circuit scheme. However, in the Measurement-Based Quantum
Computation (MBQC) scheme, multi-qubit rotation operations are implemented with
a straightforward approach that only requires a constant number of single-qubit
measurements, providing potential advantages in both resource cost and
fidelity. The structure of Hamiltonian Variational Ansatz (HVA) aligns well
with this characteristic. In this paper, we propose an efficient
measurement-based quantum algorithm for quantum many-body system simulation
tasks, alluded to as Measurement-Based Hamiltonian Variational Ansatz (MBHVA).
We then demonstrate its effectiveness, efficiency, and advantages with two
quantum many-body system models. Numerical experiments show that MBHVA is
expected to reduce resource overhead compared to the construction of quantum
circuits especially in the presence of large-scale multi-qubit rotation
operations. Furthermore, when compared to measurement-based Hardware Efficient
Ansatz (MBHEA) on quantum many-body system problems, MBHVA also demonstrates
superior performance. We conclude that the MBQC scheme is potentially better
suited for quantum simulation than the circuit-based scheme in terms of both
resource efficiency and error mitigation
Efficient Algorithms for Universal Quantum Simulation
A universal quantum simulator would enable efficient simulation of quantum
dynamics by implementing quantum-simulation algorithms on a quantum computer.
Specifically the quantum simulator would efficiently generate qubit-string
states that closely approximate physical states obtained from a broad class of
dynamical evolutions. I provide an overview of theoretical research into
universal quantum simulators and the strategies for minimizing computational
space and time costs. Applications to simulating many-body quantum simulation
and solving linear equations are discussed
Using Quantum Computers for Quantum Simulation
Numerical simulation of quantum systems is crucial to further our
understanding of natural phenomena. Many systems of key interest and
importance, in areas such as superconducting materials and quantum chemistry,
are thought to be described by models which we cannot solve with sufficient
accuracy, neither analytically nor numerically with classical computers. Using
a quantum computer to simulate such quantum systems has been viewed as a key
application of quantum computation from the very beginning of the field in the
1980s. Moreover, useful results beyond the reach of classical computation are
expected to be accessible with fewer than a hundred qubits, making quantum
simulation potentially one of the earliest practical applications of quantum
computers. In this paper we survey the theoretical and experimental development
of quantum simulation using quantum computers, from the first ideas to the
intense research efforts currently underway.Comment: 43 pages, 136 references, review article, v2 major revisions in
response to referee comments, v3 significant revisions, identical to
published version apart from format, ArXiv version has table of contents and
references in alphabetical orde
Pulse Width Modulation for Speeding Up Quantum Optimal Control Design
This paper focuses on accelerating quantum optimal control design for complex
quantum systems. Based on our previous work [{arXiv:1607.04054}], we combine
Pulse Width Modulation (PWM) and gradient descent algorithm into solving
quantum optimal control problems, which shows distinct improvement of
computational efficiency in various cases. To further apply this algorithm to
potential experiments, we also propose the smooth realization of the optimized
control solution, e.g. using Gaussian pulse train to replace rectangular
pulses. Based on the experimental data of the D-Norleucine molecule, we
numerically find optimal control functions in -qubit and -qubit systems,
and demonstrate its efficiency advantage compared with basic GRAPE algorithm
Can One Trust Quantum Simulators?
Various fundamental phenomena of strongly-correlated quantum systems such as
high- superconductivity, the fractional quantum-Hall effect, and quark
confinement are still awaiting a universally accepted explanation. The main
obstacle is the computational complexity of solving even the most simplified
theoretical models that are designed to capture the relevant quantum
correlations of the many-body system of interest. In his seminal 1982 paper
[Int. J. Theor. Phys. 21, 467], Richard Feynman suggested that such models
might be solved by "simulation" with a new type of computer whose constituent
parts are effectively governed by a desired quantum many-body dynamics.
Measurements on this engineered machine, now known as a "quantum simulator,"
would reveal some unknown or difficult to compute properties of a model of
interest. We argue that a useful quantum simulator must satisfy four
conditions: relevance, controllability, reliability, and efficiency. We review
the current state of the art of digital and analog quantum simulators. Whereas
so far the majority of the focus, both theoretically and experimentally, has
been on controllability of relevant models, we emphasize here the need for a
careful analysis of reliability and efficiency in the presence of
imperfections. We discuss how disorder and noise can impact these conditions,
and illustrate our concerns with novel numerical simulations of a paradigmatic
example: a disordered quantum spin chain governed by the Ising model in a
transverse magnetic field. We find that disorder can decrease the reliability
of an analog quantum simulator of this model, although large errors in local
observables are introduced only for strong levels of disorder. We conclude that
the answer to the question "Can we trust quantum simulators?" is... to some
extent.Comment: 20 pages. Minor changes with respect to version 2 (some additional
explanations, added references...
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