488 research outputs found
Quantum eigenvalue processing
Many problems in linear algebra -- such as those arising from non-Hermitian
physics and differential equations -- can be solved on a quantum computer by
processing eigenvalues of the non-normal input matrices. However, the existing
Quantum Singular Value Transformation (QSVT) framework is ill-suited to this
task, as eigenvalues and singular values are different in general. We present a
Quantum EigenValue Transformation (QEVT) framework for applying arbitrary
polynomial transformations on eigenvalues of block-encoded non-normal
operators, and a related Quantum EigenValue Estimation (QEVE) algorithm for
operators with real spectra. QEVT has query complexity to the block encoding
nearly recovering that of the QSVT for a Hermitian input, and QEVE achieves the
Heisenberg-limited scaling for diagonalizable input matrices. As applications,
we develop a linear differential equation solver with strictly linear time
query complexity for average-case diagonalizable operators, as well as a ground
state preparation algorithm that upgrades previous nearly optimal results for
Hermitian Hamiltonians to diagonalizable matrices with real spectra.
Underpinning our algorithms is an efficient method to prepare a quantum
superposition of Faber polynomials, which generalize the nearly-best uniform
approximation properties of Chebyshev polynomials to the complex plane. Of
independent interest, we also develop techniques to generate Fourier
coefficients with gates compared to prior
approaches with linear cost.Comment: 106 pages, 3 figure
Domain-Indexing Variational Bayes: Interpretable Domain Index for Domain Adaptation
Previous studies have shown that leveraging domain index can significantly
boost domain adaptation performance (arXiv:2007.01807, arXiv:2202.03628).
However, such domain indices are not always available. To address this
challenge, we first provide a formal definition of domain index from the
probabilistic perspective, and then propose an adversarial variational Bayesian
framework that infers domain indices from multi-domain data, thereby providing
additional insight on domain relations and improving domain adaptation
performance. Our theoretical analysis shows that our adversarial variational
Bayesian framework finds the optimal domain index at equilibrium. Empirical
results on both synthetic and real data verify that our model can produce
interpretable domain indices which enable us to achieve superior performance
compared to state-of-the-art domain adaptation methods. Code is available at
https://github.com/Wang-ML-Lab/VDI.Comment: ICLR 2023 Spotlight (notable-top-25%
The structure of -brane model
Recently, a family of interesting analytical brane solutions were found in
gravity with in Ref. [Phys. Lett. B 729, 127
(2014)]. In these solutions, inner brane structure can be turned on by tuning
the value of the parameter . In this paper, we investigate how the
parameter affects the localization and the quasilocalization of the
tensorial gravitons around these solutions. It is found that, in a range of
, despite the brane has an inner structure, there is no graviton
resonance. However, in some other regions of the parameter space, although the
brane has no internal structure, the effective potential for the graviton KK
modes has a singular structure, and there exists a series of graviton resonant
modes. The contribution of the massive graviton KK modes to the Newton's law of
gravity is discussed shortly.Comment: v2: 10 pages, 8 figures, to be published in EPJ
Natural Counterfactuals With Necessary Backtracking
Counterfactual reasoning is pivotal in human cognition and especially
important for providing explanations and making decisions. While Judea Pearl's
influential approach is theoretically elegant, its generation of a
counterfactual scenario often requires interventions that are too detached from
the real scenarios to be feasible. In response, we propose a framework of
natural counterfactuals and a method for generating counterfactuals that are
natural with respect to the actual world's data distribution. Our methodology
refines counterfactual reasoning, allowing changes in causally preceding
variables to minimize deviations from realistic scenarios. To generate natural
counterfactuals, we introduce an innovative optimization framework that permits
but controls the extent of backtracking with a naturalness criterion. Empirical
experiments indicate the effectiveness of our method
MIXGAN: Learning Concepts from Different Domains for Mixture Generation
In this work, we present an interesting attempt on mixture generation:
absorbing different image concepts (e.g., content and style) from different
domains and thus generating a new domain with learned concepts. In particular,
we propose a mixture generative adversarial network (MIXGAN). MIXGAN learns
concepts of content and style from two domains respectively, and thus can join
them for mixture generation in a new domain, i.e., generating images with
content from one domain and style from another. MIXGAN overcomes the limitation
of current GAN-based models which either generate new images in the same domain
as they observed in training stage, or require off-the-shelf content templates
for transferring or translation. Extensive experimental results demonstrate the
effectiveness of MIXGAN as compared to related state-of-the-art GAN-based
models.Comment: Accepted by IJCAI-ECAI 2018, the 27th International Joint Conference
on Artificial Intelligence and the 23rd European Conference on Artificial
Intelligenc
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 ad
Quantum Simulation of Boson-Related Hamiltonians: Techniques, Effective Hamiltonian Construction, and Error Analysis
Elementary quantum mechanics proposes that a closed physical system
consistently evolves in a reversible manner. However, control and readout
necessitate the coupling of the quantum system to the external environment,
subjecting it to relaxation and decoherence. Consequently, system-environment
interactions are indispensable for simulating physically significant theories.
A broad spectrum of physical systems in condensed-matter and high-energy
physics, vibrational spectroscopy, and circuit and cavity QED necessitates the
incorporation of bosonic degrees of freedom, such as phonons, photons, and
gluons, into optimized fermion algorithms for near-future quantum simulations.
In particular, when a quantum system is surrounded by an external environment,
its basic physics can usually be simplified to a spin or fermionic system
interacting with bosonic modes. Nevertheless, troublesome factors such as the
magnitude of the bosonic degrees of freedom typically complicate the direct
quantum simulation of these interacting models, necessitating the consideration
of a comprehensive plan. This strategy should specifically include a suitable
fermion/boson-to-qubit mapping scheme to encode sufficiently large yet
manageable bosonic modes, and a method for truncating and/or downfolding the
Hamiltonian to the defined subspace for performing an approximate but highly
accurate simulation, guided by rigorous error analysis. In this paper, we aim
to provide such an exhaustive strategy. Specifically, we emphasize two aspects:
(1) the discussion of recently developed quantum algorithms for these
interacting models and the construction of effective Hamiltonians, and (2) a
detailed analysis regarding a tightened error bound for truncating the bosonic
modes for a class of fermion-boson interacting Hamiltonians
Quantum singular value transformation and beyond: Exponential improvements for quantum matrix arithmetics
An n-qubit quantum circuit performs a unitary operation on an exponentially large, 2n-dimensional, Hilbert space, which is a major source of quantum speed-ups. We develop a new “Quantum singular value transformation” algorithm that can directly harness the advantages of exponential dimensionality by applying polynomial transformations to the singular values of a block of a unitary operator. The transformations are realized by quantum circuits with a very simple structure – typically using only a constant number of ancilla qubits – leading to optimal algorithms with appealing constant factors. We show that our framework allows describing many quantum algorithms on a high level, and enables remarkably concise proofs for many prominent quantum algorithms, ranging from optimal Hamiltonian simulation to various quantum machine learning applications. We also devise a new singular vector transformation algorithm, describe how to exponentially improve the complexity of implementing fractional queries to unitaries with a gapped spectrum, and show how to efficiently implement principal component regression. Finally, we also prove a quantum lower bound on spectral transformations
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