Related papers: On generating direct powers of dynamical Lie algeb…
Unlike most classical algorithms that take an input and give the solution directly as an output, quantum algorithms produce a quantum circuit that works as an indirect solution to computationally hard problems. In the full quantum computing…
Quantum computing has recently emerged as a transformative technology. Yet, its promised advantages rely on efficiently translating quantum operations into viable physical realizations. In this work, we use generative machine learning…
Lie algebras provide a useful framework for theoretical analysis in quantum machine learning, particularly in hybrid quantum-classical learning. From the viewpoint of function approximation, expectation values of parameterized quantum…
To each directed acyclic graph (this includes some D-dimensional lattices) one can associate some abelian algebras that we call directed abelian algebras (DAA). On each site of the graph one attaches a generator of the algebra. These…
Dynamical Lie algebras, i.e. Lie subalgebras of $\mathfrak{su}(2^n)$, generated by Pauli strings have recently been studied intensively. They are also called Pauli Lie algebras or Hamiltonian Lie algebras. In this paper we provide a uniform…
In this work, we introduce a quantum-control-inspired method for the characterization of variational quantum circuits using the rank of the dynamical Lie algebra associated with the hermitian generator(s) of the individual layers.…
In the first part of the paper we present a short review of applications of digital differential analyzers (DDA) to generation of circles showing that they can be treated as one-step numerical schemes. In the second part we present and…
Qubits are a great way to build a quantum computer, but a limited way to program one. We replace the usual "states and gates" formalism with a "props and ops" (propositions and operators) model in which (a) the C*-algebra of observables…
We present a theory of parameterized dynamic logic, namely DLp, for specifying and reasoning about a rich set of program models based on their transitional behaviours. Different from most dynamic logics that deal with regular expressions or…
For the standard Drinfeld-Jimbo quantum group ${\rm U}_q(\mathfrak{g})$ associated with a simple Lie algebra $\mathfrak{g}$, we construct explicit generators of the centre $Z({\rm U}_q(\mathfrak{g}))$, and determine the relations satisfied…
Many standard linear algebra problems can be solved on a quantum computer by using recently developed quantum linear algebra algorithms that make use of block encodings and quantum eigenvalue/singular value transformations. A block encoding…
In this short communication we show how the Lie algebra $\mathfrak{g}_2$ can easily be described as a free Lie algebra on 3 generators, subject to some simple quadruple relations for these generators.
Generative AI (GEN AI) models have revolutionized diverse application domains but present substantial challenges due to reliability concerns, including hallucinations, semantic drift, and inherent biases. These models typically operate as…
We show the applicability of the Cartan decomposition of Lie algebras to quantum circuits. This approach can be used to synthesize circuits that can efficiently implement any desired unitary operation. Our method finds explicit quantum…
When the system is linear, why should learning be nonlinear? Linear dynamical systems, the analytical backbone of control theory, signal processing and circuit analysis, have exact closed-form solutions via the state transition matrix. Yet…
Quantum Lie algebras $\qlie{g}$ are non-associative algebras which are embedded into the quantized enveloping algebras $U_q(g)$ of Drinfeld and Jimbo in the same way as ordinary Lie algebras are embedded into their enveloping algebras. The…
We consider the problem of training generative models with deep neural networks as generators, i.e. to map latent codes to data points. Whereas the dominant paradigm combines simple priors over codes with complex deterministic models, we…
In this paper, we show how to use the analysis of the Lie algebra associated with a quantum mechanical system to study its dynamics and facilitate the design of controls. We give algorithms to decompose the dynamics and describe their…
In coherent quantum feedback control schemes, a target quantum system S is put in contact with an auxiliary system A and the coherent control can directly affect only A. The system S is controlled 'indirectly' through the interaction with…
A formalism for describing the dynamics of Genetic Algorithms (GAs) using methods from statistical mechanics is applied to the problem of generalization in a perceptron with binary weights. The dynamics are solved for the case where a new…