We investigate two fundamental data encoding strategies for gate-based quantum computers: amplitude encoding and angle encoding. In amplitude encoding, a normalized -dimensional classical vector is mapped into the probability amplitudes of a -qubit quantum state, whereas angle encoding uses one or more single-qubit rotation gates per feature to encode data as angles.[1, 2] We construct a toy binary classification dataset with four real-valued features and compare these encoding schemes using a variational quantum classifier (VQC) approach. Our results indicate that angle encoding often yields slightly higher classification accuracy (e.g. 82% vs. 75% in one experiment) at the cost of requiring more qubits, while amplitude encoding compresses features into fewer qubits (2 vs. 4) but demands deeper state-preparation circuits. These findings align with previous studies that report higher accuracy for rotation encodings (e.g. vs. in MNIST experiments [3]) and note that the encoding choice acts as a hyperparameter affecting VQC performance. [4]We investigate two fundamental data encoding strategies for gate-based quantum computers: amplitude encoding and angle encoding. In amplitude encoding, a normalized -dimensional classical vector is mapped into the probability amplitudes of a -qubit quantum state, whereas angle encoding uses one or more single-qubit rotation gates per feature to encode data as angles.[1, 2] We construct a toy binary classification dataset with four real-valued features and compare these encoding schemes using a variational quantum classifier (VQC) approach. Our results indicate that angle encoding often yields slightly higher classification accuracy (e.g. 82% vs. 75% in one experiment) at the cost of requiring more qubits, while amplitude encoding compresses features into fewer qubits (2 vs. 4) but demands deeper state-preparation circuits. These findings align with previous studies that report higher accuracy for rotation encodings (e.g. vs. in MNIST experiments [3]) and note that the encoding choice acts as a hyperparameter affecting VQC performance. [4]

International Journal of Physics and MathematicsJul 1, 20252 citations

PRELIMINARIES: PROBABILITIES VERSUS AMPLITUDES

WORLD SCIENTIFIC eBooksJan 1, 20220 citations

Continuous-time quantum walks over connected graphs, amplitudes and invariants

Phillip Dukes

We examine the time dependent amplitude $ ϕ_{j}\left( t\right)$ at each vertex $j$ of a continuous-time quantum walk on the cycle $C_{n}$ . In many cases the Lissajous curve of the real vs. imaginary parts of each $ ϕ_{j}\left( t\right)$ reveals interesting shapes of the space of time-accessible amplitudes. We find two invariants of continuous-time quantum walks. First, considering the rate at which each amplitude evolves in time we find the quantity $T = \displaystyle\sum_{j=0}^{n-1} \lvert\dfrac{d ϕ_{j}\left( t\right)}{d t}\rvert^{2}$ is time invariant. The value of $T$ for any initial state can be minimized with respect to a global phase factor $e^{i θt}$ to some value $T_{min}$ . An operator for $T_{min}$ is defined. For any simply connected graph $g$ the highest possible value of $T_{min}$ with respect to the initial state is found to be $T_{min}^{max}=\left( \frac{λ_{max}}{2}\right)^{2}$ where $λ_{max}$ is the maximum eigenvalue in the Laplace spectrum of $g$ . A second invariant is found in the time-dependent probability distribution $P_{j}\left(t\right) = \lvertϕ_{j}\left(t\right)\rvert^{2}$ of any initial state satisfying $T_{min}^{max}$ , with these conditions $\displaystyle\sum_{j=0}^{n-1}\left(P_{j}^{max} - P_{j}^{min}\right)^{2} = \dfrac{4}{n}$ for all simply connected graphs of $n$ vertices.We examine the time dependent amplitude $ ϕ_{j}\left( t\right)$ at each vertex $j$ of a continuous-time quantum walk on the cycle $C_{n}$. In many cases the Lissajous curve of the real vs. imaginary parts of each $ ϕ_{j}\left( t\right)$ reveals interesting shapes of the space of time-accessible amplitudes. We find two invariants of continuous-time quantum walks. First, considering the rate at which each amplitude evolves in time we find the quantity $T = \displaystyle\sum_{j=0}^{n-1} \lvert\dfrac{d ϕ_{j}\left( t\right)}{d t}\rvert^{2}$ is time invariant. The value of $T$ for any initial state can be minimized with respect to a global phase factor $e^{i θt}$ to some value $T_{min}$. An operator for $T_{min}$ is defined. For any simply connected graph $g$ the highest possible value of $T_{min}$ with respect to the initial state is found to be $T_{min}^{max}=\left( \frac{λ_{max}}{2}\right)^{2}$ where $λ_{max}$ is the maximum eigenvalue in the Laplace spectrum of $g$. A second invariant is found in the time-dependent probability distribution $P_{j}\left(t\right) = \lvertϕ_{j}\left(t\right)\rvert^{2}$ of any initial state satisfying $T_{min}^{max}$, with these conditions $\displaystyle\sum_{j=0}^{n-1}\left(P_{j}^{max} - P_{j}^{min}\right)^{2} = \dfrac{4}{n}$ for all simply connected graphs of $n$ vertices.

arXiv:1506.03086Jun 9, 20150 citations

TRANSITION AMPLITUDES AND PROBABILITIES

H. Muirhead

Elsevier eBooksJan 1, 19710 citations

TSS Graphs for Hadamard Matrices: Real vs Complex

Wesley Lewis, Darsh Pareek, Ravi Janjam

We have observed that Hadamard matrices whether real or complex lead to a dense output of identical probabilities for any given single input state. Complex Hadamard matrices have unusual orders and introduce phase which prove useful when controlled phase transformation gates are used. By taking a superposed combinations of input states, we discovered that the Hadamards generate non-uniform probabilities which is practically significant towards amplitude amplification without the need for manual parameterization like Grover's operator~\cite{grover1996}. A variety of graph theoretic properties are applied to TSS graphs and their trends were explored. Additionally, graphs turn out to be nearly isomorphic for the same set of input states which has potential applications for developing Quantum Algorithms. Keywords: Quantum Algorithms, Hadamard Matrices, Unitary Matrix Approach, Topological Structure of Superpositions (TSS), Graph TheoryWe have observed that Hadamard matrices whether real or complex lead to a dense output of identical probabilities for any given single input state. Complex Hadamard matrices have unusual orders and introduce phase which prove useful when controlled phase transformation gates are used. By taking a superposed combinations of input states, we discovered that the Hadamards generate non-uniform probabilities which is practically significant towards amplitude amplification without the need for manual parameterization like Grover's operator~\cite{grover1996}. A variety of graph theoretic properties are applied to TSS graphs and their trends were explored. Additionally, graphs turn out to be nearly isomorphic for the same set of input states which has potential applications for developing Quantum Algorithms. Keywords: Quantum Algorithms, Hadamard Matrices, Unitary Matrix Approach, Topological Structure of Superpositions (TSS), Graph Theory

quant-pharXiv:2609.16813Sep 16, 20260 citations

Functional methods in nonequilibrium QFT

Esteban Calzetta, B. L. Hu

One of the major goals in the establishment of a quantum field theory for nonequilibrium systems is to study dynamical problems, following the evolution of the expectation value of a physical variable with respect to an in state. This is different from a scattering problem characterized by the transition amplitude between the in and out states, as is treated in every textbook on quantum field theory. This problem is usually coupled with how one could identify a relevant sector of the theory as the system (light fields vs. heavy fields, slow modes vs. fast modes, long wavelength modes vs. short ones, etc.) and determine the effect of its other sectors as the environment on this system, as we have discussed in the last chapter. Given a classical nonequilibrium system, described, for example, by a Langevin equation, there are essentially three possible strategies to follow. One may attempt to solve it, usually numerically. In the quantum field case, this gets difficult beyond the linear case. Second, one may try to transform it into an equation for the evolution of a probability distribution function in the system's configuration space. In the quantum field case the relevant object is the reduced density matrix, and the relevant equation is the Liouville-von Neumann equation. This is also infeasible beyond the linear case, unless under restrictive approximations (such as Gaussianity) which in fact reduce this approach to the third and coarsest.One of the major goals in the establishment of a quantum field theory for nonequilibrium systems is to study dynamical problems, following the evolution of the expectation value of a physical variable with respect to an in state. This is different from a scattering problem characterized by the transition amplitude between the in and out states, as is treated in every textbook on quantum field theory. This problem is usually coupled with how one could identify a relevant sector of the theory as the system (light fields vs. heavy fields, slow modes vs. fast modes, long wavelength modes vs. short ones, etc.) and determine the effect of its other sectors as the environment on this system, as we have discussed in the last chapter. Given a classical nonequilibrium system, described, for example, by a Langevin equation, there are essentially three possible strategies to follow. One may attempt to solve it, usually numerically. In the quantum field case, this gets difficult beyond the linear case. Second, one may try to transform it into an equation for the evolution of a probability distribution function in the system's configuration space. In the quantum field case the relevant object is the reduced density matrix, and the relevant equation is the Liouville-von Neumann equation. This is also infeasible beyond the linear case, unless under restrictive approximations (such as Gaussianity) which in fact reduce this approach to the third and coarsest.

Cambridge University Press eBooksJul 24, 20081 citation

We examine the time dependent amplitude $ ϕ_{j}\left( t\right)$ at each vertex

j

of a continuous-time quantum walk on the cycle

C_{n}

. In many cases the Lissajous curve of the real vs. imaginary parts of each $ ϕ_{j}\left( t\right)$ reveals interesting shapes of the space of time-accessible amplitudes. We find two invariants of continuous-time quantum walks. First, considering the rate at which each amplitude evolves in time we find the quantity

T = \displaystyle\sum_{j=0}^{n-1} \lvert\dfrac{d ϕ_{j}\left( t\right)}{d t}\rvert^{2}

is time invariant. The value of

T

for any initial state can be minimized with respect to a global phase factor

e^{i θt}

to some value

T_{min}

. An operator for

T_{min}

is defined. For any simply connected graph

g

the highest possible value of

T_{min}

with respect to the initial state is found to be

T_{min}^{max}=\left( \frac{λ_{max}}{2}\right)^{2}

where

λ_{max}

is the maximum eigenvalue in the Laplace spectrum of

g

. A second invariant is found in the time-dependent probability distribution

P_{j}\left(t\right) = \lvertϕ_{j}\left(t\right)\rvert^{2}

of any initial state satisfying

T_{min}^{max}

, with these conditions

\displaystyle\sum_{j=0}^{n-1}\left(P_{j}^{max} - P_{j}^{min}\right)^{2} = \dfrac{4}{n}

for all simply connected graphs of

n

vertices.