# Dimensions of quantum cell automata's state space

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• Jaime_mc2
In summary: This would make it difficult to use the qubits as a single unit, because the energy levels would be too spread out.
Jaime_mc2
In the paper

C. S. Lent and P. D. Tougaw, "A device architecture for computing with quantum dots," in Proceedings of the IEEE, vol. 85, no. 4, pp. 541-557, April 1997, doi: 10.1109/5.573

about quantum dots, it is stated that the basis vectors in the space of quantum states for a single cell (four quantum dots) are of the form $$|\phi_1\rangle = \left|\begin{array}{cccc}0&0&0&1 \\ 0&0&0&1\end{array}\right> \quad\cdots\quad |\phi_{16}\rangle = \left|\begin{array}{cccc}1&0&0&0 \\ 1&0&0&0\end{array}\right>$$ where the columns are related to the dot in which there is an electron, and the rows tell the projection of the spin (first row meaning that the spin points upwards). Therefore, the authors state that there are ##16## different basis states and that the dimension of the state space for ##N## cells is ##16^N##.

However, I don't see why they only take into account states in which the electrons have opposite spin projections, and they are ignoring basis states like $$\left|\begin{array}{cccc}1&1&0&0 \\ 0&0&0&0\end{array}\right>$$

Of course, because of the exclusion principle, the unique possibility for having two electrons in the same dot is that they have opposite spins, like in state $|\phi_1\rangle$, but I don't see why there should be such a restriction for two electrons being in different dots. If we take into account these extra basis states, the dimension of the state space for a single cell would be $$\dfrac{8!}{2! \cdot 6!} = 28\ ,$$ so we have ##28^N## for ##N## cells.

Why aren't these states taken into account?

Jaime_mc2 said:
Of course, because of the exclusion principle, the unique possibility for having two electrons in the same dot is that they have opposite spins, like in state $|\phi_1\rangle$, but I don't see why there should be such a restriction for two electrons being in different dots.
You certainly "want" your basis states to have nearly identical energy levels (and being close to "some" ground state). Otherwise their relative phase will spin (much) faster than your "classical" control can handle (and if you are not close to "some" ground state, you risk that your qubits will decay to "some" ground state before you can perform useful computations on them).

Jaime_mc2 said:
If we take into account these extra basis states, the dimension of the state space for a single cell would be $$\dfrac{8!}{2! \cdot 6!} = 28\ ,$$ so we have ##28^N## for ##N## cells.

Why aren't these states taken into account?
Because then you would have different types of basis states, and those different types would almost certainly sit at completely different energy levels.

## 1. What are dimensions in the context of quantum cell automata's state space?

Dimensions refer to the number of independent variables needed to describe the state space of a quantum cell automata. In other words, it represents the number of distinct properties or characteristics that can be measured or observed in the system.

## 2. How many dimensions are typically present in a quantum cell automata's state space?

The number of dimensions can vary depending on the specific system being studied. However, in most cases, quantum cell automata's state space is described by three dimensions: time, space, and quantum state.

## 3. Can the dimensions of a quantum cell automata's state space change?

Yes, the dimensions of a quantum cell automata's state space can change depending on the conditions and interactions within the system. For example, the dimensions may change when the system undergoes a phase transition or when new variables are introduced.

## 4. How do dimensions affect the behavior of quantum cell automata?

The number of dimensions in a quantum cell automata's state space can significantly impact its behavior and dynamics. For instance, a higher number of dimensions may result in more complex and unpredictable behavior, while fewer dimensions can lead to more stable and predictable patterns.

## 5. Are there any practical applications of studying the dimensions of quantum cell automata's state space?

Yes, understanding the dimensions of quantum cell automata's state space can have practical applications in various fields, such as quantum computing and simulation. It can also help us gain a deeper understanding of complex systems and their behavior.

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