Although quantum computing has many different definitions, we can roughly define it here as the process of information processing and computation by leveraging the fundamental principles of quantum mechanics. In this context, understanding the state spaces of quantum systems, their time evolution functions, and measurement processes holds significant importance.

In classical computational paradigms, the state space is generally limited to structures representing numerical records, and state changes are realized via Boolean functions. The measurement operation is defined as a simple projection function.

We have written quite a long and information-dense sentence here, but let’s discuss it a bit now. What did we mean by the sentence: “In classical computational paradigms, the state space is generally limited to structures representing numerical records, and state changes are realized via Boolean functions. The measurement operation is defined as a simple projection function.”? Let’s elaborate on this.

In digital computers, the state of the system is generally expressed numerically, and changes are made using logical rules such as true or false. Measuring or observing the current state of this system is done by simply reflecting the state. That is, in digital systems, everything is controlled and measured using numbers and logical rules.

In quantum mechanics, state spaces are generally defined as infinite-dimensional vector spaces. However, the knowledge and competencies required for quantum computing do not necessitate understanding all the details of general quantum mechanics. Therefore, it is more efficient to focus on the subspace of state spaces needed in the computational context. The evaluation of time evolution and measurement processes will be discussed later.

To explain what we have said here in a simpler way, we can express it as follows. In quantum mechanics, the states of systems are generally defined in infinite-dimensional, and thus very complex, structures. However, understanding such complex structures is not always necessary for quantum computing. Therefore, when dealing with quantum computing, it is more beneficial to focus only on the simpler parts that are useful to us.

In terms of the fundamental principles and dynamics of quantum computing, we need to consider the following points, which are quite important for us.

• Finite-Dimensional State Spaces

As mentioned above, the states of systems in quantum mechanics are generally defined in infinite-dimensional, highly complex structures. However, in quantum computing, the state spaces used are typically finite-dimensional. This simplifies the complex structures presented by general quantum mechanics and makes the computation process more comprehensible. Finite-dimensional state spaces form a vector space that represents specific states of a particular system.

To enhance understanding, let us summarize the above explanation with an example. A finite-dimensional state space implies that the number of possible states a system can possess is limited. For instance, consider a die. This die has only six distinct faces; therefore, the state space of the die is a finite-dimensional space containing only six states. We can represent each of these states with a specific vector in a vector space. The state of a six-sided die consists of six outcomes, which together form a vector space. Each dimension of this space corresponds to one face of the die. That is:

• The outcome “1” is represented by a vector (for example, [1, 0, 0, 0, 0, 0]),
• The outcome “2” is represented by another vector (for example, [0, 1, 0, 0, 0, 0]), and so on.

This six-dimensional vector space represents a finite-dimensional state space that encompasses all possible outcomes of the die.

• Pure and Mixed Quantum States

In quantum systems, there are two types of states: pure states and mixed states. Often, it is sufficient to focus solely on pure states.

Pure States refer to a quantum system being able to exist in multiple states simultaneously, a phenomenon known as superposition. For example, in the Stern-Gerlach experiment, a particle can exist in a state that points in different directions at the same time. This indicates that the particle is not in a single, definite state but can be in multiple states simultaneously. Pure states represent the properties of a quantum system more accurately and do not involve uncertainty. In other words, a pure state allows us to understand more clearly the condition of a quantum system.

Mixed States signify that a quantum system is not in a single pure state but can be in multiple possible states, which are in a mixture with specific probabilities. If we know that a quantum system is not in a pure state, it may be in a mixed state. In a mixed state, the system is not in a specific superposition; rather, it is a combination of different states. This implies that the system can exist in various states, but unlike superposition, it is not simultaneously in multiple states. In a mixed state, each possible state can emerge with a certain probability.

In the quantum realm, mixed states are typically applicable to systems that have interacted with their external environment, effectively “losing” their quantum properties. Mixed states can be thought of as instances where pure states are combined and expressed with probabilities.

To illustrate the concepts of mixed and pure states, let us consider the example of a die, which can simplify understanding.

When a die is in a pure state, it represents a specific face or can represent multiple faces simultaneously. If we think of quantum superposition, we can imagine that before rolling the die, it is “simultaneously” 1, 6, and all other faces. This means that as long as we do not roll the die, all faces are included in the probabilities, and this situation is referred to as a “pure state” in the quantum world.

For example:

• Before rolling the die, there is a possibility that it exists on faces 1, 2, 3, 4, 5, and 6 at the same time. This resembles a pure state in superposition.
• However, when we look at the die, we only see one of its faces.

In the case of quantum superposition, we cannot know the specific state of the die until an observation is made.

In a mixed state, there is no superposition of the die; rather, there are probabilities associated with certain faces. The die is not in a pure superposition but is represented solely by a mixture of probabilities for the different faces.

For example:

• Suppose the die is set such that there is a 50% probability of it landing on 1, a 25% probability on 3, and a 25% probability on 5. In this case, we know the die could land on 1, 3, or 5, but we cannot precisely state which face will appear. This represents a mixed state.

In a mixed state, the die does not represent a definite face, and there is no superposition; rather, it reflects the situation of specific faces emerging with different probabilities.

• Structure of Multiparticle Quantum Systems

Multiparticle quantum systems are represented by the tensor product of constituent systems. Understanding the structure of these state spaces is critically important, especially for investigating the interactions of multiparticle systems. This constitutes one of the more complex aspects of quantum computing, which, while simple in certain respects, becomes more intricate due to the requirements of multiparticle state spaces.

In simpler terms:

• If we consider a single quantum particle, it is relatively easy to represent the state of that particle (for instance, as “0” or “1”). However, when we think about two or more particles simultaneously, the state of each particle becomes much more complex when considered in conjunction with the others. To represent these combined states, a method known as the “tensor product” is employed. This complicates quantum computation because as the number of particles increases, the number of possible states grows rapidly. Therefore, the state of multiparticle systems is significantly more complex than that of a single particle, which in turn complicates the understanding of quantum computation.

Understanding the fundamental principles mentioned here is quite important. Unlike classical computing paradigms, quantum computing aims to develop new computational models by leveraging the unique properties offered by quantum mechanics. In this context, a profound understanding of fundamental concepts such as the structure of state spaces, time evolution, and measurement processes is essential for the successful execution of quantum computing research.