物理代写|量子计算代写Quantum computer代考|Physics 421

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

物理代写|量子计算代写Quantum computer代考|Quantum Computing Model

Quantum computers use processes of a quantum nature manifested with atoms, molecules, molecular clusters, etc. The description of such processes is based on the application of complex numbers and complex matrices.

As is well known, the basic notion of classical information theory is a bit [1]. A classical bit takes the values 0 or 1 (and no other).

A qubit (quantum bit) is the smallest element that executes the information storage function in a quantum computer [2].

Qubit is a quantum system $|\psi\rangle$ that allows two states: $|0\rangle$ and $|1\rangle$. In accordance with the so-called “bra-ket” Dirac ” notation (from word bra(c)ket), the symbols $\mid 0$ ) and $|1\rangle$ are read as “Ket $0 “$ and “Ket 1 “, respectively. The brackets $|. .\rangle\rangle$ show that $\psi$ is some state of the quantum system.

The fundamental difference between the classical bit and the qubit consists in that, the qubit can be in a state different from $|0\rangle$ or $|1\rangle$. The arbitrary state of the qubit is defined by the linear combination of basis states:
$$|\psi\rangle=u|0\rangle+v|1\rangle,$$
where the complex coefficients $u$ and $v$ satisfy the following normalization condition:
$$|u|^{2}+|v|^{2}=1 .$$
The mathematical description of basis states reduces to their representation in matrix form:
$$|0\rangle=\left[\begin{array}{l} 1 \ 0 \end{array}\right], \quad|1\rangle=\left[\begin{array}{l} 0 \ 1 \end{array}\right]$$
Based on the presentation (1.3), the arbitrary state of the qubit is written as
$$|\psi\rangle=\left[\begin{array}{l} u \ v \end{array}\right]$$

物理代写|量子计算代写Quantum computer代考|The orthogonality condition

The orthogonality condition of two states $\left|\psi^{\prime}\right\rangle$ and $\left|\psi^{\prime \prime}\right\rangle$ is written as follows:
$$\left\langle\psi^{\prime} \mid \psi^{\prime \prime}\right\rangle=\sum_{i=1}^{2^{n}} u_{i}^{*} v_{i}=0 .$$
Note that the states of the computational base (1.6) are orthonormalized.
To change the state of a quantum system, quantum operations are used, which are called quantum logic gates, or, for short, simply gates. Thus, gates perform logical operations on qubits. Note that the change of state $|\psi\rangle$ in time is also referred to as the evolution of the quantum system.

An important step of quantum algorithms is the procedure of measurement of a state. When the qubit state is measured, it randomly passes to one of its states: $|0\rangle$ or $|1\rangle$. Therefore, the complex coefficients $u$ and $v$ from the qubit definition (1.1) are associated with the probability to get the value 0 or 1 when its state is measured. According to the postulates of quantum theory, the probabilities of passing to the states $|0\rangle$ and $|1\rangle$ are equal to $|u|^{2}$ and $|v|^{2}$, respectively. In this connection, the equality (1.2) reflects the probability conservation law. After the measurement, the qubit passes to the basis state, complying with the classical result of the measurement. Generally speaking, the probabilities of getting the result 0 and 1 are different for different states of the quantum system.

In other words, quantum computing is a sequence of simple form operations with the collection of the interacting qubits. In the final step of the quantum computing procedure, the state of the quantum system is measured and a conclusion about the computing result is made. The measurement makes it possible to obtain, at a macroscopic level, the information about the quantum state. The peculiarity of the quantum measurements is their irreversibility, which radically differentiates quantum computing from the classical one.

Despite the fact that the number of qubit states is infinite, with the help of measurement it is possible to obtain only one bit of classical information. The measurement procedure transfers the qubit state to one of the basis states, so a second measurement will produce the same result.

Quantum computer is a set of $n$ qubits controlled by external (classic) signals [812]. Often, an ordered set of some qubits is called a register. The main elements of a quantum computer are shown in Fig. 1.1.

The classical quantum computer setting consists of the controlling classical computer and impulse generators controlling the qubit evolution, as well as measurement instruments of the qubit state. The system from $n$ qubits in the initial state, e.g. $\left|\psi_{\text {in }}\right\rangle=|00 \ldots 0\rangle$, forms a memory register prepared to record input data and perform computations. The data are recorded by an external action on each of the system’s qubits. The solution of the problem is determined by a measurement of the final state qubits $\left|\psi_{\text {out }}\right\rangle[2,12]$.

物理代写|量子计算代写Quantum computer代考|Main elements of a quantum computer

In this record, the state $|\psi\rangle$ of the $n$-qubit register of a quantum computer is expressed through the superposition of vectors of computational basis $B={|0\rangle,|1\rangle, \ldots$, $\left.\left|2^{n}-1\right\rangle\right}$ by the formula
$$|\psi\rangle=\sum_{k=0}^{2^{n}-1} c_{k}|k\rangle$$
where the normalization condition
$$\sum_{k=0}^{2^{k}-1}\left|c_{k}\right|^{2}=1$$
is met.
A quantum system, formed by $N$ two-level elements, has $\Sigma(N)=2^{N}$ independent states. The key point of the functioning of such a system is the interaction of separate qubits with each other. The number of states $\Sigma(N)$ grows exponentially with the growth of the quantum system, which allows solving practical problems of a very high asymptotic complexity. For example, an efficient quantum algorithm of prime factorization is known, which is very important for cryptography [13]. As a result, the quantum algorithms provide exponential or polynomial speedup in comparison with the classical solution methods for many problems.

Unfortunately, no full-function quantum computer has been created yet, although many of its elements have already been built and studied at the world’s leading laboratories $[11,14,15]$. The main obstacle to the development of quantum computing is the instability of a system of many qubits. The more the qubits are united into an entangled system, the more the effort is required to ensure the smallness of the number of measurement errors. Nevertheless, the history of quantum computer development demonstrates an enormous potential laid in the uniting of quantum theory and algorithm theory.

物理代写|量子计算代写Quantum computer代考|Quantum Computing Model

qubit（量子位）是量子计算机中执行信息存储功能的最小元素[2]。

Qubit是一个量子系统|ψ⟩允许两种状态：|0⟩和|1⟩. 根据所谓的“bra-ket”狄拉克“符号（来自单词 bra(c)ket），符号∣0） 和|1⟩读作“凯0“和“Ket 1”，分别。括号|..⟩⟩显示ψ是量子系统的某种状态。

|ψ⟩=在|0⟩+在|1⟩,

|在|2+|在|2=1.

|0⟩=[1 0],|1⟩=[0 1]

|ψ⟩=[在 在]

物理代写|量子计算代写Quantum computer代考|The orthogonality condition

⟨ψ′∣ψ′′⟩=∑一世=12n在一世∗在一世=0.

物理代写|量子计算代写Quantum computer代考|Main elements of a quantum computer

|ψ⟩=∑ķ=02n−1Cķ|ķ⟩

∑ķ=02ķ−1|Cķ|2=1

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