Author: Zen and the Art of Computer Programming
1 Introduction
With the rapid development and widespread application of human information technology, breakthroughs and advances in physics, chemistry, biology and other sciences have contributed to in-depth exploration and improvement of people's understanding of the world. Quantum mechanics is an important research direction, which interprets various quantum phenomena existing in the universe to a certain extent and provides an understanding of diverse quantum phenomena from high-temperature superconductivity to powerful astrophysics. new method. Quantum computing, as a cutting-edge topic of quantum computers, has also experienced rapid development in the past two years. Although the emergence of high-dimensional entangled states and circuit qubits has made quantum computers increasingly powerful, their true significance requires more in-depth and extensive practice and application. It is believed that with the rise of the field of artificial intelligence and the continued development of quantum computing, future quantum computers will become a major new field of human-computer interaction and information processing.
In order to better understand and master the principles and applications of quantum computers, this article will explain it in a simple and easy-to-understand manner by learning relevant theoretical knowledge of quantum computing technologies such as quantum basics, classical simulation algorithms, hybrid algorithms, and randomized algorithms, as well as specific code implementations. Its architecture, features, optimization, and prospects.
2.1 Basic concepts and terminology of quantum computing
2.1.1 Overview of Quantum Computers
Quantum Computer, also known as "quantum chip", is a computer system that uses quantum electrons (or qubits) as the main computing unit. Quantum computers can be built adversarially with classical computers, that is, they have the same performance as classical computers in actual use without any obvious performance bottlenecks. This computer model has been widely used. Quantum computers are composed of several qubits and logic gates. These qubits have statistical eigenstates and energy levels. In a quantum computer, any quantum state can be regarded as a "quantum system" composed of an infinite number of qubits.
The principle of quantum computer is based on the laws of physics, which mathematically describes quantum theory. Quantum state describes the elements in it, including particles, atomic nuclei, and even the subspace formed by hydrogen nuclei. Quantum systems are in the exponential order of magnitude of the wave function. Therefore, it is difficult to be completely represented by numbers. Therefore, quantum computers usually use discrete numerical representation methods to process quantum states. Computing tasks run on quantum computers generally consist of the following three stages:
- Building stage: Qubits and logic gates form a quantum circuit and are connected to form a complete quantum computer.
- Operating stage: By adjusting input parameters (such as providing information, initial conditions, selecting operation modes, etc.), the corresponding quantum state is input to the quantum circuit, and then the quantum logic gate is used to change the quantum state (evolution from the initial state) to the target state).
- Collection stage: The results of quantum computer operations are usually obtained through observation and collection. Qubits are often required to perform measurements or output data. Since quantum states have statistical laws, different input parameters will lead to different output results. The computing power of a quantum computer is determined by the size of the problem it can solve. For example, it can calculate some problems that cannot be calculated in the microscopic world or qualitatively understand some complex problems.
2.1.2 Basic concepts of quantum computing
2.1.2.1 Heisenberg Conjecture
The Heisenberg conjecture is a famous principle of quantum mechanics. The first quantum computing experiment, Shor's algorithm, is also an example of calculation under this assumption. The Heisenberg conjecture states that if the quantum state of a closed system satisfies the singular form of the unitary operator, then the system must be decomposed into two pure states with singular characteristics (pure + unitary) through a projection operation on a line. ).
So, logically, the Heisenberg Hypothesis solves the problem of how to use quantum computers to solve singular forms. However, strictly speaking, the Heisenberg conjecture is about a certain fixed quantum system and only proves the proposition that the system has a strange form. Therefore, for some specific problems, it still cannot be directly proved by the Heisenberg conjecture. In addition, the Heisenberg conjecture itself has many limitations, such as whether a quantum system exists (or is it assumed to exist?); what operators need to satisfy the singular form (does it only require a certain operator?), etc. Therefore, more general quantum computing theories and algorithms are the main research hotspots.
2.1.2.2 Quantum state
Quantum state refers to the wave function of a system composed of many interactions. In quantum mechanics, a special kind of wave function is described - the wave function is a statistical law, therefore, we must determine the statistics of the system. A quantum state is an n-dimensional vector. Each component in the vector corresponds to a qubit, and the value represents whether the corresponding bit is in an activated state (1) or an inactive state (0).
To sum up, for a quantum system with n qubits, its quantum state is an n-dimensional vector, indicating that the system is in its respective quantum state. Because of all the possible combinations of ordered arrangements of quantum states, their dimensions typically grow exponentially within a finite set of defined wave functions.
2.1.2.3 Quantum Gate
Quantum Gates refer to transformation operations that act on quantum states. The scope of action has nothing to do with the size of the quantum state. Usually, its operating object is only a specific qubit. Quantum gates can be basic gates or higher-order gates, with the former having a simple structure and the latter consisting of basic gates. At present, quantum computing technology has achieved a relatively mature theoretical foundation, but there are still many shortcomings. Therefore, the development of new quantum gates is still a very promising research direction.
2.1.2.4 Two classifications of quantum computing
- Quantum search algorithm: Find the appropriate quantum state by performing a certain search and transfer method on the quantum state. Quantum search algorithms are usually based on graph search algorithms, which have the advantage of effectively searching in quantum state space and finding solutions within a certain time.
- Quantum optimization algorithm: Find the quantum state that minimizes the expected objective function value by adjusting the structure, order or parameters of the quantum state. When an objective function is given, the quantum optimization algorithm can be solved using many heuristic algorithms or gradient descent methods. The effect depends on the initial value of the quantum state, the accuracy of the search, the number of iterations, etc.
2.1.2.5 Quantum communication and quantum encryption
Quantum Communication is the process of encoding and transmitting information to another party using quantum information processing technology. In quantum communication, two quantum computers exchange information through bilateral communication links (Bell states or entanglements) to complete information sharing between them.
Quantum Encryption is a process that uses quantum communication protocols to ensure information security. The quantum encryption scheme relies on the theoretical basis of quantum state cryptography, which mainly includes quantum sea mines and quantum random number generators.
2.2 Basic algorithms of quantum computing
2.2.1 Shor's algorithm
Shor's algorithm is the first quantum computing experiment and a classic example of quantum algorithms. This algorithm can apply the singular matrix U on the line and decompose it into two singular vectors e1, e2.
Specifically, assuming there is an n-bit quantum circuit, each bit corresponds to a qubit, we hope to find two angles a, b, which correspond to two different quantum states, and both satisfy the calculation basis [|0>+ |1>, |0>-|1>] properties. In other words, we hope to find two quantum states A and B, which satisfy the following conditions: $$\lvert A \rangle = a|0>\rangle+\frac{1}{sqrt(N)}|1>\rangle ,\quad\lvert B \rangle=b|0>\rangle-\frac{1}{sqrt(N)}|1>\rangle.$$
We can do this by doing the following:
Use Hadamard gates to initialize two qubits to a uniform superposition state.
Apply U to the first qubit, and then record the probability of the first bit transforming from |0> to |1>, denoted as p.
Repeat the following steps k times:
- Transform the second qubit from |0> to |1>, and measure the first qubit and the second qubit at the same time to obtain the random number a.
- Act on the first qubit U^(a^(2^j)), j=0,...,k-1, and then record the probability p' of the first bit transforming from |0> to |1>.
- Based on the above calculation, the singular angle a is calculated.
By verifying the angle a obtained in the previous step through the Hessian model, the singular angles e1 and e2 can be determined.
For a given n, Shor's algorithm can successfully find a singular root of N=2^n, where N is a large prime number. This algorithm has been a leader in the field of quantum computing, and there are already multiple algorithms that can solve the same problem in the same time.
2.2.2 Grover’s search algorithm
Grover's search algorithm is an improvement on Shor's algorithm, adding optimizations such as repeated queries and reverse order. Its basic idea is to search for the correct answer to the expected objective function value until it is found. The specific steps are as follows:
Initializing a qubit in a quantum circuit to a uniform superposition state.
Perform Grover iteration k times, each iteration includes the following operations:
- Perform the X operation on the i-th qubit, and then apply the Hadamard gate to all qubits.
- Perform the Z operation on the i-th qubit, and then apply the Hadamard gate to all qubits.
- Perform X operations on the i-th qubit.
Apply the Hadamard gate to the jth qubit, and then apply the X operation to all other qubits. Repeat the above operation k/2 times.
For the i-th qubit of any uniform superposition state, its corresponding Hamiltonian can be expressed as: $$\hat H_{i}=\sum_{x}\left(-\frac{1}{\pi} \right)^{\lvert x \rangle\rvert X_i\otimes I^{\otimes n}-I^{\otimes n}Y_ix\rangle\langle x\rvert,$$
Among them, X_i and Y_i represent the X gate and Y gate on the i-th qubit respectively, I^{\otimes n} represents the identity operator, and X^{\otimes k} represents the X gate acting on the k-th bit.
For the i-th qubit, we can perform a Fourier transform on its above-mentioned Hamiltonian. The amplitude of this transformation corresponds to the probability that the i-th qubit changes from |0> to |1>. Using its transformation, we can estimate the corresponding probability without changing the input state, which is equivalent to applying a single-bit measurement to the bit.
In the above steps, we estimate the possible probabilities corresponding to all i values. So how to find the best objective function based on this string of probabilities? Grover's idea is to select the estimated value that maximizes it as the final result based on the probability distribution of the expected objective function value. For a random variable, its distribution function can be defined as: $$F_{\beta}(x)=P{X\leqslant x}|c_0+c_1\cos(\alpha_0x)+c_2\cos(\alpha_1x)\cdots c_d\cos(\alpha_dx),$$
Among them, α_0, α_1,..., α_d are different frequencies, c_0, c_1,..., c_d are coefficients, $\beta=\frac{c_0}{\sqrt{|\Sigma|}}$ is the standard deviation.
If we can find the values of α_0, α_1,..., α_d that maximize the probability distribution function corresponding to the estimated probability distribution, then this value will correspond to the best estimate of the objective function value θ.
2.2.3 German–Jozsa algorithm
The Deutsch-Jozsa algorithm is used to determine whether a function is a constant function, that is, whether it can obtain an output of 0, 1, or infinite types by performing some operations on the input. Specific steps are as follows:
Initialize two qubits to the 0 state.
Apply the Hadamard gate to the first qubit, and then apply the X gate to all qubits.
Apply the Hadamard gate to the second qubit, and then apply the Z gate to all qubits.
Apply the Hadamard gate to the second qubit, and then apply the X gate to all other qubits.
For the first qubit of any uniform superposition state, its corresponding Hamiltonian can be expressed as: $$\hat H=\frac{1}{4}\sum_{x,y}\left[-I^ {\otimes n}Z^{\otimes m} 1xZY^{\otimes m}_2y+\lvert y\rvert-1\right]\rvert^{m} { n}\rangle\langle xy\rvert.$$
Here, I^{\otimes n} represents the identity operator, Z^{\otimes m}_1 represents the m Z gates that act on the 1st bit, and Y^{\otimes m}_2 represents the 2nd bit. m Y gates on bits. The value of m here is 2, which is equal to the input length of function f(x,y).
When considering the input as x, the corresponding measurement result is xZY(f(x)), observing its probability distribution function, it is equal to 1/4 N (N-1) (N-2) ...*(N-m +1), N is the input set size. This probability represents the probability that when the input is x, the resulting output is 0, 1, or infinite.
Determine whether there is a constant function. For example, if the function f(x)=0 or f(x)=1, then the corresponding result must be 0 or 1, and the result is unique.