On a distant planet, there lived a group of intelligent creatures with supernatural powers. They can teleport objects in space at will, and even predict the future. One day, an earth scientist accidentally traveled to this planet. After some understanding, scientists were surprised to find that the magical power they mastered came from a deep understanding of the laws of motion of microscopic particles in the universe. This law is precisely quantum mechanics.
Quantum mechanics is a branch of physics emerging in the early 20th century, which studies the laws of motion of microscopic particles such as atoms, molecules, and electrons. Its emergence and development have had a profound impact on human understanding of the material world, revealing the principles of many mysterious phenomena in the universe. A central concept of quantum mechanics is the wave function, which describes the state of microscopic particles. The wave function plays a key role in quantum computing, providing humans with a new way of computing.
Next, we introduce the concept of quantum through the double-slit interference experiment of electrons. In the experiment, a beam of electrons was fired onto a screen between two slits. When we observed the results, we found that the electrons formed interference fringes on the screen, which indicated that the electrons exhibited volatility. However, when we try to observe which slit an electron passes through, the interference fringes disappear and the electron behaves like a particle. The experiment revealed a central principle of quantum mechanics: wave-particle duality.
Schrödinger equation
Wave functions play a crucial role in quantum mechanics. The Schrödinger equation is the basic equation describing the wave function, and its form is: Hψ = Eψ.
Among them, H is the Hamiltonian operator, which represents the total energy of the system; ψ is the wave function, which describes the state of the microscopic particle; E is the energy eigenvalue, which represents the energy of the particle.
The Schrödinger equation has the following important properties:
Linearity: If ψ1 and ψ2 are solutions to the equation, then their linear combination is also a solution;
overdetermination: for a given Hamiltonian operator, there may be multiple solutions, which correspond to different energy eigenvalues;
wave function The square of the modulus of represents the probability density, which is the probability of finding a particle at a certain location.
Based on these properties, we can describe and predict the behavior of microscopic particles through wave functions.
First, let's explain in detail what the Schrödinger equation means. The Schrödinger equation is a partial differential equation that describes the evolution of the wave function over time and space. In one dimension, the Schrödinger equation is:
iħ(∂ψ/∂t) = [-ħ²/(2m)(∂²ψ/∂x²) + V(x)ψ]
where i is the imaginary unit, ħ is the reduced Planck constant, m is the mass of the particle, V(x) is the potential energy, t is time, and x is position.
From this equation, it can be seen that the time evolution of the wave function is affected by two factors: one is the kinetic energy of the particle (the second term, which is related to the second-order spatial derivative of the wave function), and the other is the potential energy of the particle (the third term , related to the potential energy function V(x)). When we solve the Schrödinger equation, we need to consider the combined effect of these two factors.
Quantum computers perform calculations using the phenomena of quantum mechanics, the most important of which are superposition and entanglement. These two phenomena endow quantum computers with powerful computing capabilities.
Superposition state : A qubit can be in a superposition state of 0 and 1, that is, ψ = α|0⟩ + β|1⟩, where α and β are complex numbers, satisfying |α|² + |β|² = 1. One qubit can represent two states at the same time, two qubits can represent four states, and so on. Therefore, as the number of qubits increases, the computing power of a quantum computer increases exponentially.
Entanglement : Quantum entanglement is a special phenomenon in quantum mechanics, which refers to the existence of a very close relationship between two or more quantum states. A typical example of an entangled state is the Bell state, which has the form: Φ = (|00⟩ + |11⟩)/√2. When two qubits are entangled, an operation or measurement on one qubit immediately affects the other. Using entanglement, quantum computers can achieve highly parallel computing processes.
To take advantage of these quantum mechanical phenomena, quantum computers need to design special quantum gates to operate. Quantum gates are fundamental elements in quantum computing, similar to logic gates in conventional computers. Commonly used quantum gates include: Hadamard gate (H gate), Pauli-X gate (X gate), Pauli-Y gate (Y gate), Pauli-Z gate (Z gate), CNOT gate, etc. These quantum gates enable various operations on qubits, including the creation of superposition states, the generation of entangled states, and logic operations.
Quantum algorithms are the core application of quantum computers, which use quantum mechanical phenomena and quantum gates to solve practical problems. Here are some well-known quantum algorithms:
Shor's Algorithm : This is a quantum factorization algorithm proposed by Peter Shor in 1994. Shor's algorithm can find the factors of a large integer in polynomial time, far faster than the best algorithms for classical computers. This makes Shor's algorithm a breakthrough application for quantum computing, as it poses a potential threat to the existing RSA encryption system.
Grover's Algorithm : Proposed by Lov Grover in 1996, this is a quantum search algorithm. When looking up target elements in an unordered database, Grover's algorithm is about √N times faster than a classical computer (N is the size of the database). Although this speedup is not as dramatic as Shor's algorithm, Grover's algorithm has broad applications in many search and optimization problems.
Quantum Phase Estimation Algorithm (QPEA) : QPEA is used to estimate the quantum state phase, which is the basic component of many quantum algorithms (such as Shor's algorithm, quantum simulation algorithm). QPEA's ability to estimate the phase with exponential precision gives quantum computers a significant advantage in certain problems.
Despite the enormous potential of quantum computing, there are still many challenges to realizing a usable quantum computer. One big challenge is the coherence of qubits. Qubits are easily disturbed by the external environment, resulting in loss of information. To solve this problem, scientists are working on various error correction methods and quantum protection techniques. In addition, the scalability, control precision and programmability of quantum computers also need breakthroughs.
Application direction
Quantum computing is developed based on the principles of quantum mechanics and wave functions. Traditional computers use bits to represent information, and each bit can only be 0 or 1. Quantum computers, on the other hand, use quantum bits (qubits), which can be in a superposition of 0 and 1. This enables quantum computers to process large amounts of information at the same time, enabling efficient calculations.
The application prospects of quantum computing are very broad. The following are some specific cases:
Quantum cryptography: a communication encryption method based on the principles of quantum mechanics, which can achieve absolutely secure information transmission; the representative theory is quantum teleportation and quantum key distribution Quantum simulation: the use of quantum computers to simulate quantum
systems Behavior, so as to study the problem of quantum mechanics, can be used for in-depth study of natural sciences such as physics and material science.
Optimization problems: Quantum computers can quickly solve complex optimization problems, such as cargo scheduling, energy distribution, etc. The parallelism and high efficiency of quantum computing determine that these directions are feasible.
Artificial intelligence: Quantum computers have great potential in the fields of machine learning and artificial intelligence. At present, quantum machine learning technology has gradually matured.