What is a Quantum Computer? A Beginner’s Guide for Hack

作者：禅与计算机程序设计艺术

1.简介

A quantum computer (QC) is a type of computer that utilizes quantum mechanical properties to perform calculations at unprecedented speeds and in unexpected ways. These new computational paradigms have the potential to revolutionize various industries such as finance, healthcare, energy industry, defense, transportation, etc. Quantum computers are widely believed to possess exceptional computational power compared to traditional computers based on transistors. The basic concept behind these advances lie in the fact that they allow for computation using quanutm logic gates or elements rather than classical bits or transistors. Quatum mechanics provides the physical laws governing how electrons behave when interact with one another, which enables quantum computing technologies to be created. This article will provide an overview of quantum computing technologies through a beginners guide for hackers approach. We will cover key concepts, algorithms, operations, code examples, future trends and challenges.

The target audience of this article are software developers who want to understand the basics of quantum computing technology and its applications.

This article assumes readers have some experience working with programming languages like Python or C++. If you're completely new to programming, we recommend starting with introductory courses before moving forward with this article.

We assume readers have knowledge of elementary linear algebra terminology including vectors and matrices, and some familiarity with differential equations and physics. It also helps if readers are familiar with probability theory and statistical mechanics.

To follow along with this article, you'll need access to a quantum simulator or hardware-based QC device. You can use the IBM Q Experience, an open source project that allows users to simulate and execute their programs on real quantum computers over the cloud platform. Alternatively, you could purchase a real quantum computer from your local tech company or online vendor.

2.概览 In this article, we will discuss the following topics:

• Introduction to quantum computing
• Basic concepts, terms and definitions
• Core algorithmic principles and techniques
• Specific implementation steps and math formulas
• Code example(s), explanation(s) and benchmarking results

Before diving into each topic, let's quickly review what makes up a quantum computer. We can break down a quantum computer into two main components: the quantum circuit, and the quantum processor. The quantum circuit is where quantum gates operate together to manipulate and transform qubits—our base unit of information in quantum computing. The quantum processor takes instructions from the quantum circuit and executes them using quantum algorithms to process data. Each component plays a critical role in making possible the unique capabilities offered by a quantum computer.

3.1 Introduction to quantum computing Let us start our discussion on quantum computing by understanding its history and origins. In the early days of quantum computing research, physicists were interested in creating machines capable of processing complex mathematical functions without relying on conventional digital circuits. One way to do so was by considering how quantum mechanics operates. Here's how it works:

1. Two identical systems separated by large distances create entanglement, meaning they share quantum states despite being far apart. When combined, the two systems become more powerful and able to affect each other indirectly, leading to exponentially faster processing rates.

2. The ability to control individual quantum particles through interactions with other electrons leads to emergent behavior exhibited by certain classes of problems, such as superposition and entanglement. For instance, entangled qubits can be used to create shared memory between different parts of a program running simultaneously. Another advantage of using entangled qubits is that they can be manipulated individually but retain their interconnection throughout the system, allowing for greater flexibility and scalability in quantum computing systems.

3. Finally, because qubits can act as independent variables in quantum mechanics, quantum computations are generally understood to depend only weakly on their initial state and subsequent measurements, leading to massively parallel processing architectures with very high throughput capacity.

Now, let's dive deeper into quantum computing and its defining features.

4.2 Basic concepts, terms and definitions There are several fundamental concepts involved in quantum computing, including quantum logic gate, quantum measurement, and quantum information. Let's briefly define each term.

Qubit (quantum bit): A quantum bit is the smallest unit of quantum information and is equivalent to a standard binary digit or logical value. They consist of two quantum states called |0⟩ and |1⟩, which represent the two competing possibilities that a single qubit can take on. By performing quantum operations on these states, we can convert them to alternate values, thereby encoding additional information within them.

Quantum logic gate: Quantum logic gates are universal devices that apply transformations to quantum states to encode and manipulate information. There are three types of quantum logic gates: Hadamard gate, CNOT gate, and U3 gate. The Hadamard gate creates an equal superposition of all possible input combinations, while the CNOT gate performs a conditional flip operation depending on whether a particular input condition is true. Similarly, the U3 gate applies rotations around the x, y, and z axes to the input qubit, with three adjustable parameters controlling the angle of rotation.

Measurement: Measuring a quantum state involves converting it into either a classical zero or one based on the outcome of a random event occurring during the interaction between the particle and its environment. In other words, measuring a qubit causes it to collapse into a single definite state, regardless of its intrinsic spin and/or position relative to others. Since quantum computations are probabilistic, multiple measurements may lead to different outcomes. Therefore, quantum error correction techniques play an important role in ensuring reliable results across different runs of the same program.

Entanglement: Entanglement refers to a property in quantum mechanics where multiple quantum systems sharing entangled particles are transformed into a joint state once measured. This means that even though the original subsystems are not directly connected, any operation performed on one subsystem affects both subsystems and vice versa. This property can be exploited in quantum computing by coupling separate subsystems that require communication or coordination.

Superposition: Superposition refers to the property where a quantum system exists in a combination of all possible quantum states, resulting in an arbitrary phase relationship among its constituent subatomic particles. This gives rise to the idea of amplitude amplification, which uses repeated measurements to extract meaningful information from a highly mixed quantum state.

Quantum machine learning: Quantum machine learning combines quantum computing with classical machine learning algorithms to train models on data stored in quantum systems. Advantages of this approach include enhanced accuracy due to quantum correlations between samples, lower costs due to decreasing resource requirements, and increased scalability via parallelization and fault tolerance. However, this requires sophisticated optimization algorithms and advanced modeling techniques to handle the exponential growth in sample size required to train large deep neural networks.

5.3 Core algorithmic principles and techniques Now that we have a better understanding of the core concepts and architecture of a quantum computer, let's move on to discussing the specific algorithmic principles and techniques.

5.3.1 Quantum Fourier Transform The quantum Fourier transform (QFT) is a well known algorithm used to map quantum states to classical ones. It is based on the Shor's decomposition method, which allows us to factorize polynomials into prime factors. The QFT maps an n-qubit input state to an n/2-qubit output state where the coefficients of the input and output states satisfy the relations:

|j> = |0...0> + (-1)^j * |1...0> * sqrt(1/2^(n/2))

where j ranges from 0 to n-1. The inverse QFT maps an n/2-qubit input state back to an n-qubit output state satisfying the relation:

|i> = SUM_j [w^(n)(-1)ij] * |j>, w = exp(-2 pi i/2^n).

These relations give us insight into the mathematics underlying quantum computing and how it achieves exponential scaling beyond traditional digital circuits.

5.3.2 Quantum Phase Estimation Algorithm One of the most useful quantum algorithms is quantum phase estimation (QPE). This algorithm allows us to estimate the eigenvalue of a unitary operator given an oracle implementing a controlled unitary transformation. To achieve this task efficiently, we first select a suitable parameterized ansatz wavefunction that has relatively few nonzero coefficients, then measure its overlap with the desired eigenvector using the projection operator P. Then we repeat the procedure with reduced basis sizes until convergence. Once we know the number of iterations needed, we can obtain the approximate value of the eigenvalue by applying continued fractions.

Here is a brief description of the entire QPE algorithm flowchart:

1. Initialize an eigenvector v and find its eigenvalue lambda via brute force search or the QPE algorithm.
2. Implement a quantum oracle O_lambda on the system state psi = vpsi.
3. Measure the overlap between psi and vpsi under the projection operator P.
4. Repeat step 2 and 3 with smaller and smaller basis sets until convergence or maximum iteration limit is reached.
5. Use the converged result to calculate the continued fraction expansion of lambdainitia / i and recover the decimal approximation of the eigenvalue.

Note that QPE requires us to choose a suitable parameterized ansatz wavefunction with low degree of non-identity terms to reduce the computational complexity of the problem, otherwise it becomes prohibitively expensive to evaluate the expectation value of the projection operator under the oracle.

5.3.3 Grover's Search Algorithm Grover's search algorithm is a well-known quantum algorithm designed to solve database searches of unknown structure and content. The basic principle behind this algorithm is to focus on those items whose membership in a collection satisfies a particular pattern, rather than scanning the entire database in sequence. More specifically, the algorithm starts with an initially empty database D and a query function f, representing the criteria for a valid solution. Next, we randomly select a subset S of D consisting of k items matching the query function. We set S as the input for the oracle function O, which simulates the necessary pre-processing steps to identify a relevant item within the set. After that, we initialize the search space S' as all elements in D except for S, since we cannot modify the database after initialization. Finally, we repeatedly apply the Grover iterative searching algorithm, repeating the following actions:

1. Apply O to S'.
2. Construct a marked version of S', marking all occurrences of the relevant item in S with an auxiliary qubit.
3. Reflect about the center of the marked region to cancel out errors introduced by the reflection.
4. Unmark all positions that did not contain the relevant item.
5. Set S as the intersection of the current search space S' and the marked version of S.

Once we have identified the relevant item within S, we terminate the search and return its index as the answer. Although the time complexity of this algorithm depends on the number of elements in D and the number of times we repeat the iteration loop, it is still polynomial in the total number of possible solutions found, enabling efficient execution of large databases. Overall, grover's search algorithm demonstrates the potential for quantum computing to offer significant improvements in search performance and scale capabilities beyond classical algorithms.

5.3.4 Quantum Amplitude Amplification Quantum amplitude amplification (QAA) is a technique for solving optimization problems with quadratic constraints. The basic idea is to prepare two copies of a state and ask the participant to optimize one copy by applying a fixed number of operations. Then, we feed the optimized state into a detector that measures the deviation from the optimal objective function. Finally, we iterate the above cycle until convergence is achieved, yielding an approximate solution to the optimization problem.

Here is a brief summary of the overall QAA algorithm:

1. Choose a variational ansatz wavefunction that produces a state with good overlap with the target solution and low coherence between the copies.
2. Generate a random guess for the amplitude vector θ containing the amplitudes of the copies.
3. Prepare the two copies of the wavefunction using the amplitude vector θ.
4. Participate in the optimization process by applying fixed number of operations to one of the copies.
5. Send the second copy to a detector that measures the deviation from the optimal objective function.
6. Update the amplitude vector θ according to the deviation obtained in step 5, obtaining a refined estimate of the optimal solution.
7. Iterate the above steps until convergence is achieved.

Note that the choice of the variational ansatz wavefunction and the number of iterations required for convergence typically involve tradeoffs between computational efficiency and effectiveness of the final result.

5.3.5 Quantum Teleportation Protocol Quantum teleportation is a technique for transmitting quantum information through classical communication channels. It is commonly applied to send a quantum signal between distant locations over long distance links without losing quantum entanglement. The protocol involves the following four stages:

1. Alice wants to transfer her quantum state q to Bob. She generates a pair of entangled photons E and E_prime, both pointing towards her own entangled state q. Her message m is encrypted using her secret key K and sent to Bob.

2. Bob receives the pair of entangled photons, unscrambles them to obtain the message m, and reconstructs his own entangled state r by interacting the received photons with the identity operator I. He sends back the measurement outcomes of the received photons and m.

3. Alice decrypts the message m using her secret key K and obtains the entangled state q_hat. She uses the correlation of the incoming photons to estimate the entangled state q and applies a phase correction accordingly.

4. Alice continues her transmission by sending her entangled state q_hat alongside the previously transmitted pair of entangled photons to Bob. Bob calculates the difference between his own entangled state r and q_hat and applies a correction based on the previous measurement outcomes to obtain the corrected entangled state r_tilde.

Overall, quantum teleportation demonstrated the potential for quantum computing to enable secure and private communications over long distances with minimal loss of quantum entanglement.

5.3.6 Quantum Key Distribution Protocol Quantum key distribution protocols (QKDPs) help establish a shared quantum key between parties in a distributed quantum network. The simplest protocol consists of five phases:

1. Authenticator (AKA Base station): A trusted third party conducts the authentication procedure to ensure that the participants are authorized to participate in the key exchange.

2. Preparation: The user initiates the QKDP session by generating a public-private keypair (PK, SK). The PK is transmitted to the receiver via a channel secured by authenticated encryption.

3. Key generation: The sender of the key selects a random string of messages {m1,..., mk} and encodes them into entangled states using PK. He sends the resulting pairs of entangled states {Eik,...} to the authenticator.

4. Authentication: The authenticator verifies that the receivers of the keys correctly encoded their respective messages into corresponding entangled states and collects all the pairs of entangled states.

5. Key dissemination: The authenticator computes the common entropy H using the collected entangled states, encrypts it with SK, and distributes it to the recipients of the keys. Recipients of the keys generate their own random number Rk, compute Kr = E(Rk, H)/E(SK, H), and decode the received messages into plaintexts.

Overall, quantum key distribution protocols demonstrate the potential for quantum computing to enable secure and private key exchange in distributed quantum networks.

5.3.7 Quantum Error Correction Quantum error correction (QEC) is essential for securing quantum communication systems against errors generated by noise or imperfect quantum processes. The goal of QEC is to make sure that no part of the quantum communication gets corrupted due to errors in intermediate steps. Several methods of QEC exist, including parity check codes, linear codes, and topological codes. Parity check codes work by checking the parity of every individual bit in the message to determine the correctness of the whole message. Linear codes use linear combinations of the bits to detect and correct errors. Topological codes use local symmetries and global decoding rules to minimize the number of errors detected.

6.4 Future Trends and Challenges With the advent of quantum computers, we have entered a new era of quantum computing. However, it is worth noting that there are many challenges ahead as well. Some of the major challenges and opportunities facing quantum computing are:

1. Scalability challenge: With the development of quantum computers that process hundreds of qubits at once, companies are expecting to build ever larger quantum computers. Thus, scaling up existing quantum algorithms poses an increasing challenge.

2. Classical hardware acceleration challenge: While quantum processors have already been developed and deployed, quantum computing still faces a bottleneck due to slow theoretical scaling limits imposed by the constraints of classical hardware design. As a result, quantum computing researchers are exploring alternative approaches to address this issue, such as accelerating classical simulations using quantum annealing techniques and specialized hardware chips.

3. Application diversity challenge: Despite the promise of quantum computing, practical applications of quantum technologies are still limited. Different industries and businesses, particularly those involved in financial services, health care, energy, and security, are finding it difficult to come up with practical use cases for quantum technologies.

4. Regulatory compliance challenge: In order to regulate quantum technologies effectively and responsibly, governments must invest heavily in education, policy development, and enforcement mechanisms. Current regulations addressing quantum technologies tend to be fairly conservative and focused on technical specifications and verification efforts. As a result, compliance efforts remain an ongoing challenge for government agencies seeking to promote safe, privacy-preserving, and trustworthy quantum technologies.