Quantum Computing Basics: Understanding Qubits and Superposition

Quantum computing is an emerging technology that leverages the principles of quantum mechanics to perform calculations exponentially faster than classical computers for specific problems. This field has gained significant attention due to its potential to revolutionize various domains, including cryptography, optimization, and simulation. Quantum computers rely on qubits, which are the fundamental units of quantum information, and can exist in multiple states simultaneously, enabling parallel processing.

One of the major challenges in building a large-scale quantum computer is maintaining control over the qubits while minimizing errors caused by decoherence. Decoherence occurs when the qubits interact with their environment, causing loss of quantum coherence and introducing errors into the computation. To mitigate this issue, researchers are developing new techniques for quantum error correction, such as surface codes and concatenated codes.

Quantum computing also has the potential to simulate complex quantum systems, which could lead to breakthroughs in fields like chemistry and materials science. Researchers have used quantum computers to simulate the behavior of molecules and chemical reactions, which could lead to the discovery of new materials with unique properties. Additionally, quantum algorithms are being developed to solve specific problems more efficiently than classical algorithms.

The development of quantum computing is an active area of research, with many different approaches being explored. While significant technical challenges remain to be overcome, the potential rewards are substantial, and researchers are making rapid progress towards realizing the promise of quantum computing. Quantum computing has the potential to revolutionize various domains and solve complex problems that are currently unsolvable with classical computers.

Researchers are working on developing reliable methods for fabricating and characterizing qubits, as well as designing scalable architectures for quantum computing. The development of quantum computing is a multidisciplinary effort, requiring expertise in physics, materials science, computer science, and engineering. As the field continues to advance, we can expect significant breakthroughs and innovations that will bring us closer to realizing the full potential of quantum computing.

What Is Quantum Computing

Quantum computing is a revolutionary technology that leverages the principles of quantum mechanics to perform calculations exponentially faster than classical computers. At its core, quantum computing relies on the concept of qubits, which are the fundamental units of quantum information. Unlike classical bits, which can exist in only two states (0 or 1), qubits can exist in multiple states simultaneously due to a phenomenon known as superposition.

In a superposition state, a qubit can represent not just 0 and 1 but also any linear combination of these values, allowing for an exponential increase in the number of possible states. This property enables quantum computers to process vast amounts of information in parallel, making them particularly useful for complex calculations such as simulating molecular interactions or optimizing complex systems.

Another key feature of qubits is entanglement, which allows two or more qubits to become correlated in such a way that the state of one qubit cannot be described independently of the others. This property enables quantum computers to perform operations on multiple qubits simultaneously, further increasing their computational power.

Quantum computing also relies on the concept of interference, where the phases of different qubits can either reinforce or cancel each other out. By carefully controlling the phases of qubits, quantum computers can perform complex calculations and even correct errors that may occur during computation.

The development of quantum computing has been an active area of research in recent years, with significant advancements made in the creation of stable qubits and the control of their interactions. However, much work remains to be done before quantum computers become practical tools for solving real-world problems.

One of the main challenges facing the development of quantum computing is the fragile nature of qubits, which can easily lose their quantum properties due to interactions with their environment. To overcome this challenge, researchers are exploring new materials and technologies that can help to stabilize qubits and improve their coherence times.

Introduction To Qubits

Qubits are the fundamental units of quantum information, analogous to classical bits in computing. A qubit can exist in multiple states simultaneously, known as a superposition, which allows it to process vast amounts of information in parallel (Nielsen & Chuang, 2010). This property is a result of the principles of quantum mechanics, where particles can exist in multiple energy states at the same time.

In a classical system, a bit can only be in one of two states: 0 or 1. However, a qubit can exist as a linear combination of both 0 and 1, represented by the equation |ψ= α|0+ β|1, where α and β are complex coefficients satisfying the normalization condition |α|^2 + |β|^2 = 1 (Mermin, 2007). This means that a qubit can process multiple possibilities simultaneously, making it an incredibly powerful tool for certain types of computations.

Qubits can also become “entangled,” meaning that their properties are correlated in such a way that the state of one qubit cannot be described independently of the others (Einstein et al., 1935). This phenomenon allows for quantum teleportation and superdense coding, which have potential applications in quantum communication and cryptography.

The no-cloning theorem states that it is impossible to create a perfect copy of an arbitrary qubit (Wootters & Zurek, 1982). This has significant implications for quantum computing, as it means that errors cannot be simply copied out of existence. Instead, sophisticated error correction techniques must be developed to mitigate the effects of decoherence and other sources of noise.

Qubits can be physically realized using a variety of systems, including superconducting circuits, trapped ions, and photons (Ladd et al., 2010). Each of these implementations has its own advantages and challenges, and researchers are actively exploring new architectures and materials to improve the coherence times and scalability of qubit arrays.

The manipulation of qubits is typically achieved through the application of carefully calibrated pulses of electromagnetic radiation, which can induce rotations and entanglement operations (Chow et al., 2012). The precise control of these pulses is crucial for maintaining the fragile quantum states required for reliable computation.

Properties Of Qubits Explained

Qubits are the fundamental units of quantum information, analogous to classical bits in classical computing. A qubit can exist in a superposition of states, meaning it can represent both 0 and 1 simultaneously, unlike classical bits which can only be in one definite state. This property is known as quantum parallelism, allowing a single qubit to process multiple possibilities simultaneously (Nielsen & Chuang, 2010; Mermin, 2007).

The properties of qubits are governed by the principles of quantum mechanics, specifically the Schrödinger equation. The time-evolution of a qubit is described by a unitary operator, which preserves the norm of the state vector (Sakurai & Napolitano, 2011). Qubits can also become entangled, meaning their properties are correlated in such a way that measuring one qubit affects the state of the other. This property is essential for quantum computing and quantum information processing (Bennett et al., 1993).

Qubits have several key characteristics, including <a href=”https://quantumzeitgeist.com/decoherence-impact-on-flying-qubits-a-step-forward-in-quantum-computing/”>coherence, which refers to the ability of a qubit to maintain its quantum state over time. Decoherence occurs when a qubit interacts with its environment, causing it to lose its quantum properties (Zurek, 2003). Qubits also exhibit quantum noise, which is the random fluctuation of their quantum states due to interactions with the environment (Preskill, 1998).

The no-cloning theorem states that it is impossible to create a perfect copy of an arbitrary qubit. This has significant implications for quantum computing and quantum information processing, as it means that quantum information cannot be copied or replicated perfectly (Wootters & Zurek, 1982). Qubits can also be used for quantum teleportation, which allows for the transfer of quantum information from one location to another without physical transport of the qubit itself (Bennett et al., 1993).

Quantum error correction is essential for large-scale quantum computing, as it allows for the correction of errors that occur due to decoherence and other noise sources. Qubits can be encoded in a way that allows for the detection and correction of errors, using techniques such as quantum error correction codes (Gottesman, 1996). This is crucial for maintaining the integrity of quantum information over time.

The manipulation of qubits requires precise control over their quantum states. Quantum gates are the quantum equivalent of logic gates in classical computing, and they allow for the manipulation of qubits to perform specific operations (Barenco et al., 1995). Qubits can be manipulated using a variety of techniques, including microwave pulses and laser light.

Quantum Mechanics Basics

In quantum mechanics, the fundamental unit of information is the qubit, which is a two-state quantum system that can exist in multiple states simultaneously (Nielsen & Chuang, 2010). This property, known as superposition, allows a single qubit to process multiple possibilities simultaneously, making it a powerful tool for quantum computing. The qubit’s state is described by a complex vector in a two-dimensional Hilbert space, which can be represented using the Bloch sphere (Bennett et al., 1993).

The principles of superposition and entanglement are fundamental to quantum mechanics and have been experimentally verified numerous times ( Aspect, 1982; Hensen et al., 2015). In a superposition state, the qubit exists in both states simultaneously, which is represented by a linear combination of the two basis states. This property allows for the creation of quantum gates, which are the building blocks of quantum algorithms (DiVincenzo, 1995).

Quantum entanglement is another fundamental aspect of quantum mechanics, where two or more qubits become correlated in such a way that the state of one qubit cannot be described independently of the others (Einstein et al., 1935). Entangled states are essential for quantum computing and have been experimentally demonstrated using various physical systems, including photons and ions (Bouwmeester et al., 1997; Sackett et al., 2000).

The no-cloning theorem is a fundamental result in quantum mechanics that states that it is impossible to create a perfect copy of an arbitrary qubit state (Wootters & Zurek, 1982). This theorem has far-reaching implications for quantum computing and cryptography. The no-cloning theorem implies that any attempt to measure or copy a qubit state will inevitably introduce errors, which must be corrected using quantum error correction techniques.

Quantum measurement is another fundamental aspect of quantum mechanics, where the act of measurement causes the qubit’s state to collapse from a superposition to one of the basis states (von Neumann, 1932). This process is known as wave function collapse and has been experimentally verified numerous times. The measurement problem in quantum mechanics remains an open question, with various interpretations attempting to explain the nature of wave function collapse.

Quantum computing relies heavily on the principles of superposition, entanglement, and interference (Feynman, 1982). Quantum algorithms, such as Shor’s algorithm for factorization and Grover’s algorithm for search, rely on these principles to achieve exponential speedup over classical algorithms. The development of quantum computing is an active area of research, with various physical systems being explored for their potential to implement large-scale quantum computation.

Understanding Superposition Principle

In quantum mechanics, the superposition principle states that any two or more quantum states can be added together to form another valid quantum state. This means that a qubit, which is the fundamental unit of quantum information, can exist in multiple states simultaneously, such as both 0 and 1 at the same time (Nielsen & Chuang, 2010). The superposition principle is mathematically represented by the wave function, which describes the probability amplitude of each possible state. When a qubit is measured, its wave function collapses to one of the possible states, illustrating the probabilistic nature of quantum mechanics.

The concept of superposition has been experimentally verified in various systems, including photons ( Aspect et al., 1982), electrons (Kleinpoppen & Krause, 1992), and atoms (Monroe et al., 1996). These experiments demonstrate that particles can exist in multiple states simultaneously, which is a fundamental aspect of quantum mechanics. The superposition principle has also been applied to the study of quantum computing, where it is used to perform operations on qubits.

In the context of quantum computing, the superposition principle allows for the creation of quantum gates, which are the building blocks of quantum algorithms (Barenco et al., 1995). Quantum gates operate by manipulating the wave function of a qubit, allowing it to exist in multiple states simultaneously. This enables the performance of operations on multiple bits simultaneously, which is not possible with classical computing.

The superposition principle also has implications for quantum error correction, as it allows for the creation of quantum error-correcting codes (Shor, 1995). These codes work by encoding qubits in a way that allows them to exist in multiple states simultaneously, enabling the detection and correction of errors. This is essential for large-scale quantum computing, where errors can quickly accumulate and destroy the fragile quantum states.

The study of superposition has also led to advances in our understanding of quantum entanglement (Einstein et al., 1935), which is a fundamental aspect of quantum mechanics. Entanglement occurs when two or more particles become correlated in such a way that their properties are dependent on each other, even when separated by large distances.

The superposition principle has been extensively studied and experimentally verified, demonstrating its validity as a fundamental concept in quantum mechanics. Its applications in quantum computing and error correction have the potential to revolutionize our understanding of information processing and storage.

Entanglement And Its Significance

Entanglement is a fundamental concept in quantum mechanics, where two or more particles become correlated in such a way that the state of one particle cannot be described independently of the others (Einstein et al., 1935). This means that measuring the state of one particle will instantaneously affect the state of the other entangled particles, regardless of the distance between them. Entanglement is a key feature of quantum systems and has been experimentally confirmed in various studies ( Aspect, 1982).

The significance of entanglement lies in its potential to revolutionize the way we process information. In classical computing, information is represented as bits, which can have a value of either 0 or 1. However, in quantum computing, information is represented as qubits, which can exist in multiple states simultaneously due to superposition (Nielsen & Chuang, 2010). Entangled particles can be used to create a shared quantum state between two parties, enabling secure communication and quantum cryptography (Bennett et al., 1993).

Entanglement has also been shown to play a crucial role in the study of quantum many-body systems. In these systems, entanglement is responsible for the emergence of complex phenomena such as superconductivity and superfluidity (Laughlin, 2005). Furthermore, entanglement has been proposed as a resource for quantum computing, enabling the creation of robust quantum gates and quantum algorithms (Jozsa & Linden, 2003).

The study of entanglement has also led to a deeper understanding of the foundations of quantum mechanics. The concept of entanglement has been used to test the principles of quantum non-locality and the violation of Bell’s inequalities (Bell, 1964). These experiments have consistently confirmed the predictions of quantum mechanics, demonstrating the reality of entanglement and its implications for our understanding of space and time.

In recent years, entanglement has also been explored in the context of quantum gravity and cosmology. Some theories suggest that entanglement may play a key role in the holographic principle, which proposes that the information contained in a region of spacetime is encoded on its surface (Susskind, 1995). This idea has led to new insights into the nature of black holes and the behavior of matter at very small distances.

The study of entanglement continues to be an active area of research, with ongoing experiments and theoretical work aimed at understanding its properties and applications. As our knowledge of entanglement grows, so too does our appreciation for the strange and counterintuitive world of quantum mechanics.

Quantum Gates And Operations

Quantum gates are the fundamental building blocks of quantum computing, enabling the manipulation of qubits to perform specific operations. A quantum gate is a unitary transformation that acts on one or more qubits, modifying their state in a controlled manner (Nielsen & Chuang, 2010). The most common quantum gates include the Hadamard gate, Pauli-X gate, Pauli-Y gate, and Pauli-Z gate, each with distinct properties and applications.

The Hadamard gate is a fundamental quantum gate that creates a superposition of states in a single qubit. It is defined as H = 1/√2 [1 1; 1 -1], where the matrix elements represent the probability amplitudes for the qubit to be in the |0or |1state (Mermin, 2007). The Hadamard gate is essential for creating a superposition of states, which is a critical resource for quantum computing.

Quantum gates can also be combined to form more complex operations. For example, the CNOT gate is a two-qubit gate that applies an X gate to the target qubit if the control qubit is in the |1state (Barenco et al., 1995). The CNOT gate is a fundamental component of many quantum algorithms and is used extensively in quantum computing.

In addition to single-qubit gates, multi-qubit gates are also essential for quantum computing. These gates enable the manipulation of multiple qubits simultaneously, allowing for more complex operations to be performed (DiVincenzo, 1995). Multi-qubit gates can be constructed using a combination of single-qubit gates and two-qubit gates.

Quantum gate operations must be carefully controlled to maintain the coherence of the qubits. Any errors or decoherence in the quantum gates can lead to a loss of quantum information (Unruh, 1995). Therefore, precise control over the quantum gates is essential for reliable quantum computing.

The implementation of quantum gates and operations relies on various physical systems, such as superconducting circuits, trapped ions, and photons. Each system has its unique characteristics and challenges, requiring careful consideration in the design and implementation of quantum gates (Ladd et al., 2010).

Quantum Circuit Model Overview

The Quantum Circuit Model is a theoretical framework used to describe the behavior of quantum systems, particularly in the context of quantum computing. It provides a mathematical representation of quantum circuits, which are composed of quantum gates and wires that connect them (Nielsen & Chuang, 2010). This model allows researchers to analyze and design quantum algorithms, as well as study the properties of quantum systems.

In the Quantum Circuit Model, quantum information is represented by qubits, which are two-level quantum systems. These qubits can exist in a superposition of states, meaning they can represent both 0 and 1 simultaneously (Bennett et al., 1993). The model also includes quantum gates, which are the quantum equivalent of logic gates in classical computing. These gates perform operations on qubits, such as rotations and entanglement, to manipulate the quantum information.

The Quantum Circuit Model is based on a set of postulates that describe how quantum systems evolve over time (Deutsch, 1989). These postulates include the notion of unitary evolution, which states that quantum systems evolve according to linear, invertible transformations. This means that any operation performed on a qubit can be reversed by applying the inverse operation.

Quantum circuits are composed of multiple quantum gates and wires that connect them. The model allows researchers to analyze the behavior of these circuits by representing them as matrices (Mermin, 2007). These matrices describe how the quantum information is transformed as it passes through the circuit. By analyzing these matrices, researchers can study the properties of quantum systems and design new quantum algorithms.

The Quantum Circuit Model has been used to study a wide range of phenomena in quantum computing, including quantum error correction (Shor, 1995) and quantum simulation (Lloyd, 1996). It provides a powerful tool for understanding the behavior of quantum systems and designing new quantum technologies.

Quantum circuits can be classified into different types based on their structure and function. For example, some circuits are designed to perform specific tasks, such as quantum teleportation or superdense coding (Bennett et al., 1993). Others are designed to simulate complex quantum systems, such as many-body systems (Lloyd, 1996).

Quantum Measurement And Collapse

Quantum measurement and collapse are fundamental concepts in quantum mechanics, describing the process by which a quantum system’s state is determined upon observation. The act of measurement causes the system’s wave function to collapse from a superposition of states to one definite state. This concept was first introduced by Werner Heisenberg in 1927, who proposed that the act of measurement itself causes the wave function to collapse (Heisenberg, 1927).

The mathematical framework for understanding quantum measurement and collapse is based on the Copenhagen interpretation, which posits that a quantum system’s wave function collapses upon interaction with a measuring device. This collapse is described by the Born rule, which relates the probability of finding a system in a particular state to the square of the absolute value of its wave function (Born, 1926). The Copenhagen interpretation has been widely accepted as the standard framework for understanding quantum measurement and collapse.

However, alternative interpretations have also been proposed, such as the Many-Worlds Interpretation, which suggests that the universe splits into multiple branches upon measurement, each corresponding to a different possible outcome (Everett, 1957). Another alternative is the pilot-wave theory, also known as de Broglie-Bohm theory, which posits that particles have definite positions and trajectories, even when not measured (de Broglie, 1926).

Quantum measurement and collapse have been experimentally verified through numerous studies, including the famous double-slit experiment, which demonstrates the wave-particle duality of quantum systems (Davisson & Germer, 1927). More recent experiments have also confirmed the predictions of quantum mechanics regarding measurement and collapse, such as the Quantum Eraser experiment (Kim et al., 2000).

The implications of quantum measurement and collapse are far-reaching, with potential applications in fields such as quantum computing and cryptography. Understanding these concepts is essential for developing new technologies that harness the power of quantum mechanics.

In summary, quantum measurement and collapse are fundamental aspects of quantum mechanics, describing the process by which a quantum system’s state is determined upon observation. The Copenhagen interpretation provides a widely accepted framework for understanding this process, although alternative interpretations have also been proposed.

Quantum Error Correction Techniques

Quantum Error Correction Techniques are essential for the development of reliable quantum computers. One such technique is Quantum Error Correction Codes (QECCs), which can detect and correct errors that occur during quantum computations. QECCs work by encoding a qubit into multiple physical qubits, allowing errors to be detected and corrected through measurements on these physical qubits (Gottesman, 1996). For example, the surface code is a type of QECC that encodes a logical qubit into a two-dimensional array of physical qubits, enabling errors to be detected and corrected through local measurements (Fowler et al., 2012).

Another technique for quantum error correction is Dynamical Decoupling (DD), which aims to suppress decoherence by applying a sequence of pulses to the qubits. DD can effectively reduce the effects of unwanted interactions between qubits and their environment, thereby prolonging the coherence time of the qubits (Viola et al., 1998). This technique has been experimentally demonstrated in various quantum systems, including superconducting qubits (Bylander et al., 2011) and trapped ions (Biercuk et al., 2009).

Quantum error correction techniques can also be applied to protect quantum information during quantum communication. Quantum Key Distribution (QKD) protocols, for example, rely on the principles of quantum mechanics to encode and decode cryptographic keys between two parties. QKD protocols are designed to detect any eavesdropping attempts by an adversary, ensuring secure key exchange over long distances (Bennett et al., 1993). However, errors can still occur during the transmission process, which is where quantum error correction techniques come into play.

One such technique for correcting errors in QKD is the use of entanglement-based quantum error correction codes. These codes work by encoding the quantum information onto entangled particles, allowing errors to be detected and corrected through measurements on these particles (Bennett et al., 1996). This approach has been experimentally demonstrated in various QKD systems, including those based on optical fibers (Hwang, 2003) and free-space optics (Jennewein et al., 2000).

In addition to QECCs and DD, other techniques for quantum error correction include Topological Quantum Error Correction Codes (TQECCs), which encode qubits onto a two-dimensional lattice of physical qubits. TQECCs have been shown to be robust against various types of errors, including bit-flip and phase-flip errors (Dennis et al., 2002). Another technique is the use of concatenated quantum codes, which involve encoding qubits into multiple layers of QECCs (Knill et al., 1998).

The development of robust quantum error correction techniques is crucial for the realization of reliable quantum computers. By combining different techniques and approaches, researchers aim to create a comprehensive framework for protecting quantum information against errors and decoherence.

Quantum Algorithms And Applications

Quantum algorithms are designed to take advantage of the unique properties of quantum mechanics, such as superposition and entanglement, to solve specific problems more efficiently than classical algorithms. One of the most well-known quantum algorithms is Shor’s algorithm, which can factor large numbers exponentially faster than the best known classical algorithm (Shor, 1997). This has significant implications for cryptography, as many encryption schemes rely on the difficulty of factoring large numbers.

Another important quantum algorithm is Grover’s algorithm, which can search an unsorted database of N entries in O(sqrt(N)) time, whereas the best classical algorithm requires O(N) time (Grover, 1996). This has potential applications in areas such as data mining and machine learning. Quantum algorithms have also been developed for solving linear systems of equations (Harrow et al., 2009), simulating quantum systems (Lloyd, 1996), and approximating the Jones polynomial (Aharonov et al., 2007).

Quantum algorithms can be broadly classified into two categories: simulation-based algorithms and optimization-based algorithms. Simulation-based algorithms aim to simulate complex quantum systems, such as many-body systems or chemical reactions, whereas optimization-based algorithms aim to find the optimal solution to a problem by exploiting the principles of quantum mechanics (Bennett et al., 2000). Quantum algorithms have also been developed for solving specific problems in areas such as chemistry and materials science.

One of the key challenges in developing practical quantum algorithms is the need for robust error correction mechanisms. Quantum computers are prone to errors due to the noisy nature of quantum systems, and developing robust methods for correcting these errors is essential for large-scale quantum computing (Gottesman, 1997). Another challenge is the need for efficient quantum control and calibration techniques, as small errors in the control parameters can lead to significant deviations from the desired behavior.

Quantum algorithms have also been developed for solving specific problems in areas such as machine learning and optimization. For example, the Quantum Approximate Optimization Algorithm (QAOA) has been shown to be effective for solving certain types of optimization problems (Farhi et al., 2014). Another example is the Quantum Support Vector Machine (QSVM), which has been shown to be effective for solving certain types of machine learning problems (Rebentrost et al., 2014).

The development of practical quantum algorithms requires a deep understanding of both quantum mechanics and computer science. Researchers in this field must have expertise in areas such as quantum information theory, computational complexity theory, and software engineering.

Future Of Quantum Computing

Quantum computing has the potential to revolutionize various fields, including cryptography, optimization problems, and simulation of complex systems. The future of quantum computing relies heavily on the development of robust and scalable qubits, which are the fundamental units of quantum information. Currently, most qubits are based on superconducting circuits, but other approaches, such as topological quantum computing and ion trap quantum computing, are also being explored (Nielsen & Chuang, 2010; Ladd et al., 2010).

One of the major challenges in building a large-scale quantum computer is maintaining control over the qubits while minimizing errors caused by decoherence. Decoherence occurs when the qubits interact with their environment, causing loss of quantum coherence and introducing errors into the computation (Unruh, 1995; Zurek, 2003). To mitigate this issue, researchers are developing new techniques for quantum error correction, such as surface codes and concatenated codes (Gottesman, 1996; Knill & Laflamme, 1997).

Another area of active research is the development of quantum algorithms that can solve specific problems more efficiently than classical algorithms. One notable example is Shor’s algorithm for factorizing large numbers, which has been shown to be exponentially faster than the best known classical algorithm (Shor, 1994). Other examples include Grover’s algorithm for searching an unsorted database and the Harrow-Hassidim-Lloyd (HHL) algorithm for solving linear systems of equations (Grover, 1996; Harrow et al., 2009).

Quantum computing also has the potential to simulate complex quantum systems, which could lead to breakthroughs in fields such as chemistry and materials science. For example, researchers have used quantum computers to simulate the behavior of molecules and chemical reactions (Aspuru-Guzik et al., 2005; Whitfield et al., 2011). This could lead to the discovery of new materials with unique properties.

In addition to these technical challenges, there are also significant engineering and practical considerations that must be addressed in order to build a large-scale quantum computer. For example, researchers need to develop reliable methods for fabricating and characterizing qubits, as well as designing scalable architectures for quantum computing (Devoret & Schoelkopf, 2013; Monroe et al., 2014).

The development of quantum computing is an active area of research, with many different approaches being explored. While significant technical challenges remain to be overcome, the potential rewards are substantial, and researchers are making rapid progress towards realizing the promise of quantum computing.