Quantum volume measures the complexity and power of quantum systems, particularly in quantum computing and simulation. It represents the amount of information that can be processed and stored within a quantum system, considering the number of qubits and their connectivity.
The concept of quantum volume has been explored through various experimental demonstrations, including those using superconducting qubits, trapped ions, and photonic systems. These experiments have shown that increasing the quantum volume of a system can lead to improved performance in tasks such as quantum simulation and metrology. For instance, Google’s 53-qubit Sycamore processor demonstrated a quantum volume of 2^20, showcasing its ability to perform complex calculations beyond the capabilities of classical computers.
The study of quantum volume is an active area of research, with ongoing efforts to develop more efficient methods for calculating it and exploring its connections to other quantum metrics. Additionally, researchers are investigating the relationship between quantum volume and other areas of physics, including quantum gravity and cosmology. As our understanding of quantum volume continues to evolve, it is likely to play an increasingly important role in the development of new quantum technologies and our understanding of complex quantum systems.
Defining Quantum Volume
Quantum Volume is a measure of the largest quantum circuit that can be reliably executed on a quantum computer, taking into account both the number of qubits and the quality of their interactions. It was first introduced by IBM researchers in 2017 as a way to benchmark the performance of different quantum computing architectures . The Quantum Volume metric is based on the idea that a quantum computer’s ability to perform complex calculations depends not only on the number of qubits it has, but also on how well those qubits can interact with each other.
The Quantum Volume metric is calculated by running a series of benchmarking tests on a quantum computer, including randomized benchmarking and quantum Fourier transform (QFT) circuits. These tests are designed to stress different aspects of the quantum computer’s performance, such as its ability to perform single-qubit rotations and two-qubit entangling gates. The results of these tests are then used to calculate a Quantum Volume score, which is a measure of the largest circuit that can be reliably executed on the device .
One of the key advantages of the Quantum Volume metric is that it provides a more comprehensive picture of a quantum computer’s performance than other metrics, such as the number of qubits or the quantum gate fidelity. This is because Quantum Volume takes into account not only the raw computing power of the device but also its ability to perform complex calculations reliably . For example, a quantum computer with a large number of qubits but poor gate fidelity may have a lower Quantum Volume score than a smaller device with higher-quality gates.
Quantum Volume has been widely adopted as a benchmarking metric in the quantum computing community, and is now used by many researchers and companies to evaluate the performance of different quantum computing architectures. For example, IBM uses Quantum Volume to benchmark its own quantum computers, including the 53-qubit quantum processor that it announced in 2019 . Other companies, such as Rigetti Computing and IonQ, also use Quantum Volume to evaluate their own devices.
The widespread adoption of Quantum Volume has helped to drive progress in the development of more powerful and reliable quantum computers. By providing a clear and comprehensive metric for evaluating performance, Quantum Volume has enabled researchers and engineers to focus on improving specific aspects of their devices, such as gate fidelity and qubit coherence times .
Origins Of Quantum Volume Concept
The concept of quantum volume originated in the context of quantum computing, specifically in the study of quantum error correction and fault-tolerant quantum computation. Quantum volume is a measure of the largest number of qubits that can be reliably controlled and manipulated within a quantum computer. It was first introduced by researchers at IBM Research in 2017 as a way to quantify the capabilities of different quantum computing architectures.
The idea behind quantum volume is to provide a single metric that captures the performance of a quantum computer, taking into account both the number of qubits and their quality. Quantum volume is defined as the product of the number of qubits (n), the coherence time (T1), and the gate fidelity (F). This means that even if two quantum computers have the same number of qubits, they can still differ significantly in terms of quantum volume due to differences in coherence times and gate fidelities.
Quantum volume has been used as a benchmark for comparing different quantum computing architectures, including superconducting qubit-based systems, trapped ion systems, and topological quantum computers. For example, researchers have used quantum volume to compare the performance of IBM’s 53-qubit quantum computer with that of Google’s 72-qubit Bristlecone processor.
The importance of quantum volume lies in its ability to provide a clear and concise way to communicate the capabilities of different quantum computing systems. This is particularly important for applications such as quantum simulation, where the accuracy of the computation depends on the quality of the qubits. By providing a single metric that captures both the number of qubits and their quality, quantum volume enables researchers and developers to make informed decisions about which quantum computing architectures are best suited for specific tasks.
Quantum volume has also been used as a tool for guiding the development of new quantum computing technologies. For example, researchers have used quantum volume to identify areas where improvements in coherence times or gate fidelities could lead to significant increases in overall performance.
The concept of quantum volume continues to evolve and be refined by researchers in the field. As quantum computing technology advances, it is likely that new metrics will emerge that provide even more insight into the capabilities and limitations of different quantum computing architectures.
Relationship To Quantum Computing Power
Quantum computing power is closely related to the concept of Quantum Volume, which measures the largest quantum circuit that can be executed on a quantum computer with high fidelity. The Quantum Volume metric takes into account both the number of qubits and the quality of the quantum gates, providing a more comprehensive picture of a quantum computer’s capabilities. According to IBM Research, “Quantum Volume is a measure of the complexity of quantum circuits that can be reliably executed on a quantum processor” (IBM Research, 2020). This statement is supported by a paper published in the journal Physical Review X, which states that “the Quantum Volume metric provides a more complete picture of the capabilities of a quantum computer than other metrics such as the number of qubits or the quantum gate fidelity” (Moll et al., 2018).
The relationship between Quantum Volume and quantum computing power is further elucidated by considering the concept of quantum error correction. As noted in a paper published in the journal Nature, “quantum error correction is essential for large-scale quantum computing, but it requires a significant overhead in terms of qubits and quantum gates” (Gottesman, 2009). This means that a higher Quantum Volume is required to achieve reliable quantum computation with a larger number of qubits. According to a paper published in the journal IEEE Transactions on Information Theory, “the Quantum Volume metric can be used to estimate the resources required for quantum error correction” (Cross et al., 2020).
The importance of Quantum Volume in determining quantum computing power is also highlighted by its relationship with quantum algorithms. As noted in a paper published in the journal Science, “many quantum algorithms require a large number of qubits and high-quality quantum gates to achieve a quantum advantage” (Arute et al., 2019). This means that a higher Quantum Volume is required to run these algorithms reliably. According to a paper published in the journal Physical Review Letters, “the Quantum Volume metric can be used to estimate the performance of quantum algorithms on near-term quantum devices” (Dugas et al., 2020).
In addition to its relationship with quantum error correction and quantum algorithms, Quantum Volume is also related to the concept of quantum noise. As noted in a paper published in the journal Physical Review X, “quantum noise can significantly impact the performance of quantum computers, but it can be mitigated by increasing the Quantum Volume” (Preskill, 2018). This means that a higher Quantum Volume is required to achieve reliable quantum computation in the presence of noise. According to a paper published in the journal Nature Physics, “the Quantum Volume metric can be used to estimate the impact of quantum noise on quantum computing performance” (Aliferis et al., 2006).
The relationship between Quantum Volume and quantum computing power has important implications for the development of large-scale quantum computers. As noted in a paper published in the journal Science, “achieving a high Quantum Volume is essential for demonstrating quantum supremacy and achieving practical quantum computing applications” (Boixo et al., 2018). This means that researchers and developers must focus on increasing the Quantum Volume of their devices to achieve reliable and scalable quantum computation.
The importance of Quantum Volume in determining quantum computing power is also highlighted by its relationship with quantum control. As noted in a paper published in the journal Physical Review Letters, “quantum control is essential for achieving high-fidelity quantum gates and large-scale quantum computation” (Ball et al., 2016). This means that a higher Quantum Volume requires more precise control over the quantum states of the qubits.
Measuring Quantum Volume Effectively
Measuring Quantum Volume Effectively requires a deep understanding of the underlying quantum mechanics principles. The concept of Quantum Volume (QV) is closely related to the notion of quantum entanglement, which is a fundamental aspect of quantum computing. QV measures the amount of entanglement in a quantum system, and it has been shown that a higher QV is essential for achieving reliable quantum computation.
The most common method for measuring Quantum Volume is through the use of randomized benchmarking protocols. These protocols involve applying a series of random quantum gates to a qubit register and then measuring the resulting output. By analyzing the output, researchers can infer the amount of entanglement present in the system, which is directly related to the QV. This method has been widely adopted in the field of quantum computing and has been used to measure the QV of various quantum systems.
Another approach for measuring Quantum Volume involves using machine learning algorithms to analyze the output of a quantum circuit. This method has been shown to be particularly effective for large-scale quantum systems, where traditional methods may become impractical. By training a machine learning model on a dataset of quantum circuits, researchers can learn to recognize patterns in the output that are indicative of high QV.
The importance of measuring Quantum Volume effectively cannot be overstated. As quantum computing continues to advance, it is essential to have reliable methods for characterizing the performance of quantum systems. By accurately measuring QV, researchers can identify areas where improvements need to be made and optimize their systems for better performance.
Recent studies have demonstrated that measuring Quantum Volume can also provide insights into the underlying physics of quantum systems. For example, a study published in Physical Review X used QV measurements to investigate the role of entanglement in quantum many-body systems. The results showed that high QV is essential for achieving reliable quantum computation and provided new insights into the interplay between entanglement and quantum coherence.
Impact On Quantum Error Correction
Quantum error correction is crucial for the development of reliable quantum computing systems. The impact of quantum volume on quantum error correction is significant, as it affects the accuracy and reliability of quantum computations. Quantum volume is a measure of the number of qubits that can be used to perform a computation with low error rates (Cross et al., 2019). As the quantum volume increases, so does the complexity of the computations that can be performed, which in turn requires more sophisticated error correction techniques.
One of the primary challenges in quantum error correction is the fragility of quantum states. Quantum states are prone to decoherence due to interactions with the environment, which causes errors in the computation (Nielsen & Chuang, 2010). To mitigate this, quantum error correction codes such as surface codes and Shor codes have been developed. These codes work by encoding qubits into a larger Hilbert space, allowing for the detection and correction of errors.
However, as the number of qubits increases, so does the complexity of the error correction codes. This is where quantum volume comes in – it provides a measure of the resources required to implement these codes (Gottesman, 1996). For example, surface codes require a large number of physical qubits to achieve high accuracy, which can be challenging to implement with current technology.
Recent advances in quantum error correction have led to the development of more efficient codes such as concatenated codes and topological codes. These codes have been shown to be more robust against errors and require fewer resources (Fowler et al., 2012). However, their implementation still relies on achieving high quantum volume, which is a significant challenge.
In summary, quantum error correction is essential for reliable quantum computing, and quantum volume plays a critical role in determining the accuracy and reliability of these corrections. As research continues to advance in this field, it is likely that new techniques will be developed to improve the efficiency and effectiveness of quantum error correction codes.
The development of robust quantum error correction codes has significant implications for the future of quantum computing. With the ability to perform reliable computations on a large number of qubits, quantum computers can tackle complex problems that are currently unsolvable with classical computers (Aaronson & Arkhipov, 2013). This could lead to breakthroughs in fields such as chemistry and materials science.
Comparison To Other Quantum Metrics
Quantum Volume is often compared to other quantum metrics, such as Quantum Circuit Depth and Quantum Gate Count. However, these metrics are not directly comparable, as they measure different aspects of a quantum computer’s performance. Quantum Circuit Depth measures the number of layers in a quantum circuit, while Quantum Gate Count measures the total number of gates required to implement a quantum algorithm (Nielsen & Chuang, 2010). In contrast, Quantum Volume takes into account both the number of qubits and the quality of the quantum gates, providing a more comprehensive measure of a quantum computer’s capabilities.
Another metric that is sometimes compared to Quantum Volume is the Number of Qubits. However, this metric only provides information about the size of the quantum computer, without taking into account its performance or quality (DiVincenzo, 2000). In contrast, Quantum Volume provides a more nuanced measure of a quantum computer’s capabilities, as it takes into account both the number of qubits and their quality.
Quantum Volume has also been compared to other metrics such as Quantum Entanglement and Quantum Fidelity. However, these metrics are not directly comparable, as they measure different aspects of a quantum system’s behavior (Horodecki et al., 2009). Quantum Entanglement measures the degree of correlation between two or more qubits, while Quantum Fidelity measures the accuracy with which a quantum gate can be implemented. In contrast, Quantum Volume provides a more comprehensive measure of a quantum computer’s capabilities.
In terms of its relationship to other quantum metrics, Quantum Volume has been shown to be closely related to Quantum Circuit Depth and Quantum Gate Count (Atos & Williams, 2018). This is because all three metrics are sensitive to the quality of the quantum gates and the number of qubits. However, Quantum Volume provides a more comprehensive measure of a quantum computer’s capabilities, as it takes into account both the number of qubits and their quality.
Quantum Volume has also been compared to classical metrics such as Floating-Point Operations Per Second (FLOPS) and Million Instructions Per Second (MIPS). However, these metrics are not directly comparable, as they measure different aspects of a computer’s performance (Hennessey & Patterson, 2019). FLOPS measures the number of floating-point operations that can be performed per second, while MIPS measures the number of instructions that can be executed per second. In contrast, Quantum Volume provides a more nuanced measure of a quantum computer’s capabilities.
Quantum Volume has been recognized as an important metric for evaluating the performance of quantum computers (Cross et al., 2019). This is because it takes into account both the number of qubits and their quality, providing a comprehensive measure of a quantum computer’s capabilities. As such, Quantum Volume is likely to play an increasingly important role in the development and evaluation of quantum computing technologies.
Role In Quantum Circuit Optimization
Quantum Circuit Optimization is a crucial step in the development of quantum computing, as it enables the efficient execution of quantum algorithms on noisy intermediate-scale quantum (NISQ) devices. The goal of Quantum Circuit Optimization is to minimize the number of quantum gates required to implement a specific quantum algorithm, thereby reducing the overall error rate and improving the reliability of the computation.
One key approach to Quantum Circuit Optimization is the use of quantum circuit transpilation, which involves transforming a given quantum circuit into an equivalent but more efficient form. This can be achieved through various techniques, such as gate merging, gate cancellation, and circuit rewriting. For example, a study published in the journal Physical Review X demonstrated that quantum circuit transpilation can reduce the number of gates required to implement the Quantum Approximate Optimization Algorithm (QAOA) by up to 50% . Another approach is the use of machine learning algorithms to optimize quantum circuits, as demonstrated in a paper published in the journal Nature Physics .
Quantum Circuit Optimization also involves the optimization of specific quantum circuit components, such as quantum error correction codes and quantum gates. For instance, researchers have developed optimized versions of the surface code, a popular quantum error correction code, which can reduce the number of physical qubits required to implement the code by up to 30% . Similarly, studies have shown that optimized quantum gate implementations can reduce the error rate of quantum computations by up to an order of magnitude .
The optimization of quantum circuits is also closely related to the concept of Quantum Volume, which characterizes the computational power of a quantum computer. Specifically, Quantum Volume takes into account both the number of qubits and the quality of the quantum gates in a given quantum circuit. As such, optimizing quantum circuits for Quantum Volume can lead to improved overall performance and reliability .
In addition to these approaches, researchers have also explored the use of automated tools and software frameworks to optimize quantum circuits. For example, the Qiskit framework developed by IBM provides a range of tools and techniques for optimizing quantum circuits, including transpilation, gate optimization, and circuit rewriting . Similarly, the Cirq framework developed by Google provides a range of tools and libraries for optimizing quantum circuits, including automated gate optimization and circuit rewriting .
Overall, Quantum Circuit Optimization is a critical component of quantum computing research, as it enables the efficient execution of quantum algorithms on NISQ devices. By leveraging various techniques and approaches, researchers can optimize quantum circuits to reduce error rates, improve reliability, and increase overall computational power.
Connection To Quantum Noise Reduction
Quantum noise reduction is a crucial aspect of quantum computing, as it directly impacts the accuracy and reliability of quantum operations. Quantum noise refers to the random fluctuations in the quantum states of qubits, which can cause errors in quantum computations. Reducing quantum noise is essential to achieve high-fidelity quantum gates and maintain the coherence of qubits.
One approach to reducing quantum noise is through the use of quantum error correction codes. These codes work by encoding quantum information in a highly entangled state, which allows for the detection and correction of errors caused by quantum noise. For example, the surface code is a popular quantum error correction code that has been shown to be effective in reducing quantum noise in superconducting qubits (Gottesman, 1997; Fowler et al., 2012). Another approach is through the use of dynamical decoupling techniques, which involve applying sequences of pulses to qubits to suppress the effects of quantum noise (Viola & Lloyd, 1998; Uhrig, 2007).
Quantum noise reduction can also be achieved through the optimization of quantum control pulses. By carefully designing the shape and duration of control pulses, it is possible to minimize the impact of quantum noise on qubit dynamics. For example, optimal control theory has been used to design pulses that reduce the effects of quantum noise in ion trap qubits (Doria et al., 2011). Additionally, machine learning algorithms have been applied to optimize quantum control pulses for noise reduction (Kelly et al., 2014).
The connection between quantum noise reduction and quantum volume is critical. Quantum volume is a measure of the number of qubits that can be controlled with high fidelity, and reducing quantum noise is essential to achieving high quantum volume. By minimizing the effects of quantum noise, it is possible to increase the coherence times of qubits, allowing for more complex quantum operations to be performed (Cross et al., 2019). Furthermore, reducing quantum noise can also improve the accuracy of quantum error correction codes, which are essential for large-scale quantum computing.
In summary, quantum noise reduction is a critical aspect of quantum computing, and various approaches have been developed to minimize its effects. By optimizing quantum control pulses, using quantum error correction codes, and employing dynamical decoupling techniques, it is possible to reduce quantum noise and increase the coherence times of qubits. This, in turn, can improve the accuracy of quantum operations and increase the achievable quantum volume.
The development of robust methods for reducing quantum noise will be essential for the realization of large-scale quantum computing. As quantum systems become increasingly complex, the effects of quantum noise will become more pronounced, making it essential to develop effective strategies for mitigating its impact (Preskill, 2018). By continuing to advance our understanding of quantum noise and developing innovative methods for reducing its effects, we can move closer to realizing the full potential of quantum computing.
Implications For Quantum Algorithm Design
Quantum algorithm design is heavily influenced by the concept of quantum volume, which quantifies the largest quantum circuit that can be reliably executed on a given quantum computing device. The implications for quantum algorithm design are significant, as it highlights the need to optimize algorithms for specific hardware architectures (Nielsen & Chuang, 2010). This means that quantum algorithm designers must consider the limitations of the underlying hardware when developing new algorithms.
One key implication is that quantum algorithms should be designed with a focus on reducing the number of two-qubit gates required, as these are typically the most error-prone operations in current quantum computing architectures (Gottesman, 1997). This has led to the development of new quantum algorithms and techniques, such as quantum teleportation-based circuits, which aim to minimize the number of two-qubit gates required (Bennett et al., 1993).
Another important consideration is the need for robustness against noise and errors in quantum computations. Quantum algorithm designers must develop methods to mitigate the effects of decoherence and other sources of error, such as quantum error correction codes (Shor, 1995). This has led to significant advances in the development of fault-tolerant quantum computing architectures.
The concept of quantum volume also highlights the importance of considering the trade-offs between different quantum algorithm design parameters, such as circuit depth, width, and number of qubits required. Quantum algorithm designers must carefully balance these competing demands to develop algorithms that are both efficient and reliable (Barenco et al., 1995).
Furthermore, the study of quantum volume has also led to a greater understanding of the importance of quantum control and calibration in achieving high-fidelity quantum computations. This has significant implications for the development of practical quantum computing architectures, as it highlights the need for advanced control systems and calibration techniques (Koch et al., 2016).
In summary, the concept of quantum volume has far-reaching implications for quantum algorithm design, highlighting the need to optimize algorithms for specific hardware architectures, reduce two-qubit gate counts, develop robust methods against noise and errors, balance competing design parameters, and prioritize quantum control and calibration.
Quantum Volume And Scalability Tradeoffs
Quantum Volume is a measure of the largest quantum circuit that can be executed on a quantum computer with high fidelity, taking into account both the number of qubits and the quality of their interactions. It was introduced by IBM researchers in 2017 as a way to quantify the power of a quantum computer . The Quantum Volume metric is defined as the product of the number of qubits, the coherence time, and the gate fidelity, which measures how accurately quantum gates can be applied.
The scalability tradeoff refers to the challenge of increasing the number of qubits in a quantum computer while maintaining control over their interactions. As the number of qubits increases, the complexity of the system grows exponentially, making it harder to maintain coherence and control . This is because each additional qubit introduces new sources of noise and error, which can quickly overwhelm the system’s ability to correct them.
One key challenge in scaling up quantum computers is the need for high-fidelity quantum gates. Quantum gates are the basic operations that manipulate qubits, but they are prone to errors due to decoherence and other sources of noise . To achieve high fidelity, researchers must carefully optimize the design of the quantum gates, taking into account factors such as pulse shaping and calibration.
Another challenge is the need for robust error correction mechanisms. As the number of qubits increases, so does the likelihood of errors occurring during computation. Quantum error correction codes can help mitigate this problem, but they require a significant overhead in terms of additional qubits and control systems .
Researchers are actively exploring new architectures and technologies to overcome these scalability tradeoffs. For example, topological quantum computing uses exotic materials called topological insulators to create robust quantum bits that are less prone to decoherence . Other approaches include the use of superconducting qubits with built-in error correction mechanisms or the development of novel quantum algorithms that can tolerate higher levels of noise.
The Quantum Volume metric provides a useful framework for evaluating the performance of different quantum computing architectures and identifying areas where further research is needed. By understanding the scalability tradeoffs involved in building large-scale quantum computers, researchers can develop more effective strategies for overcoming these challenges and realizing the full potential of quantum computing.
Experimental Demonstrations Of Quantum Volume
Experimental demonstrations of quantum volume have been conducted in various systems, including superconducting qubits, trapped ions, and photonic systems. One such demonstration was performed by the group of John Preskill at Caltech, who used a 53-qubit superconducting quantum processor to demonstrate a quantum volume of 2^53 (Arute et al., 2019). This experiment showcased the ability to control and manipulate a large number of qubits, which is essential for demonstrating quantum supremacy.
Another experimental demonstration of quantum volume was performed by the group of Christopher Monroe at the University of Maryland, who used a trapped-ion quantum computer to demonstrate a quantum volume of 2^20 (Wright et al., 2019). This experiment highlighted the ability to control and manipulate individual ions in a large-scale quantum system. The results from these experiments have been verified by multiple independent sources, including theoretical simulations and experimental measurements.
Theoretical models have also been developed to describe the behavior of quantum systems with large volumes. For example, the concept of “quantum volume” was introduced by physicists at Google AI Lab, who used a mathematical framework to describe the scaling behavior of quantum systems (Bremner et al., 2016). This work provided a theoretical foundation for understanding the behavior of quantum systems with large volumes.
Experimental demonstrations of quantum volume have also been performed in photonic systems. For example, the group of Jian-Wei Pan at the University of Science and Technology of China used a photonic quantum computer to demonstrate a quantum volume of 2^14 (Zhong et al., 2019). This experiment showcased the ability to control and manipulate photons in a large-scale quantum system.
The results from these experimental demonstrations have been verified by multiple independent sources, including theoretical simulations and experimental measurements. These experiments have provided strong evidence for the concept of quantum volume and its relevance to the study of quantum systems.
Future Directions For Quantum Volume Research
Quantum volume research has made significant progress in recent years, with various studies exploring its applications in quantum computing, simulation, and metrology. One of the future directions for quantum volume research is to investigate its relationship with other quantum metrics, such as quantum entanglement and quantum discord. This line of inquiry aims to better understand how different quantum properties interact and influence one another (Bennett et al., 1996; Horodecki et al., 2009).
Another area of focus for future research is the development of more efficient methods for calculating quantum volume. Currently, most approaches rely on numerical simulations or approximations, which can be computationally intensive and limited in their accuracy. Researchers are exploring alternative techniques, such as machine learning algorithms and analytical models, to improve the efficiency and precision of quantum volume calculations (Liu et al., 2020; Wang et al., 2019).
The study of quantum volume also has implications for our understanding of quantum many-body systems and phase transitions. By analyzing the behavior of quantum volume in different regimes, researchers can gain insights into the underlying physics of these complex systems and identify potential applications in fields such as materials science and condensed matter physics (Sachdev, 2011; Hastings et al., 2007).
Furthermore, research on quantum volume has the potential to inform the development of new quantum technologies, such as quantum sensors and quantum communication devices. By understanding how quantum volume affects the performance of these systems, researchers can design more efficient and robust protocols for quantum information processing (Giovannetti et al., 2011; Degenhardt et al., 2020).
In addition to these areas, future research on quantum volume may also explore its connections to other fields, such as quantum gravity and cosmology. By investigating the behavior of quantum volume in different gravitational and cosmological contexts, researchers can gain a deeper understanding of the interplay between quantum mechanics and general relativity (Ashtekar et al., 2015; Rovelli, 2004).
Theoretical models of quantum volume have also been proposed to study its behavior in various systems, including spin chains and fermionic systems. These models provide a framework for understanding the underlying physics of quantum volume and can be used to make predictions about its behavior in different regimes (Vidal et al., 2003; Hastings et al., 2010).
