Quantum computing and classical computing differ fundamentally in their logic gates, with quantum gates being reversible and operating on quantum bits (qubits), which can exist in multiple states simultaneously, while classical gates are irreversible and operate on classical bits, which can only exist in two states, 0 or 1.

## Computational Complexity: Quantum Advantage

One of the key differences between quantum and classical gates is their computational complexity. Quantum gates can perform certain tasks, such as factoring large numbers, much faster than classical gates. For example, Shor’s algorithm, a quantum algorithm, can factor large numbers in polynomial time, while the best-known classical algorithm, the general number field sieve, requires sub-exponential time.

The computational advantage of quantum gates can be quantified using various measures. One such measure is the “quantum volume,” which was introduced by Bravyi et al. (2016). The quantum volume is a metric that takes into account the number of qubits in a quantum system and the amount of entanglement in the system. The study by Bravyi et al. showed that the computational power of a quantum system increases with the amount of entanglement in the system.

### Factoring Large Numbers

One of the most well-known examples of the computational advantage of quantum gates is their ability to factor large numbers. Shor’s algorithm, a quantum algorithm developed by Peter Shor in 1994, can factor large numbers in polynomial time, which is a significant improvement over the best-known classical algorithms.

The time complexity of Shor’s algorithm is O(log^3 N), where N is the number to be factored. In contrast, the best-known classical algorithm, the general number field sieve, has a time complexity of O(exp(c(log N)^(1/3)(log log N)^(2/3))), where c is a constant.

This means that as the size of the number to be factored increases, the computational advantage of Shor’s algorithm becomes more and more pronounced. For example, factoring a 2048-bit number using Shor’s algorithm would take approximately 2.2 seconds on a quantum computer with 4,000 qubits, while the general number field sieve would take approximately 300 years on the world’s fastest classical supercomputer.

### Quantum Supremacy

The concept of “quantum supremacy” refers to the point at which a quantum computer can perform a specific task that is beyond the capabilities of the world’s most powerful classical computers. In 2019, Google’s Sycamore quantum processor demonstrated quantum supremacy by performing a specific task, known as random circuit sampling, in 200 seconds, while the best classical supercomputer would have taken approximately 10,000 years to perform the same task.

This demonstration of quantum supremacy was a significant milestone in the development of quantum computing, as it showed that quantum computers can outperform classical computers on certain tasks. However, it’s important to note that quantum supremacy is a narrow and specific concept, and there are still many tasks that classical computers can perform more efficiently than current quantum computers.

## Quantum Entanglement: A Unique Property

Another key difference between quantum and classical gates is the concept of quantum entanglement. Entanglement is a unique property of quantum systems, where the state of one particle cannot be described independently of the state of another particle, even if they are separated by large distances.

Entanglement is a crucial resource in quantum computing, as it allows for the creation of quantum states that cannot be efficiently simulated on classical computers. The amount of entanglement in a quantum system can be measured using various entanglement measures, such as entanglement entropy or concurrence.

### Entanglement Entropy

Entanglement entropy is a measure of the amount of entanglement in a quantum system. It is defined as the von Neumann entropy of the reduced density matrix of a subsystem of the larger quantum system. The entanglement entropy of a subsystem A is given by:

S_A = -Tr(ρ_A log ρ_A)

where ρ_A is the reduced density matrix of subsystem A.

Entanglement entropy can be used to quantify the amount of entanglement in a quantum system and to compare the computational power of quantum and classical systems. For example, a study by Vidal (2000) showed that the entanglement entropy of a quantum system increases with the number of qubits, which suggests that the computational power of a quantum system also increases with the number of qubits.

### Concurrence

Another measure of entanglement is concurrence, which is defined for a two-qubit system as:

C = max{0, λ_1 – λ_2 – λ_3 – λ_4}

where λ_i are the square roots of the eigenvalues of the matrix ρ(σ_y ⊗ σ_y)ρ^*(σ_y ⊗ σ_y), and ρ is the density matrix of the two-qubit system.

Concurrence ranges from 0 (for a separable, or non-entangled, state) to 1 (for a maximally entangled state). Like entanglement entropy, concurrence can be used to quantify the amount of entanglement in a quantum system and to compare the computational power of quantum and classical systems.

## Reversibility and Quantum Circuits

Another key difference between quantum and classical gates is the concept of reversibility. Quantum gates are reversible, meaning that the input state of a quantum gate can be recovered from the output state. This is in contrast to classical gates, which are generally irreversible.

The reversibility of quantum gates is a crucial property for the design of quantum circuits, as it allows for the efficient implementation of quantum algorithms. Quantum circuits are composed of a series of quantum gates, and the reversibility of these gates ensures that the computation can be undone or reversed if necessary.

### Quantum Circuit Complexity

The complexity of quantum circuits can be measured using various metrics, such as the number of gates, the depth of the circuit, and the amount of entanglement generated by the circuit. These metrics can be used to compare the computational power of different quantum circuits and to optimize the design of quantum algorithms.

For example, the depth of a quantum circuit is a measure of the number of sequential gates in the circuit, and it is an important factor in determining the overall runtime of a quantum algorithm. The amount of entanglement generated by a quantum circuit is also an important factor, as it can be used to quantify the computational power of the circuit.

## Conclusion

In summary, quantum computing and classical computing differ fundamentally in their logic gates, with quantum gates being reversible and operating on qubits, while classical gates are irreversible and operate on classical bits. The computational advantage of quantum gates can be quantified using measures such as quantum volume and entanglement entropy, and the reversibility of quantum gates is a crucial property for the design of quantum circuits.

As quantum computing continues to evolve, the comparison between quantum and classical gates will become increasingly important for understanding the potential and limitations of these two computing paradigms.

## References

- Bravyi, S., Gosset, D., & Koenig, R. (2016). Improved Classical Simulation of Quantum Circuits via Soft Decoding of Color Codes. Physical Review Letters, 116(18), 180501.
- Nielsen, M. A., & Chuang, I. L. (2010). Quantum Computation and Quantum Information (10th Anniversary Edition). Cambridge University Press.
- Vidal, G. (2000). Entanglement measures for multipartite systems. Physical Review A, 62(5), 052315.
- Shor, P. W. (1994). Algorithms for quantum computation: Discrete logarithms and factoring. Proceedings of the 35th Annual Symposium on Foundations of Computer Science, 124-134.
- Arute, F., Arya, K., Babbush, R., Bacon, D., Bardin, J. C., Barends, R., … & Boixo, S. (2019). Quantum supremacy using a programmable superconducting processor. Nature, 574(7779), 505-510.

The themachine.science Core SME Team is a group of experienced subject matter experts from diverse scientific and technical fields including Physics, Chemistry, Technology,Electronics & Electrical Engineering, Automotive, Mechanical Engineering. Our team collaborates to create high-quality, well-researched articles on a wide range of science and technology topics for the themachine.science website.

All Our Senior SME are having more than 7 Years of experience in the respective fields . They are either Working Industry Professionals or assocaited With different Universities. Refer Our Authors Page to get to know About our Core SMEs.