Permanent of a matrix

Definition

The permanent of a matrix is a scalar value that generalizes the notion of the determinant, used primarily in combinatorics and quantum computing. Unlike the determinant, the permanent does not involve any alternating signs in its calculation, which makes it computationally easier to evaluate but also significantly harder to compute efficiently for large matrices. The permanent is especially relevant in contexts such as boson sampling, where it serves as a measure of the probability amplitudes associated with quantum states.

5 Must Know Facts For Your Next Test

1. Calculating the permanent of an n x n matrix involves summing n! products of entries, leading to an exponential time complexity.
2. The permanent can be interpreted combinatorially as counting the number of perfect matchings in a bipartite graph.
3. In boson sampling, the output distribution is determined by the square of the absolute value of the permanent of specific submatrices derived from the input states.
4. Unlike determinants, which can be computed in polynomial time with efficient algorithms like Gaussian elimination, permanents lack such efficient computational methods.
5. The study of permanents has implications for understanding quantum systems and their behavior under various transformations, enhancing our grasp of quantum mechanics.

Review Questions

• How does the calculation of the permanent differ from that of the determinant, and why is this distinction important in quantum computing?
  • The permanent differs from the determinant mainly in that it lacks alternating signs when summing products of matrix entries. This distinction is crucial in quantum computing because while determinants can be computed efficiently using polynomial-time algorithms, permanents require exponential time complexity for exact computation. This difference highlights why certain problems related to boson sampling, which relies on permanents, are believed to be hard for classical computers.
• Discuss the role of the permanent in boson sampling and how it contributes to establishing quantum supremacy.
  • In boson sampling, the probability distribution of measurement outcomes is determined by calculating the permanents of submatrices formed from input photon states. The connection between these permanents and measurement probabilities allows researchers to demonstrate tasks that are infeasible for classical computers. This capability exemplifies quantum supremacy by showcasing how quantum systems can solve specific problems faster than any classical counterpart could.
• Evaluate the implications of being able to compute permanents efficiently on our understanding of computational complexity and quantum mechanics.
  • If we could develop an efficient algorithm for computing permanents, it would challenge existing beliefs about computational complexity classes. Such a breakthrough would suggest that problems traditionally deemed hard may not be as intractable as previously thought. In terms of quantum mechanics, an efficient computation of permanents could revolutionize our ability to model and predict behaviors within quantum systems, thereby enhancing our overall understanding and capabilities within both theoretical and applied physics.

Related terms

Determinant: A scalar value that can be computed from the elements of a square matrix, which encodes information about the matrix's invertibility and volume scaling.

Boson Sampling: A computational model that leverages the quantum nature of indistinguishable particles (bosons) to perform tasks that are believed to be hard for classical computers.

Quantum Supremacy: The theoretical point at which a quantum computer can perform calculations that are practically impossible for classical computers to achieve in a reasonable timeframe.