Quantum-inspired algorithm for estimating the permanent of positive semidefinite matrices

We construct a quantum-inspired classical algorithm for computing the permanent of Hermitian positive semidefinite matrices by exploiting a connection between these mathematical structures and the boson sampling model. Specifically, the permanent of a Hermitian positive semidefinite matrix can be expressed in terms of the expected value of a random variable, which stands for a specific photon-counting probability when measuring a linear-optically evolved random multimode coherent state. Our algorithm then approximates the matrix permanent from the corresponding sample mean and is shown to run in polynomial time for various sets of Hermitian positive semidefinite matrices, achieving a precision that improves over known techniques. This work illustrates how quantum optics may benefit algorithm development.

Figure 1

Boson sampling setup with an $M$-mode thermal state ${\rho }_{\mathrm{in}}^{\text{th}}={⨂}_{i=1}^M{\rho }_i^{\text{th}}$ injected into an $M$-mode linear-optical network that is characterized by the unitary matrix $\mathbf{U}$ (acting on bosonic-mode operators), followed by the detection of a pattern of photons $\{m_1,...,m_M\}$. The problem is to sample from the probability distribution given by Eq. (2).