A Faster Hafnian Formula for Complex Matrices and Its Benchmarking on a Supercomputer

Abstract

We introduce new and simple algorithms for the calculation of the number of perfect matchings of complex weighted, undirected graphs with and without loops. Our compact formulas for the hafnian and loop hafnian of n × n complex matrices run in O(n 3 2 n/2 ) time, are embarrassingly parallelizable and, to the best of our knowledge, are the fastest exact algorithms to compute these quantities. Despite our highly optimized algorithm, numerical benchmarks on the Titan supercomputer with matrices up to size 56 × 56 indicate that one would require the 288000 CPUs of this machine for about a month and a half to compute the hafnian of a 100 × 100 matrix.

Speedup in classical simulation of Gaussian boson sampling

“…The Hafnian of a symmetric n × n matrix V is defined by (n must be even) Hafnian is permutation invariant from its definition, that is we can first interchange two columns of V , and then interchange the corresponding rows, and the Hafnian of the new matrix will remain the same. Similar to permanents, exact computation of matrix Hafnian is also #P -hard [31], which also implies GBS for the same squeezing parameters is also computational hard; the best classical algorithm takes time O(n 3 2 n/2 ) [5], which is closely related to the hardness of GBS [20]. Next, the matrix W s is constrcuted from another m × m matrix (here U refers to the m × m unitary acting on the single-photon subspace)…”

“…According to Ref. [5], the time complexity of computing the Hafnian of a n × n matrix is O(n 3 2 n/2 ), so computing q n (x 1 , · · · , x k ) by directly computing m n−k Haf(W x ) is of course costly. However, we shall show that one can eliminate many terms by breaking each Hafnian | Haf(W x )| 2 into many pieces of smaller sub-Hafnians.…”

“…By [5], the time complexity of Hafnian of a n × n matrix is O(n 3 2 n/2 ), and the time complexity of Per(S A,B ) is O (µ).…”

“…GBS utilizes single-mode squeezed states (SMSS) as input, and there is no need for post-selection. The output probability of GBS is related to a matrix function called Hafnian, which, like permanent, is also #P -hard to compute [5]. This model is proved to be computationally hard based on the #P hardness of computing Hafnian [17,20].…”

“…The second step uses a similar idea of the algorithm in Ref. [12], but the form of marginal probabilities in our case is more complicated, since calculating Hafnian might be harder than calculating permanent [5]. Specifically, we give a method to break a large Hafnian (and the square of its modulus) into pieces of smaller Hafnians and permanents, so that we can speed up the calculation of marginal probabilities.…”