| Type: | Package | 
| Title: | Classical Boson Sampling | 
| Version: | 0.1.5 | 
| Date: | 2023-10-10 | 
| Description: | Classical Boson Sampling using the algorithm of Clifford and Clifford (2017) <doi:10.48550/arXiv.1706.01260>. Also provides functions for generating random unitary matrices, evaluation of matrix permanents (both real and complex) and evaluation of complex permanent minors. | 
| Maintainer: | Raphaël Clifford <clifford@cs.bris.ac.uk> | 
| License: | GPL-2 | 
| Imports: | Rcpp (≥ 0.12.12) | 
| LinkingTo: | Rcpp, RcppArmadillo | 
| Encoding: | UTF-8 | 
| NeedsCompilation: | yes | 
| Packaged: | 2023-10-10 16:33:12 UTC; Peter Clifford | 
| Author: | Peter Clifford [aut], Raphaël Clifford [cre, aut] | 
| Repository: | CRAN | 
| Date/Publication: | 2023-10-10 17:50:02 UTC | 
Classical Boson Sampling
Description
Classical Boson Sampling using the algorithm of Clifford and Clifford (2017) <arXiv:1706.01260>. Also provides functions for generating random unitary matrices, evaluation of matrix permanents (both real and complex) and evaluation of complex permanent minors.
Details
Index of help topics:
BosonSampling-package   Classical Boson Sampling
Permanent-functions     Functions for evaluating matrix permanents
bosonSampler            Function for independently sampling from the
                        Boson Sampling distribution
randomUnitary           Random unitary
Author(s)
Peter Clifford <peter.clifford@jesus.ox.ac.uk> and Raphaël Clifford <clifford@cs.bris.ac.uk>
Maintainer: Raphaël Clifford <clifford@cs.bris.ac.uk>
Functions for evaluating matrix permanents
Description
These three functions are used in the classical Boson Sampling problem
Usage
cxPerm(A)
rePerm(B)
cxPermMinors(C)
Arguments
| A | a square complex matrix. | 
| B | a square real matrix. | 
| C | a rectangular complex matrix where  | 
Details
Permanents are evaluated using Glynn's formula (equivalently that of Nijenhuis and Wilf (1978))
Value
cxPerm(A) returns a complex number: the permanent of the complex matrix A.
rePerm(B) returns a real number: the permanent of the real matrix B.
cxPermMinors(C) returns a complex vector of length ncol(C)+1: the permanents of all 
ncol(C)-dimensional square matrices constructed by removing individual rows from C.
References
Glynn, D.G. (2010) The permanent of a square matrix. European Journal of Combinatorics, 31(7):1887–1891.
Nijenhuis, A. and Wilf, H. S. (1978). Combinatorial algorithms: for computers and calculators. Academic press.
Examples
  set.seed(7)
  n <- 20
  A <- randomUnitary(n)
  cxPerm(A)
  #
  B <- Re(A)
  rePerm(B)
  #
  C <- A[,-n]
  v <- cxPermMinors(C)
  #
  # Check Laplace expansion by sub-permanents
  c(cxPerm(A),sum(v*A[,n]))
Function for independently sampling from the Boson Sampling distribution
Description
The function implements the Boson Sampling algorithm defined in Clifford and Clifford (2017) https://arxiv.org/abs/1706.01260
Usage
bosonSampler(A, sampleSize, perm = FALSE)
Arguments
| A | the first  | 
| sampleSize | the number of independent sample values required for given  | 
| perm | TRUE if the permanents and pmfs of each sample value are required | 
Details
Let the matrix A be the first n columns of an (m x m) random unitary matrix, then 
X <- bosonSampler(A, sampleSize = N, perm = TRUE) provides X$values, X$perms and X$pmfs, 
 The component X$values is an (n x N) matrix with columns that are 
independent sample values from the Boson Sampling distribution. 
Each sample value is a vector of n integer-valued output modes in random order. The elements of the vector can be sorted in 
increasing order to provide a multiset representation of the sample value. 
 
 The outputs X$perms and X$pmfs are 
vectors of the permanents and probability mass functions (pmfs) associated with the sample values.
The permanent associated with a sample value v = (v_1,...,v_n) is the permanent of an (n x n) matrix constructed with rows 
v_1,...,v_n of A. Note the constructed matrix, M, may have repeated rows since v_1,...,v_n 
are not necessarily distinct.  
The pmf is calculated as Mod(pM)^2/prod(factorial(tabulate(c)) where pM is the permanent of M.
Value
X = bosonSampler(A, sampleSize = N, perm = TRUE) provides X$values, X$perms and X$pmfs. See Details.
References
Clifford, P. and Clifford, R. (2017) The Classical Complexity of Boson Sampling, https://arxiv.org/abs/1706.01260
Examples
set.seed(7)
n <- 20  # number of photons
m <- 200 # number of output modes
A <- randomUnitary(m)[,1:n]
# sample of output vectors
valueList <- bosonSampler(A, sampleSize = 10)$values 
valueList
# sample of output multisets 
apply(valueList,2, sort) 
#
set.seed(7)
n <- 12  # number of photons
m <- 30 # number of output modes
A <- randomUnitary(m)[,1:n]
# sample of output vectors
valueList = bosonSampler(A, sampleSize = 1000)$values 
# Compare frequency of output modes at different
# positions in the output vectors 
matplot(1:m,apply(valueList,1,tabulate), pch =20, t = "p", 
xlab = "output modes", ylab = "frequency")
Random unitary
Description
Returns a square complex matrix sampled from the Haar random unitary distribution.
Usage
randomUnitary(size)
Arguments
| size | dimension of matrix | 
Value
A square complex matrix.
Examples
m <- 25 # size of matrix (m x m)
set.seed(7)
U <- randomUnitary(m)
#  
n <- 5 # First n columns
A <- U[,1:n]