bosonSampler {BosonSampling} R Documentation

## 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 `n` columns of an `(m x m)` random unitary matrix, see `randomUnitary` `sampleSize` the number of independent sample values required for given `A` `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,1, 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")
```

[Package BosonSampling version 0.1.3 Index]