Math

Collection common (vectorized) math functions.

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comb

 comb (n, k)

Vectorized combination: comb(n,k) = n! / ((n-k)!k!)

	Type	Details
n	int or np.array of int	First parameter of combination
k	int or np.array of int	Second parameter of combination
Returns	np.array	Combination (choose k out of n)

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  else: warn(msg)

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binom

 binom (k, n, p)

Vectorized binomial distribution: binom(k,n,p)=comb(n,k) p^k (1-p)^n-k

	Type	Details
k	int or list of int	Second parameter of combination
n	int or list of int	First parameter of combination
p	float or list of float	Probability
Returns	np.array	Value(s) of binomial distribution evaluated at k,n,p.

test_close(binom(k=[1,2], n=[3,4], p=0.1), [0.243 , 0.0486], eps=1e-05)

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joint_binom

 joint_binom (k, n, p)

Product of independent binomial distributions with parameters k, n and p (can be list of lists), i.e.:

joint_binom(k,n,p)=binom(k[0],n[0],p[0])×…×binom(k[-1],n[-1],p[-1])

see Eq. 26 in paper

	Type	Details
k	list of int	List of second parameters of combination
n	list of int	List of first parameters of combination
p	list of list, list of float, or float	Probability
Returns	np.array	Joint probability

test_close(joint_binom(k=[1,2], n=[3,4], p=[0.1,0.2]), binom(k=1,n=3,p=0.1) * binom(k=2,n=4,p=0.2))
test_close(joint_binom(k=[1,2], n=[2,3], p=[[0.1,0.2],[.3,.4]]), [binom(1,2,0.1)*binom(2,3,0.2), binom(1,2,0.3)*binom(2,3,0.4)])

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Wilson_var

 Wilson_var (p, N)

Wilson estimator of binomial variance

The formula for the Wilson interval is:

CI = p+z^2/(2n) \pm z\sqrt{pq/n + z^2/(4n^2)}/(1 + z^2/n)

we can extract the var (z=1) as:

Var(p) = (CI/2)^2 = (npq + 0.25) / (1 + n)^2

see Eq. C12 in paper

	Type	Details
p	float	Estimator of probability
N	int	Sample size
Returns	float	Estimated variance of Wilson CI

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Wald_var

 Wald_var (p, N)

Wald estimation of binomial variance

see Eq. C11 in paper

	Type	Details
p	float	Estimator of probability
N	int	Sample size
Returns	float	Estimated variance of Wald CI

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subset_cards

 subset_cards (superset)

Calculate cardinalities of all possible subsets of superset

	Type	Details
superset	set	Input set
Returns	list of int	All possible cardinalities of subsets in superset

assert(subset_cards({1,2,3}) == {0,1,2,3})
assert(subset_cards({(0,0), (0,1), (0,2)}) == {0,1,2,3})

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cartesian_product

 cartesian_product (list_of_sets)

Calculate cartesian product between all members of sets

	Type	Details
list_of_sets	list	List of sets between which to calculate Cartesian product
Returns	list of tuple	Cartesian products

assert(cartesian_product([{1,2},{3,4}]) == [(1,3), (1,4), (2,3), (2,4)])

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subset_probs

 subset_probs (circuit, error_model, prob)

Calculate occurence probability of subsets in circuit with physical error rate prob. error_model defines how the circuit is to be partitioned before occurence probabilities are calculated.

	Type	Details
circuit	Circuit	Circuit wrt. which subset probabilities are calculated
error_model	ErrorModel	Error model by which to partition circuit
prob	float or list of float	Physical error probabilities