The coefficients λ appearing in Eq. (2) are inner products between vectors, multiplied by a power of the singular values in the case of linear systems. Quantuminspired algorithms compute approximations of these coefficients using Monte Carlo estimation. In general, a coefficient λ is given by
for some appropriate vectors y,z, as in Eqs. (3) and (4). The strategy to estimate this inner product is to perform lengthsquare sampling from one of the vectors. Without loss of generality, we take this vector to be y
. Define the random variable
χ that takes values
χi=yizi/px(i), where the indices
i are sampled from the lengthsquare distribution
py(i)=y2i/∥y∥2. The expectation value of the random variable satisfies

E(χ)=n∑i=1yizipy(i)py(i)=⟨y,z⟩=λ. 

(8) 
Similarly, the second moment is

E(χ2)=n∑i=1(yizipy(i))2py(i)=∥y∥2∥z∥2, 

(9) 
and σ2χ=E(χ2)−E(χ)2
is the variance of
χ. The strategy to estimate coefficients
λ is to draw
N samples
χ(1),χ(2),…,χ(N) from
χand compute the unbiased estimator
^λ=1N∑Nj=1χ(j)≈λ. This constitutes a form of importance sampling in Monte Carlo estimation: large entries of
yare preferably selected since they contribute more significantly to the inner product. The error in the estimation is quantified by the ratio between the standard deviation and the mean of the estimator:
ϵ:=√Var(^λ)/λ. The variance of the estimator is
Var(^λ)=σ2χ/N, leading to a precision
ϵ=σχ/(λ√N) in the estimator. This implies that the number of samples
N needed to, with high probability, achieve a precision
ϵ is

N= 
σ2χλ2ϵ2=1ϵ2[∥y∥2∥z∥2⟨y,z⟩2−1] 

(10) 

= 
O(∥y∥2∥z∥2ϵ2⟨y,z⟩2)=O(1ϵ2cos2θ), 

(11) 
where θ is the angle between the vectors y and z.
Comments
There are no comments yet.