We present a method of fast determining a homogeneous linear function $f(x):= s.x=s_1x_1+ s_2x_2+\dots+s_Nx_N$ from $\{0,1,\dots,d-1\}^N$ with coefficients $s=(s_1,\dots,s_N)$. Here $x=(x_1,\dots,x_N)$ and $x_j\in{\bf R}$. Given the interpolation values $(f(1), f(2),...,f(N))=\vec{y}$, we shall determine the unknown coefficients $s = (s_1(\vec{y}),\dots, s_N(\vec{y}))$ of the linear function, simultaneously. The speed of determining the values is shown to outperform the classical case by a factor of $N$. Our method is based on the generalized Bernstein-Vazirani algorithm \cite{BVG} to qudit systems \cite{BVD}.
We present a method of fast determining a homogeneous linear function f(x):=s.x=s1x1+s2x2+⋯+sNxN from {0,1,…,d−1}N with coefficients s=(s1,…,sN). Here x=(x1,…,xN) and xj∈R. Given the interpolation values (f(1),f(2),...,f(N))=y, we shall determine the unknown coefficients s=(s1(y),…,sN(y)) of the linear function, simultaneously. The speed of determining the values is shown to outperform the classical case by a factor of N. Our method is based on the generalized Bernstein-Vazirani algorithm \cite{BVG} to qudit systems \cite{BVD}.
We suggest a method of computing many functions in the same time by using many parallel quantum systems. We use the Bernstein-Vazirani algorithm. Given the set of real values {a1,a2,a3,…,aN}, and the function g:R→{0,1}, we shall determine the following values {g(a1),g(a2),g(a3),…,g(aN)} simultaneously. By using M parallel quantum systems, we can compute M functions g1,g2,...,gM simultaneously. The speed of determining the N×M values will be shown to outperform the classical case by a factor of N.
First, we review the Bernstein-Vazirani algorithm for determining a bit string. Next, we discuss the generalized Bernstein-Vazirani algorithm for determining a natural number string. Finally, we discuss the generalized Bernstein-Vazirani algorithm for determining an integer string. All of the generalized algorithms presented here have the following structure. Given the set of real values {a1,a2,a3,…,aN} and a special function g, we determine N values of the function g(a1),g(a2),g(a3),…,g(aN) simultaneously. The speed of determining the strings is shown to outperform the best classical case by a factor of N in every case.
We present generalization of the Bernstein-Vazirani algorithm. Suppose there are many natural numbers: a1,a2,a3,…,aN. Here, we introduce a function: $ g:{\bf N}\rightarrow \{0,1\}. $ Our goal is to determine the following values simultaneously: $ g(a_1),g(a_2),g(a_3),\ldots, g(a_N). $ The speed to determine N values improves by a factor of N by comparing the classical case. We obtain the Bernstein-Vazirani algorithm when g:ai→ai.
Bernstein-Vazirani algorithm (the one-query algorithm) can identify a completely specified linear Boolean function using a single query to the oracle with certainty. The first aim of the paper is to show that if the provided Boolean function is affine, then one more query to the oracle (the two-query algorithm) is required to identify the affinity of the function with certainty. The second aim of the paper is to show that if the provided Boolean function is incompletely defined, then the one-query and the two-query algorithms can be used as bounded-error quantum polynomial algorithms to identify certain classes of incompletely defined linear and affine Boolean functions respectively with probability of success at least 2/3.
Linear threshold functions are fw,θ(x)=sign(⟨x,w⟩−θ), where the weight vector w∈Rn is a unit vector, θ∈R is a threshold, and typically x∈Rn or x∈{−1,1}n. When θ=0, the LTF is called homogeneous, and we write fw:=fw,0. Such functions are among the most important objects in machine learning, since they serve to linearly discriminate positive and negative examples. We give three positive results about learning LTFs:
1. Suppose we can make real-domain queries, meaning we can compute fw,θ(x) at any x∈Rn of our choice. We give a quantum algorithm that learns fw,θ up to Euclidean error ε using O(log(n/ε)) membership queries and O(n) other gates. Then we have also learned fw,θ up to error O(ε) when x is Gaussian. Classical algorithms need Ω(nlog(1/ε)) queries.
2. A homogeneous LTF fw on domain {−1,1}n where w has only k nonzero entries of the same value, is the Majority function on the support of w. Belovs gave a bounded-error quantum algorithm that identifies the hidden support exactly (and hence learns fw) using O(k1/4) queries. We give an exponential improvement, using O(logk) queries.
3. Suppose we have a unitary U that can produce (discretized) quantum examples under Gaussian measure, corresponding to ∫xγn(x)∣x⟩∣fw(x)⟩dx. This is a weaker access model than membership queries. We give a quantum algorithm based on the efficient Hermite transform of Jain et al. to learn homogeneous LTFs fw with error ε under the Gaussian distribution, using O(n1/4/ε) applications of U and U† and O(n2/ε4) other gates.
How many quantum queries are required to determine the coefficients of a degree- d polynomial in n variables? We present and analyse quantum algorithms for this multivariate polynomial interpolation problem over the fields F q , R and C . We show that k C and 2 k C queries suffice to achieve probability 1 for C and R , respectively, where k C = ⌈ ( 1 / ( n + 1 ) ) ( n + d d ) ⌉ except for d =2 and four other special cases. For F q , we show that ⌈( d /( n + d ))( n + d d ) ⌉ queries suffice to achieve probability approaching 1 for large field order q . The classical query complexity of this problem is ( n + d d ) , so our result provides a speed-up by a factor of n +1, ( n +1)/2 and ( n + d )/ d for C , R and F q