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The Generalized Bernstein-Vazirani Algorithm for Determining an Integer String

Koji Nagata, Tadao Nakamura, Han J.F. Geurdes, Josep Amengual i Batle, Ahmed Farouk, Do Ngoc Diep

Published Mar 1, 2018• 0 citations
OpenAlex

Abstract

We present the generalized Bernstein-Vazirani algorithm for determining a restricted integer string. Given the set of real values $\{a_1,a_2,a_3,\ldots,a_N\}$ and a function $g:{\bf R}\rightarrow {\bf Z}$, we shall determine the following values $\{g(a_1),g(a_2),g(a_3),\ldots, g(a_N)\}$ simultaneously. The speed of determining the values is shown to outperform the classical case by a factor of $N$. The method determines the maximum of and the minimum of the function $g$ that the finite domain is $\{a_1,a_2,a_3,\ldots,a_N\}$.

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We present the generalized Bernstein-Vazirani algorithm for determining a restricted integer string. Given the set of real values {a1,a2,a3,…,aN}\{a_1,a_2,a_3,\ldots,a_N\}{a1​,a2​,a3​,…,aN​} and a function g:R→Zg:{\bf R}\rightarrow {\bf Z}g:R→Z, we shall determine the following values {g(a1),g(a2),g(a3),…,g(aN)}\{g(a_1),g(a_2),g(a_3),\ldots, g(a_N)\}{g(a1​),g(a2​),g(a3​),…,g(aN​)} simultaneously. The speed of determining the values is shown to outperform the classical case by a factor of NNN. The method determines the maximum of and the minimum of the function ggg that the finite domain is {a1,a2,a3,…,aN}\{a_1,a_2,a_3,\ldots,a_N\}{a1​,a2​,a3​,…,aN​}.
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}\{a_1,a_2,a_3,\ldots,a_N\}{a1​,a2​,a3​,…,aN​} and a special function ggg, we determine NNN values of the function g(a1),g(a2),g(a3),…,g(aN)g(a_1),g(a_2),g(a_3),\ldots, g(a_N)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 NNN in every case.
We present generalization of the Bernstein-Vazirani algorithm. Suppose there are many natural numbers: a1,a2,a3,…,aNa_1,a_2,a_3,\ldots,a_Na1​,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 NNN values improves by a factor of NNN by comparing the classical case. We obtain the Bernstein-Vazirani algorithm when g:ai→aig:a_i\rightarrow a_ig:ai​→ai​.
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}\{a_1,a_2,a_3,\ldots,a_N\}{a1​,a2​,a3​,…,aN​}, and the function g:R→{0,1}g:{\bf R}\rightarrow \{0,1\}g:R→{0,1}, we shall determine the following values {g(a1),g(a2),g(a3),…,g(aN)}\{g(a_1),g(a_2),g(a_3),\ldots, g(a_N)\}{g(a1​),g(a2​),g(a3​),…,g(aN​)} simultaneously. By using MMM parallel quantum systems, we can compute MMM functions g1,g2,...,gMg^1,g^2,...,g^Mg1,g2,...,gM simultaneously. The speed of determining the N×MN\times MN×M values will be shown to outperform the classical case by a factor of NNN.
We present a method of fast determining a homogeneous linear function f(x):=s.x=s1x1+s2x2+⋯+sNxNf(x):= s.x=s_1x_1+ s_2x_2+\dots+s_Nx_Nf(x):=s.x=s1​x1​+s2​x2​+⋯+sN​xN​ from {0,1,…,d−1}N\{0,1,\dots,d-1\}^N{0,1,…,d−1}N with coefficients s=(s1,…,sN)s=(s_1,\dots,s_N)s=(s1​,…,sN​). Here x=(x1,…,xN)x=(x_1,\dots,x_N)x=(x1​,…,xN​) and xj∈Rx_j\in{\bf R}xj​∈R. Given the interpolation values (f(1),f(2),...,f(N))=y⃗(f(1), f(2),...,f(N))=\vec{y}(f(1),f(2),...,f(N))=y​, we shall determine the unknown coefficients s=(s1(y⃗),…,sN(y⃗))s = (s_1(\vec{y}),\dots, s_N(\vec{y}))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 NNN. Our method is based on the generalized Bernstein-Vazirani algorithm \cite{BVG} to qudit systems \cite{BVD}.
A quantum algorithm to solve the parity problem is better than its most efficient classical counter- part with a separation that is polynomial in the number of queries. This was shown by E. Bernstein and U. Vazirani and was one of the earliest indications that the quantum information processing can outperform the classical one by a significant margin. The problem and its solution both is usually stated for a 2-level system since we generally work with bits/qubits. However, many works have been done generalizing known quantum computing techniques to higher level systems. Following this, we look at a generalization of the Bernstein-Vazirani algorithm implemented on a general qudit system.