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A Method of Computing Many Functions Simultaneously by Using Many Parallel Quantum Systems

Koji Nagata, Germano Resconi, Tadao Nakamura, Josep Amengual i Batle, S. Abdalla, Ahmed Farouk, Han J.F. Geurdes

Published Apr 1, 2017• 1 citation
OpenAlex

Abstract

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 $\{a_1,a_2,a_3,\ldots,a_N\}$, and the function $g:{\bf R}\rightarrow \{0,1\}$, we shall determine the following values $\{g(a_1),g(a_2),g(a_3),\ldots, g(a_N)\}$ simultaneously. By using $M$ parallel quantum systems, we can compute $M$ functions $g^1,g^2,...,g^M$ simultaneously. The speed of determining the $N\times M$ values will be shown to outperform the classical case by a factor of $N$.

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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 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​.
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 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.
Here, we propose a new type of quantum algorithm for determining the 2N2^N2N values of a function. By measuring the single output state, we determine all the values of f(x)f(x)f(x) for all xxx simultaneously. This is very interesting indeed: the quantum circuit gives us the ability to determine a perfect property of f(x)f(x)f(x), namely, f(x)f(x)f(x). This is faster than a classical apparatus by a factor of 2N2^N2N.
We first propose herein a novel parallel computation, even though today's algorithm methodology for quantum computing, for all of the combinations of values in variables of a logical function. Our concern so far has been to obtain an attribute of some function. In fact such a task is only for one task problem solving. However, we could treat positively the plural evaluations of some logic function in parallel instead of testing the function for finding out its attribute. In fact, these evaluations of the function are naturally included and evaluated, in parallel, in normal quantum computing discussing a function in a Boolean algebra stemmed from atoms in it. As is naturally understandable with mathematics, quantum computing with qubit systems naturally is included and exemplified by a Boolean algebra, which treats only both 0 and 1. Namely, the theory in this paper is quite natural in logical sense even though physics domain. Therefore, quantum computing has an ability to solve some mathematical problems described in a Boolean algebra. The reason why we positively introduce a Boolean algebra here is because we have multiple evaluations of a function in quantum computing general.