In this paper, we propose a bio-molecular algorithm with O( ${n}^{{2}} + {m}$ ) biological operations, O( $2^{n}$ ) DNA strands, O( ${n}$ ) tubes and the longest DNA strand, O( ${n}$ ), for solving the independent-set problem for any graph G with ${m}$ edges and ${n}$ vertices. Next, we show that a new kind of the straightforward Boolean circuit yielded from the bio-molecular solutions with ${m}$ NAND gates, ( ${m} +{n} \times $ ( ${n} + {1}$ )) AND gates and (( ${n} \times $ ( ${n} + {1}$ ))/2) NOT gates can find the maximal independent-set(s) to the independent-set problem for any graph ${G}$ with ${m}$ edges and ${n}$ vertices. We show that a new kind of the proposed quantum-molecular algorithm can find the maximal independent set(s) with the lower bound $\Omega $ ( $2^{n/{2}}$ ) queries and the upper bound O( $2^{n/{2}}$ ) queries. This work offers an obvious evidence for that to solve the independent-set problem in any graph ${G}$ with ${m}$ edges and ${n}$ vertices, bio-molecular computers are able to generate a new kind of the straightforward Boolean circuit such that by means of implementing it quantum computers can give a quadratic speed-up. This work also offers one obvious evidence that quantum computers can significantly accelerate the speed and enhance the scalability of bio-molecular computers. Next, the element distinctness problem with input of ${n}$ bits is to determine whether the given $2^{n}$ real numbers are distinct or not. The quantum lower bound of solving the element distinctness problem is $\Omega $ ( $2^{n\times {(}{2}/{3}{)}}$ ) queries in the case of a quantum walk algorithm. We further show that the proposed quantum-molecular algorithm reduces the quantum lower bound to $\Omega $ (( $2^{n/{2}}$ )/( $2^{\text {1/2}}{)}$ ) queries. Furthermore, to justify the feasibility of the proposed quantum-molecular algorithm, we successfully solve a typical independent set problem for a graph G with two vertices and one edge by carrying out experiments on the backend ibmqx4 with five quantum bits and the backend simulator with 32 quantum bits on IBM’s quantum computer.

中文翻译：

---

在 IBM 量子计算机上实现独立集问题生物分子解决方案的量子加速和数学解决方案


在本文中，我们提出了一种生物分子算法，其中 O( ${n}^{{2}} + {m}$ ) 生物操作、O( $2^{n}$ ) DNA 链、O( ${n }$ ) 管和最长的 DNA 链 O( ${n}$ )，用于解决具有 ${m}$ 边和 ${n}$ 顶点的任何图 G 的独立集问题。接下来，我们展示了一种由具有 ${m}$ 与非门的生物分子解决方案产生的新型简单布尔电路，( ${m} +{n} \times $ ( ${n} + {1} $ )) AND 门和 (( ${n} \times $ ( ${n} + {1}$ ))/2) NOT 门可以找到任意独立集问题的最大独立集具有 ${m}$ 条边和 ${n}$ 个顶点的图 ${G}$。我们表明，提出的一种新型量子分子算法可以找到具有下界 $\Omega $ ( $2^{n/{2}}$ ) 查询和上限 O( $2 ^{n/{2}}$ ) 查询。这项工作提供了明显的证据，证明要解决任何具有 ${m}$ 边和 ${n}$ 顶点的图 ${G}$ 中的独立集问题，生物分子计算机能够生成一种新的简单的布尔电路，通过实现它，量子计算机可以提供二次加速。这项工作还提供了一个明显的证据，表明量子计算机可以显着加快生物分子计算机的速度并增强其可扩展性。接下来，输入${n}$位的元素不同性问题是确定给定的$2^{n}$实数是否不同。在量子行走算法的情况下，解决元素独特性问题的量子下界是 $\Omega $ ( $2^{n\times {(}{2}/{3}{)}}$ ) 查询。我们进一步表明，所提出的量子分子算法将量子下界降低到 $\Omega $ (( $2^{n/{2}}$ )/( $2^{\text {1/2}}{)}$ ）查询。 此外，为了证明所提出的量子分子算法的可行性，我们通过在具有 5 个量子位的后端 ibmqx4 和具有 32 个量子位的后端模拟器上进行实验，成功解决了具有两个顶点和一条边的图 G 的典型独立集问题。 IBM 量子计算机上的量子比特。