Local Quantum Search Algorithm for Random $k$-SAT with $Ω(n^{1+ε})$ Clauses

TL;DR

The k-local quantum search algorithm is proposed, which extends quantum search to structured scenarios and proves that max-k-SSAT is polynomial on average when $m=\Omega(n^{2+\epsilon})$ based on the average-case complexity theory.

Abstract

The random k-SAT instances undergo a "phase transition" from being generally satisfiable to unsatisfiable as the clause number m passes a critical threshold, $r_k n$. This causes a drastic reduction in the number of satisfying assignments, shifting the problem from being generally solvable on classical computers to typically insolvable. Beyond this threshold, it is challenging to comprehend the computational complexity of random k-SAT. In quantum computing, Grover's search still yields exponential time requirements due to the neglect of structural information. Leveraging the structure inherent in search problems, we propose the k-local quantum search algorithm, which extends quantum search to structured scenarios. Grover's search, by contrast, addresses the unstructured case where k=n. Given that the search algorithm necessitates the presence of a target, we specifically focus on the problem of searching the interpretation of satisfiable instances of k-SAT, denoted as max-k-SSAT. If this problem is solvable in polynomial time, then k-SAT can also be solved within the same complexity. We demonstrate that, for small $k \ge 3$, any small $ε>0$ and sufficiently large n: $\cdot$ k-local quantum search achieves general efficiency on random instances of max-k-SSAT with $m=Ω(n^{2+δ+ε})$ using $\mathcal{O}(n)$ iterations, and $\cdot$ k-local adiabatic quantum search enhances the bound to $m=Ω(n^{1+δ+ε})$ within an evolution time of $\mathcal{O}(n^2)$. In both cases, the circuit complexity of each iteration is $\mathcal{O}(n^k)$, and the efficiency is assured with overwhelming probability $1 - \mathcal{O}(\mathrm{erfc}(n^{δ/2}))$. By modifying this algorithm capable of solving all instances of max-k-SSAT, we further prove that max-k-SSAT is polynomial on average when $m=Ω(n^{2+ε})$ based on the average-case complexity theory.

Figures (6)

• Figure 1: The distribution of success probability $p_t$ for 3-local quantum search on 100 random instances in $F_s(n, m, 3)$. The number of variables $n$ ranges from 10 to 20, and the number of clauses $m=n^2$, $2n^2$ and $4n^2$, represented by red, blue and green boxes, respectively.
• Figure 2: The distribution of success probability $p_t$ for the adiabatic 3-local quantum search on 100 random instances in $F_s(n,cn,3)$. The clause density coefficient $c$ ranges from 2.5 to 10. The number of variables $n$ is 16, 18, and 20, represented by red, blue, and green boxes, respectively.
• Figure 3: The evolution of the success probability for $k$-local quantum search with $n=20$. The cases of $k=1,2,3$ are represented by the green, blue and red line, respectively.
• Figure 4: The clause selecting process during the generation of a random $k$-SAT instance. Consider an instance $I \in U_s$ with a clause set $\{\alpha\}$ and an interpretation set $\{t\}$. For the model $F(n,m,k)$, arbitrary clause is randomly chosen from $S_1 \cup S_2 \cup S_3$. In contrast, the model $F_s(n,m,k)$ selectively chooses clauses that are satisfiable by any interpretation in $\{t\}$, specifically from $S_1 \cup S_2$. Meanwhile, the model $F_f(n,m,k)$ exclusively selects clauses from the set $S_1$.
• Figure 5: An example quantum circuit for $e^{i\theta h_\alpha}$, where $\alpha = x_1 \land \lnot x_2 \land x_4$. This clause is denoted as $\alpha = (1, -2, 4)$, indicating that the 1st, 2nd, and 4th qubits (from top to bottom in the figure) are involved. An additional pair of $X$ gates is applied to the 2nd qubit due to its inversion. In this circuit, the 4th qubit serves as the target qubit, with the phase gate $P_\theta$ activated on this qubit. The 1st and 2nd qubits are the control qubits, denoted by black point in this figure. However, due to the property of the multi-controlled phase gate, the target qubit can be any of the involved qubits, with the remaining qubits acting as the control qubits.

Theorems & Definitions (38)

• Theorem 1: main theorem
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Theorem 2
• proof
• Lemma 2.4
• proof
• Theorem 3