Alessandro Barenghi

Quantum walk-based search algorithms have demonstrated an asymptotic quadratic speedup compared to classical search methods. Formulating a generic search problem as a (quantum) search over a graph makes the efficiency of the algorithm to be closely dependent on the properties of the graph itself. In this work, we present a complete implementation of a quantum walk search procedure on Johnson graphs, speeding up the solution of the Subset Sum Problem, a well-known computational problem with applications in resource allocation, scheduling, and cryptanalysis. We provide a detailed design of each sub-circuit, quantifying their costs in terms of gate count, circuit depth, and qubit width, and exhibit all figures of merit as a function of the problem parameters. Our approach includes two distinct implementations: one that minimizes qubit usage and another focused on optimizing circuit depth. We compare our solutions against the unstructured Grover based quantum search algorithm, demonstrating a reduction of the cost on terms of T-count and T-depth, for practically solvable instances. The proposed design serves as a foundational building block for the development of efficient quantum search algorithms that can be modeled on Johnson graphs, bridging the gap with existing theoretical complexity analyses.

The Boolean matching problem via NP-equivalence requires determining whether two Boolean functions are equivalent or not up to a permutation and negation of the input binary variables. Its solution is a fundamental step in the electronic design automation (EDA) tool chains commonly used for digital circuit design. In fact, the library-mapping step of an EDA workflow requires matching parts of the gate-level design (netlist) with the components available in a technology library, considering them as Boolean functions, while taking into account that permutations and negations of input variables can be efficiently implemented through rewiring and the use of inverters. For $n$-to-$n$ vector Boolean functions, where $n$ is the number of input and output variables, the search space of possible negations and permutations is super-exponential in size, while the $\mathcal {O}(n!n2^{2n})$ time complexity of classical approaches allows solving only small instances of the NP-problem, often limited to $n$-to-1 Boolean functions (executing $\mathcal {O}(n!2^{2n})$ bit operations). This work presents a quantum algorithm for matching $n$-to-$n$ vector Boolean functions by effectively combining the Grover-meets-Simon approach with an original and novel use of the Simon solver without the constraints imposed by its usual premises. We provide a fully detailed quantum circuit implementing our proposal, calculate its cost by evaluating key performance indicators for circuit synthesis, and show an exponential speedup over classical solutions. Finally, we validate our approach on the Boolean functions included in the ISCAS benchmark suite, which are of practical interest in EDA.