Localized Quantum Chemistry on Quantum Computers

Quantum chemistry calculations of large, strongly correlated systems are typically limited by the computation cost that scales exponentially with the size of the system. Quantum algorithms, designed specifically for quantum computers, can alleviate this, but the resources required are still too large for today’s quantum devices. Here we present a quantum algorithm that combines a localization of multireference wave functions of chemical systems with quantum phase estimation (QPE) and variational unitary coupled cluster singles and doubles (UCCSD) to compute their ground state energy. Our algorithm, termed “local active space unitary coupled cluster” (LAS-UCC), scales linearly with system size for certain geometries, providing a polynomial reduction in the total number of gates compared with QPE, while providing accuracy above that of the variational quantum eigensolver using the UCCSD ansatz and also above that of the classical local active space self-consistent field. The accuracy of LAS-UCC is demonstrated by dissociating (H2)2 into two H2 molecules and by breaking the two double bonds in trans-butadiene and resources estimates are provided for linear chains of up to 20 H2 molecules.

Nonlinear Quantum Optimization Algorithms via Efficient Ising Model Encodings

Despite extensive research efforts, few quantum algorithms for classical optimization demonstrate realizable advantage. The utility of many quantum algorithms is limited by high requisite circuit depth and nonconvex optimization landscapes. We tackle these challenges to quantum advantage with two new variational quantum algorithms, which utilize multi-basis graph encodings and nonlinear activation functions to outperform existing methods with remarkably shallow quantum circuits. Both algorithms provide a polynomial reduction in measurement complexity and either a factor of two speedup a factor of two reduction in quantum resources. Typically, the classical simulation of such algorithms with many qubits is impossible due to the exponential scaling of traditional quantum formalism and the limitations of tensor networks. Nonetheless, the shallow circuits and moderate entanglement of our algorithms, combined with efficient tensor method-based simulation, enable us to successfully optimize the MaxCut of high-connectivity global graphs with up to 512 nodes (qubits) on a single GPU.

AMulet 2.0 for Verifying Multiplier Circuits

AMulet 2.0 is a fully automatic tool for the verification of integer multipliers using computer algebra. Our tool models multiplier circuits given as and-inverter graphs as a set of polynomials and applies preprocessing techniques based on elimination theory of Gröbner bases. Finally it uses a polynomial reduction algorithm to verify the correctness of the given circuit. AMulet 2.0 is a re-factorization and improved re-implementation of our previous multiplier verification tool AMulet 1.0.

On polynomial reduction of problems based on diagonal Latin squares to the exact cover problem

The paper discusses the reduction of problems based on Latin squares to the exact cover problem aiming at its subsequent solution using the dancing links algorithm. The former problems include generation of Latin squares and diagonal Latin squares of a general form/with a given normalization, generation of orthogonal Latin and diagonal Latin squares directly/through the set of transversals, obtaining a set of transversals for a given square, forming a subset of disjoint transversals. For each subproblem, we describe in detail the process of forming the corresponding binary coverage matrices. We show that the use of the proposed approach in comparison with the classical one, i.e. the formation of sets of transversals and their coverages using exhaustive enumeration, allows one to increase the eective processing pace of diagonal Latin squares by 2.5{5.6 times. The developed software implementations of the algorithms are used in computational experiments as part of the Gerasim@Home volunteer distributed computing project on the BOINC platform

NP-completeness and One Polynomial Subclass of the Two-Step Graph Colouring Problem

In this paper, we study the two-step colouring problem for an undirected connected graph. It is required to colour the graph in a given number of colours in a way, when no pair of vertices has the same colour, if these vertices are at a distance of 1 or 2 between each other. Also the corresponding recognition problem is set. The problem is closely related to the classical graph colouring problem. In the article, we study and prove the polynomial reduction of the problems to each other. So it allows us to prove NP-completeness of the problem of two-step colouring. Also we specify some of its properties. Special interest is paid to the problem of two-step colouring in application to rectangular grid graphs. The maximum vertex degree in such a graph is between 0 and 4. For each case, we elaborate and prove the function of two-vertex colouring in the minimum possible number of colours. The functions allow each vertex to be coloured independently from others. If vertices are examined in a sequence, colouring time is polynomial for a rectangular grid graph.

NP-Hardness of the Problem of Optimal Box Positioning

We consider the problem of finding a position of a d-dimensional box with given edge lengths that maximizes the number of enclosed points of the given finite set P ⊂ R d , i.e., the problem of optimal box positioning. We prove that while this problem is polynomial for fixed values of d, it is NP-hard in the general case. The proof is based on a polynomial reduction technique applied to the considered problem and the 3-CNF satisfiability problem.

Probabilistic logic over equations and domain restrictions

We propose and study a probabilistic logic over an algebraic basis, including equations and domain restrictions. The logic combines aspects from classical logic and equational logic with an exogenous approach to quantitative probabilistic reasoning. We present a sound and weakly complete axiomatization for the logic, parameterized by an equational specification of the algebraic basis coupled with the intended domain restrictions.We show that the satisfiability problem for the logic is decidable, under the assumption that its algebraic basis is given by means of a convergent rewriting system, and, additionally, that the axiomatization of domain restrictions enjoys a suitable subterm property. For this purpose, we provide a polynomial reduction to Satisfiability Modulo Theories. As a consequence, we get that validity in the logic is also decidable. Furthermore, under the assumption that the rewriting system that defines the equational basis underlying the logic is also subterm convergent, we show that the resulting satisfiability problem is NP-complete, and thus the validity problem is coNP-complete.We test the logic with meaningful examples in information security, namely by verifying and estimating the probability of the existence of offline guessing attacks to cryptographic protocols.