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Encoding procedure for integer factorization. (a) Graphical representation of the set of equations to be satisfied for integer factorization (m = p · q). The factor bits p i , q i and binary variables s i,j , c i,j are represented by copy lines to construct an effective square crossing lattice for the problem, with a filled square at crossings and the integer bits m i specifying some of the boundary conditions of the problem. (b) Each filled square represents a set of equality constraints between the binary variables associated with the adjacent legs. The final UDG-MWIS can be obtained by replacing each square with the factoring gadget that enforces the mathematical constraints relevant for the factoring problem. The unit-disk radius should be slightly larger than 2 √ 2 times the lattice constant. (c) An example of the UDG-MWIS representation of the factoring problem 6 = 3 × 2.
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Programmable quantum systems based on Rydberg atom arrays have recently been used for hardware-efficient tests of quantum optimization algorithms [Ebadi et al., Science, 376, 1209 (2022)] with hundreds of qubits. In particular, the maximum independent set problem on so-called unit-disk graphs, was shown to be efficiently encodable in such a quantum...
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... the factoring constraints F i,j . The constraints F i,j are manifestly local in two dimensions, in the sense that the variables s i,j , c i,j , p i,j , and q i,j can be arranged on a square lattice such that all factoring constraints F i,j involve only neighboring or diagonally neighboring variables. A graphical representation of this is given in Fig. 8(a), with the box at lattice point (i, j ) representing the constraints Specifically, each line represents a binary variable, and each box represents the factoring constraints that have to be satisfied by the variables connected to it. We note that each variable enters in exactly two factoring constraints, except at the ...
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... formulation of the factoring problem allows for a mapping to a UDG-MWIS problem. Specifically, we introduce a new factoring gadget consisting of a weighted 32-vertex unit-disk graph depicted in Fig. 8(b), where we identify eight of the vertices with the eight variables involved in the factoring constraints F i,j . See Appendix D for more details on the construction of the factoring gadget. In particular, we follow the design principle given in Sec. IV A: the factoring gadget is designed such that (i) the MWIS space is degenerate, (ii) ...
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... gadget is designed such that (i) the MWIS space is degenerate, (ii) every MWIS coincides with a valid solution of F i,j on the vertices that represent the involved variables, and (iii) every valid solution of F i,j is represented by at least one MWIS. All these requirements can be checked by exhaustive search for the factoring gadget depicted in Fig. 8(b). Since each variable has to satisfy two factoring constraints [see Fig. 8(a)], we design the factoring gadget such that this geometrical requirement can be easily met. Indeed, we can represent the full set of constraints {F i,j |i = 0, . . . , k − 1; j = 0, . . . , n − k − 1} as a unit-disk graph (of unit radius r = 2 √ 2 on a square ...
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... coincides with a valid solution of F i,j on the vertices that represent the involved variables, and (iii) every valid solution of F i,j is represented by at least one MWIS. All these requirements can be checked by exhaustive search for the factoring gadget depicted in Fig. 8(b). Since each variable has to satisfy two factoring constraints [see Fig. 8(a)], we design the factoring gadget such that this geometrical requirement can be easily met. Indeed, we can represent the full set of constraints {F i,j |i = 0, . . . , k − 1; j = 0, . . . , n − k − 1} as a unit-disk graph (of unit radius r = 2 √ 2 on a square lattice), by repeating the factoring gadget on a k(n − k) square lattice, as ...
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... we design the factoring gadget such that this geometrical requirement can be easily met. Indeed, we can represent the full set of constraints {F i,j |i = 0, . . . , k − 1; j = 0, . . . , n − k − 1} as a unit-disk graph (of unit radius r = 2 √ 2 on a square lattice), by repeating the factoring gadget on a k(n − k) square lattice, as depicted in Fig. 8(c). This construction therefore results in a lattice with some of the boundary conditions fixed by the values of m i , such that the MWIS of the rest of the graph reveals the values of p i and q j , satisfying Eq. (11), thus providing the solution for the factoring ...
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... this Appendix, we elaborate on the factoring gadget introduced in Fig. 8. The factoring gadget is designed such that the MWIS space corresponds to the satisfying assignments ...
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... = 2e + f . To obtain the MWIS representation of ab + c + d = 2e + f , we thus simply join the graphs in Figs. 12(a) and 12(b) at the common vertex z. Note that the total weight of the vertex z in this joint graph is the sum of its weights in each individual graph. One can easily identify this joint structure in the full factoring gadget given in Fig. 8(b). The remaining parts of this gadget are simply formed by combining it with copy and crossing gadgets that satisfy (D2) and (D3) and to route the variables to positions where they can be accessed also by neighboring factoring ...
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... For satisfying constraints of the original problem or experimental device limitations, penalties can be added to the cost function. Nevertheless, it is often not sufficient and thinking on how to encode the constraints directly in the ansatz seems to be the most promising way (Nguyen et al., 2023). This approach was implemented on a quantum emulator considering Rydberg atoms device on Maximum Independent Set (MIS) and max-k-cut problems with promising results (Dalyac et al., 2021). ...
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... Recent experimental development have resulted in renewed interest in spin glasses. In particular, the precise positioning of large numbers of Rydberg atoms with tweezers provides an avenue towards the realization of spin glasses with long-ranged interactions [53][54][55][56][57][58]. The idea is as follows, lasers are used to drive the Rabi transition between ground-state atoms and a highly excited long-lived Rydberg state. ...
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Published by the American Physical Society 2024
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