Embedding of Large Boolean Functions for Reversible Logic

Abstract

Reversible logic represents the basis for many emerging technologies and has recently been intensively studied. However, most of the Boolean functions of practical interest are irreversible and must be embedded into a reversible function before they can be synthesized. Thus far, an optimal embedding is guaranteed only for small functions, whereas a significant overhead results when large functions are considered. In this paper, we study this issue. We prove that determining an optimal embedding is coNP-hard already for restricted cases. Then, we propose heuristic and exact methods for determining both the number of additional lines as well as a corresponding embedding. For the approaches we considered sums of products and binary decision diagrams as function representations. Experimental evaluations show the applicability of the approaches for large functions. Consequently, the reversible embedding of large functions is enabled as a precursor to subsequent synthesis.

One Additional Qubit is Enough: Encoded Embeddings for Boolean Components in Quantum Circuits

“…We compare the idea of coded embeddings to approaches that do not consider coding when realizing a Boolean function f : B n → B m in quantum logic. More precisely, we compare to exact methods utilizing max(n, m + ⌈log 2 µ(p 1 )⌉) qubits [13], [14] as well as to heuristic ones that always utilize a Bennett embedding with n + m qubits [23], [24] (e.g., generated when using an ESoP based synthesis approach [25]). f51m 159 14 8 22 19 15 tial 214 14 8 22 19 15 cu 141 14 11 25 25 15 misex3 180 14 14 28 28 15 misex3c 181 14 14 28 28 15 table3 209 14 14 28 28 15 s1488 split 14 25 39 38 15 s1494 split 14 25 39 38 15 b12 15 9 24 22 16 in0 162 15 11 26 25 16 parity 188 16 1 17 16 16 ryy6 198 16 1 17 17 17 t481 208 16 1 17 17 17 cmb 134 16 4 20…”

“…In order to realize non-reversible functions in quantum logic, further qubits (often called ancillary, ancillae, or working qubits) have to be added in order to make the output patterns distinguishable and, hence, obtain a reversible function (a process called embedding [13], [14]). Moreover, such additional qubits are often used to store intermediate results and have to be restored to their initial state (by de-computing intermediate results) before "leaving" the oracle.…”

The Group Zoo of Classical Reversible Computing and Quantum Computing

“…The reader may verify that the irreversible AND function is embedded in the reversible TOFFOLI function, as presetting in Table 1b the input C to logic 0 leads to the output R being equal to A AND B, as is highlighted by boldface. In the general case, any irreversible truth table can be embedded in a reversible truth table with w = w i + w o or less bits [6]. Table 1 Truth table of two The next step in the journey from the conventional to the quantum world, is replacing the reversible truth table by a permutation matrix.…”

JKQ

“…Accordingly, there is a lively community of researchers and industrial stakeholders currently developing toolkits that allow to work with these machines (such as IBM's Qiskit [1], Google's Cirq [7], or Rigetti's Forest [9]). However, developing efficient design methods is an immensely complex task due to the combinatorial and exponential nature of many design problems considered for quantum computing-some have even been proven to be NP-complete [2], coNP-hard [16], or QMA-complete [13].…”