Abstract
In reversible as well as quantum computation, unitary matrices (so-called transformation matrices) are employed to comprehensively describe the respectively considered functionality. Due to the exponential growth of these matrices, dedicated and efficient means for their representation and manipulation are essential in order to deal with this complexity and handle reversible/quantum systems of considerable size. To this end, Quantum Multiple-Valued Decision Diagrams (QMDDs) have shown to provide a compact representation of those matrices and have proven their effectiveness in many areas of reversible and quantum logic design such as embedding, synthesis, or equivalence checking. However, the desired functionality is usually not provided in terms of QMDDs, but relies on alternative representations such as Boolean Algebra, circuit netlists, or quantum algorithms. In order to apply QMDD-based design approaches, the corresponding QMDD has to be constructed first—a gap in many of these approaches. In this paper, we show how QMDD representations can efficiently be obtained for Boolean functions, both reversible and irreversible ones, as well as general quantum functionality.
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- 1.
Initial experiments verifying the underlying link between information loss and thermodynamics have been reported in [2].
- 2.
Actually, there is a large body of research on how to derive BDD representations from various other, algebraic or netlist-based, representations of Boolean functions.
- 3.
Without loss of generality, we consider only basis states of the underlying quantum system, i.e. each qubit is assumed to be in one of its basis states. Due to the linearity of quantum operations, these are sufficient to construct the corresponding transformation matrix which yields the correct behaviour also for the case of superposed input states.
- 4.
The appropriate weights of the base transition will be incorporated later.
- 5.
If there is no further control below the current qubit, the gate inactivity is ensured by choosing a 0-edge as the initial QMDD.
References
Athas, W., Svensson, L.: Reversible logic issues in adiabatic CMOS. In: Proceedings of the Workshop on Physics and Computation, PhysComp 1994, pp. 111–118 (1994)
Berut, A., Arakelyan, A., Petrosyan, A., Ciliberto, S., Dillenschneider, R., Lutz, E.: Experimental verification of Landauer’s principle linking information and thermodynamics. Nature 483, 187–189 (2012)
Bryant, R.E.: Graph-based algorithms for Boolean function manipulation. IEEE Trans. Comput. 35(8), 677–691 (1986)
Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Theory of Computing, pp. 212–219 (1996)
Houri, S., Valentian, A., Fanet, H.: Comparing CMOS-based and NEMS-based adiabatic logic circuits. In: Conference on Reversible Computation, pp. 36–45 (2013)
Merkle, R.C.: Reversible electronic logic using switches. Nanotechnology 4(1), 21 (1993)
Mermin, N.D.: Quantum Computer Science: An Introduction. Cambridge University Press, New York (2007)
Miller, D.M., Thornton, M.A.: QMDD: a decision diagram structure for reversible and quantum circuits. In: International Symposium on Multi-Valued Logic, p. 6 (2006)
Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, New York (2000)
Niemann, P., Wille, R., Drechsler, R.: On the “Q” in QMDDs: efficient representation of quantum functionality in the QMDD data-structure. In: Conference on Reversible Computation, pp. 125–140 (2013)
Niemann, P., Wille, R., Drechsler, R.: Efficient synthesis of quantum circuits implementing Clifford group operations. In: ASP Design Automation Conference, pp. 483–488 (2014)
Niemann, P., Wille, R., Drechsler, R.: Equivalence checking in multi-level quantum systems. In: Conference on Reversible Computation, pp. 201–215 (2014)
Niemann, P., Wille, R., Miller, D.M., Thornton, M.A., Drechsler, R.: QMDDs: efficient quantum function representation and manipulation. IEEE Trans. CAD 35(1), 86–99 (2016)
Ren, J., Semenov, V., Polyakov, Y., Averin, D., Tsai, J.S.: Progress towards reversible computing with nSQUID arrays. IEEE Trans. Appl. Supercond. 19(3), 961–967 (2009)
Shor, P.W.: Algorithms for quantum computation: discrete logarithms and factoring. In: Foundations of Computer Science, pp. 124–134 (1994)
Soeken, M., Wille, R., Hilken, C., Przigoda, N., Drechsler, R.: Synthesis of reversible circuits with minimal lines for large functions. In: ASP Design Automation Conference, pp. 85–92 (2012)
Soeken, M., Wille, R., Keszocze, O., Miller, D.M., Drechsler, R.: Embedding of large Boolean functions for reversible logic. J. Emerg. Technol. Comput. Syst. 12(4), 41:1–41:26 (2015)
Somenzi, F.: Efficient manipulation of decision diagrams. Softw. Tools Technol. Transf. 3(2), 171–181 (2001)
Wille, R., Große, D., Teuber, L., Dueck, G.W., Drechsler, R.: RevLib: an online resource for reversible functions and reversible circuits. In: International Symposium on Multi-Valued Logic, pp. 220–225 (2008). RevLib is available at http://www.revlib.org
Zulehner, A., Wille, R.: Make it reversible: efficient embedding of non-reversible functions. In: Design, Automation and Test in Europe, pp. 458–463 (2017)
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This work has partially been supported by the European Union through the COST Action IC1405.
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Niemann, P., Zulehner, A., Wille, R., Drechsler, R. (2017). Efficient Construction of QMDDs for Irreversible, Reversible, and Quantum Functions. In: Phillips, I., Rahaman, H. (eds) Reversible Computation. RC 2017. Lecture Notes in Computer Science(), vol 10301. Springer, Cham. https://doi.org/10.1007/978-3-319-59936-6_17
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DOI: https://doi.org/10.1007/978-3-319-59936-6_17
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