GF(4) Based Synthesis of Quaternary Reversible/Quantum Logic Circuits
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1 GF(4) Based Synthesis of Quaternary Reversible/Quantum Logic Circuits Mozammel H. A. KhanEast West University, Dhaka, BangladeshMarek A. PerkowskiPortland State University, Portland, OR, USA
2 IntroductionD-level (multiple-valued) quantum circuits have many advantagesThere is not much published about the practical circuit realization for such circuitsMV logic functions having many inputs can be expressed as GFSOPGFSOP can be realized as cascade of Feynman and Toffoli gatesNo work has yet been done on expressing quaternary logic function as QGFSOPNo work has yet been done on realizing QGFSOP as cascade of quaternary Feynman and Toffoli gates
3 Contribution of the Paper We have developed nine QGFEs (QGFE1 – QGFE9)We show way of constructing QGFDDs using QGFEsWe show method of generating QGFSOP by flattening QGFDDWe show technique of realizing QGFSOP as a cascade of quaternary 1-qudit, Feynman, and Toffoli gates
4 Contribution of the Paper (contd) We show way of 2-bit encoded quaternary realization of binary functionsWe have developed circuit for binary-to-quaternary encodingWe have developed circuit for quaternary-to-binary decoding
5 Quaternary Galois field arithmetic Table 1. GF(4) operations+123Example: (2 x+1) 2= (2 2) x + (1 2) = 3 x + 2
6 Quaternary Galois field sum of products expression Table 2. Basic quaternary reversible-literalsInputxx+1x+2x+31232x2x+12x+22x+33x3x+13x+23x+3Inputx2x2+1x2+2x2+31232x22x2+12x2+22x2+33x23x2+13x2+23x2+3Example of one-qutrit gate3x2+1Example of one-qutrit gate
7 Quaternary Galois field sum of products expression (contd) Table 3. Products of basic quaternary reversible-literals and the constant 2literalxx+1x+2x+32(literal)2x2x+22x+32x+13x3x+23x+33x+1x2x2+1x2+2x2+32x22x2+22x2+32x2+13x23x2+23x2+33x2+1Example: (2 x+1) 2= (2 2) x + (1 2) = 3 x + 2
8 Quaternary Galois field sum of products expression (contd) Table 4. Product of basic quaternary reversible-literal and the constant 3literalxx+1x+2x+33(literal)3x3x+33x+13x+22x2x+12x+22x+3x2x2+1x2+2x2+33x23x2+33x2+13x2+22x22x2+12x2+22x2+3
9 Quaternary Galois field sum of products expression (contd) Product of two or more basic quaternary reversible-literals is called a QGFP.(2x+2)(3x2+2)(2x2)Sum of two or more QGFP is called a QGFSOP(2x+2)(3x2+2) + (3x+1)(2x) + xThese may be functions of one or more variables
11 Quaternary Galois field expansion (contd) Composite CofactorsSee notation for some composite cofactors
12 Quaternary Galois field expansions (contd) First four Quaternary Expansions – they are generalizations of the familiar Shannon and Davio expansionsCan be derived from inverted from quaternary Shannon Expansion.QGFE 1:QGFE 2:QGFE 3:QGFE 4:
13 Quaternary Galois field expansions (contd) QGFE 5:
14 Quaternary Galois field expansions (contd) QGFE 6:
15 Quaternary Galois field expansions (contd) QGFE 7:
16 Quaternary Galois field expansions (contd) QGFE 8:QGFE 9:
17 Quaternary Galois field decision diagrams Table 5. Truth Table of an example quaternary functionF = x + y (GF4)xyx y0001020310111213f(x,y)123xy2021222330313233
18 Quaternary Galois field decision diagrams (contd) xTwo expansion variables, x and yyFigure 1. QGFDD for the function of Table 5 using QGFE1 and QGFE2
19 Quaternary Galois field decision diagrams (contd) xyFigure 2. QGFDD for the function of Table 5 using QGFE9
20 Quaternary Galois Field Decision Diagrams Similarly to KFDDs, the order of variables and the choice of expansion type for every level affects the number of nodes (size) of the decision diagram.
21 Quaternary 1-qudit reversible/quantum gates Each of the 24 quaternary reversible-literals can be implemented as 1-qudit gates using quantum technologyFigure 3. Representation of quaternary reversible 1-qudit gates
22 Quaternary 2-qudit Muthukrishnan-Stroud gate family Figure 4. Quaternary Muthukrishnan-Stroud gate family
28 Are d-level quantum circuit an advantage? Benchmarking is necessary.In some cases quaternary circuit is much simpler than binary.These applications include especially circuits with many arithmetic blocks and comparators.Control should be binary, data path should be multiple-valued.We need hybrid circuits that convert from binary to d-level and vice versa. This is relatively easy in quantum.
29 Oracle for Quantum Map of Europe Coloring GermanyFranceSwitzerlandSpainquaternarySpainFranceGermanySwitzerlandGood coloring
30 Oracle for Quantum Map of Europe Coloring 1231230+1=11+1=02+1=33=1=20+0=01+0=12+0=23+0=30+3=31+3=22+3=13+3=00+2=21+2=32+2=03+2=1A+11 when A = B123123+1B+2+3+3+2Quaternary FeynmanQuaternary input/binary output comparator of equality
31 Oracle for Quantum Map of Europe Coloring Comparator for each frontier123+1+3+2AB1 1 -- when control 11 -- for controls 0,2 and 3Binary qudit =1 for frontier AB when countries A and B have different colors123+1+3+2CDBinary signal 1 when all frontiers well coloredQuaternary controlled binary target gateBinary Toffoli
32 Conclusion We have developed nine QGFEs These QGFEs can be used for constructing QGFDDsBy flattening the QGFDD we can generate QGFSOPWe have shown example of implementation of QGFSOP as cascade of quaternary 1-qudit gate, Feynman gate, and Toffoli gate
33 Conclusion (contd)For QGFSOP based quantum realization of functions with many input variables, we need to use quantum gates with many inputs.Quantum gates with more than two inputs are very difficult to realize as a primitive gateWe have shown the quantum realization of macro-level quaternary 2-qudit Feynman and 3-qudit Toffoli gates on the top of theoretically liquid ion-trap realizable 1-qudit gates and 2-qudit Muthukrishnan-Stroud primitive gatesWe also show the realization of m-qudit (m > 3) Toffoli gates using 3-qudit Toffoli gates
34 Conclusion (contd)The quaternary base is very useful for 2-bit encoded realization of binary functionWe show quantum circuit for binary-to-quaternary encoder and quaternary-to-binary decoder for this purpose
35 Conclusion (contd)The presented method is especially applicable to quantum oraclesThe developed method performs a conversion of a non-reversible function to a reversible one as a byproduct of the synthesis processOur method can be used for large functionsAs it is using Galois logic, the circuits are highly testable
36 Conclusion (contd) Our future research includes developing more QGFEs, if such expansions existdeveloping algorithms forselecting expansion for each variablevariable orderingconstructing QGFDD (Kronecker and pseudo-Kronecker types) for both single-output and multi-output functionsBuilding gates on the level of Pauli Rotations, similarly as it was done in our published paper – Soonchil Lee et al.
37 Conclusion (contd)Building gates on the level of Pauli Rotations will require deciding in which points on the Bloch Sphere are the basic statesPaper in RM 2007This will affect rotations between these states, which means complexity of single-qudit and two-qudit gates.For each of created gates we will calculate quantum cost – numbers of Pauli rotations and Interaction gates (Controlled Z).
38 Conclusion (contd)Building blocks on the levels of first these gates and next Pauli Rotations to analyse what is the real gain of using mv concepts – example is comparator in Graph Coloring Oracle for Grover.Adders, multipliers, comparators of order, counting circuits, converters between various representations, etc.Practical realization of Pauli rotations and Interaction gates (Controlled Z) in NMR, ion trap, one-way and other technologies.
39 Research Questions What is the best location of basic states? Can Galois Field mathematical theory be used for more efficient factorization or expansion, in general for synthesis?Can we generalize Galois Field to circuits with 6 basic states (for which good placement on Bloch Sphere exists)?Can we create some kind of algebra to allow expansion, factorization and other algebraic rules directly applied to easily realizable MS gates, rather than complex gates such as based on Galois Fields?