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**GF(4) Based Synthesis of Quaternary Reversible/Quantum Logic Circuits**

Mozammel H. A. Khan East West University, Dhaka, Bangladesh Marek A. Perkowski Portland State University, Portland, OR, USA

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Introduction D-level (multiple-valued) quantum circuits have many advantages There is not much published about the practical circuit realization for such circuits MV logic functions having many inputs can be expressed as GFSOP GFSOP can be realized as cascade of Feynman and Toffoli gates No work has yet been done on expressing quaternary logic function as QGFSOP No work has yet been done on realizing QGFSOP as cascade of quaternary Feynman and Toffoli gates

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**Contribution of the Paper**

We have developed nine QGFEs (QGFE1 – QGFE9) We show way of constructing QGFDDs using QGFEs We show method of generating QGFSOP by flattening QGFDD We show technique of realizing QGFSOP as a cascade of quaternary 1-qudit, Feynman, and Toffoli gates

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**Contribution of the Paper (contd)**

We show way of 2-bit encoded quaternary realization of binary functions We have developed circuit for binary-to-quaternary encoding We have developed circuit for quaternary-to-binary decoding

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**Quaternary Galois field arithmetic**

Table 1. GF(4) operations + 1 2 3 Example: (2 x+1) 2= (2 2) x + (1 2) = 3 x + 2

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**Quaternary Galois field sum of products expression**

Table 2. Basic quaternary reversible-literals Input x x+1 x+2 x+3 1 2 3 2x 2x+1 2x+2 2x+3 3x 3x+1 3x+2 3x+3 Input x2 x2+1 x2+2 x2+3 1 2 3 2x2 2x2+1 2x2+2 2x2+3 3x2 3x2+1 3x2+2 3x2+3 Example of one-qutrit gate 3x2+1 Example of one-qutrit gate

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**Quaternary Galois field sum of products expression (contd)**

Table 3. Products of basic quaternary reversible-literals and the constant 2 literal x x+1 x+2 x+3 2(literal) 2x 2x+2 2x+3 2x+1 3x 3x+2 3x+3 3x+1 x2 x2+1 x2+2 x2+3 2x2 2x2+2 2x2+3 2x2+1 3x2 3x2+2 3x2+3 3x2+1 Example: (2 x+1) 2= (2 2) x + (1 2) = 3 x + 2

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**Quaternary Galois field sum of products expression (contd)**

Table 4. Product of basic quaternary reversible-literal and the constant 3 literal x x+1 x+2 x+3 3(literal) 3x 3x+3 3x+1 3x+2 2x 2x+1 2x+2 2x+3 x2 x2+1 x2+2 x2+3 3x2 3x2+3 3x2+1 3x2+2 2x2 2x2+1 2x2+2 2x2+3

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**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) + x These may be functions of one or more variables

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**Quaternary Galois field expansions**

Cofactors

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**Quaternary Galois field expansion (contd)**

Composite Cofactors See notation for some composite cofactors

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**Quaternary Galois field expansions (contd)**

First four Quaternary Expansions – they are generalizations of the familiar Shannon and Davio expansions Can be derived from inverted from quaternary Shannon Expansion. QGFE 1: QGFE 2: QGFE 3: QGFE 4:

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**Quaternary Galois field expansions (contd)**

QGFE 5:

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**Quaternary Galois field expansions (contd)**

QGFE 6:

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**Quaternary Galois field expansions (contd)**

QGFE 7:

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**Quaternary Galois field expansions (contd)**

QGFE 8: QGFE 9:

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**Quaternary Galois field decision diagrams**

Table 5. Truth Table of an example quaternary function F = x + y (GF4) x y x y 00 01 02 03 10 11 12 13 f(x,y) 1 2 3 xy 20 21 22 23 30 31 32 33

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**Quaternary Galois field decision diagrams (contd)**

x Two expansion variables, x and y y Figure 1. QGFDD for the function of Table 5 using QGFE1 and QGFE2

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**Quaternary Galois field decision diagrams (contd)**

x y Figure 2. QGFDD for the function of Table 5 using QGFE9

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**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.

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**Quaternary 1-qudit reversible/quantum gates**

Each of the 24 quaternary reversible-literals can be implemented as 1-qudit gates using quantum technology Figure 3. Representation of quaternary reversible 1-qudit gates

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**Quaternary 2-qudit Muthukrishnan-Stroud gate family**

Figure 4. Quaternary Muthukrishnan-Stroud gate family

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**Quaternary Feynman gate**

Figure 5. Quaternary Feynman gate Figure 6. Realization of quaternary Feynman gate

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**Quaternary Toffoli gate**

Figure 7. Quaternary Toffoli gate Figure 8. Realization of quaternary Toffoli gate One ancilla bit

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**Quaternary Toffoli gate (contd)**

Figure 9. Four-input quaternary Toffoli gate Two ancilla bits

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**Synthesis of QGFSOP expressions**

Figure 10. Realization of QGFSOP expression

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**Binary-to-quaternary encoder and quaternary-to-binary decoder circuits**

Figure 11. Binary-to-quaternary encoder circuit Figure 12. Quaternary-to-binary decoder circuit garbage inputs output outputs input

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**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.

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**Oracle for Quantum Map of Europe Coloring**

Germany France Switzerland Spain quaternary Spain France Germany Switzerland Good coloring

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**Oracle for Quantum Map of Europe Coloring**

1 2 3 1 2 3 0+1=1 1+1=0 2+1=3 3=1=2 0+0=0 1+0=1 2+0=2 3+0=3 0+3=3 1+3=2 2+3=1 3+3=0 0+2=2 1+2=3 2+2=0 3+2=1 A +1 1 when A = B 1 2 3 1 2 3 +1 B +2 +3 +3 +2 Quaternary Feynman Quaternary input/binary output comparator of equality

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**Oracle for Quantum Map of Europe Coloring**

Comparator for each frontier 1 2 3 +1 +3 +2 A B 1 1 -- when control 1 1 -- for controls 0,2 and 3 Binary qudit =1 for frontier AB when countries A and B have different colors 1 2 3 +1 +3 +2 C D Binary signal 1 when all frontiers well colored Quaternary controlled binary target gate Binary Toffoli

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**Conclusion We have developed nine QGFEs**

These QGFEs can be used for constructing QGFDDs By flattening the QGFDD we can generate QGFSOP We have shown example of implementation of QGFSOP as cascade of quaternary 1-qudit gate, Feynman gate, and Toffoli gate

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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 gate We 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 gates We also show the realization of m-qudit (m > 3) Toffoli gates using 3-qudit Toffoli gates

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Conclusion (contd) The quaternary base is very useful for 2-bit encoded realization of binary function We show quantum circuit for binary-to-quaternary encoder and quaternary-to-binary decoder for this purpose

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Conclusion (contd) The presented method is especially applicable to quantum oracles The developed method performs a conversion of a non-reversible function to a reversible one as a byproduct of the synthesis process Our method can be used for large functions As it is using Galois logic, the circuits are highly testable

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**Conclusion (contd) Our future research includes**

developing more QGFEs, if such expansions exist developing algorithms for selecting expansion for each variable variable ordering constructing QGFDD (Kronecker and pseudo-Kronecker types) for both single-output and multi-output functions Building gates on the level of Pauli Rotations, similarly as it was done in our published paper – Soonchil Lee et al.

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Conclusion (contd) Building gates on the level of Pauli Rotations will require deciding in which points on the Bloch Sphere are the basic states Paper in RM 2007 This 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).

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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.

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**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?

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Thanks Questions?

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