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IEEE International Symposium on Multiple-Valued Logic (ISMVL)

Quantum circuit synthesis using classes of GF(3) reversible fast spectral transforms

Anas N. Al‐Rabadi

Published Sep 28, 2004• 9 citations
Publisher

Abstract

Novel quantum circuit synthesis, using reversible Davio expansions, is introduced. The new method uses two planes to synthesize the quantum circuits: (1) a reversible butterfly circuit plane; and (2) a plane of quantum gates to perform additions and multiplications. Since the reduction of power consumption is a major requirement for circuit design of future technologies, such as in quantum circuits, the main features of several future technologies must include reversibility, and thus the new synthesis method, using reversible butterfly circuits, can play an important role in the synthesis of circuits that consume minimal power.

DOI
10.1109/ismvl.2004.1319925

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Novel quantum circuit synthesis, using reversible Davio expansions, is introduced. The new method uses two planes to synthesize the quantum circuits: (1) a reversible butterfly circuit plane; and (2) a plane of quantum gates to perform additions and multiplications. Since the reduction of power consumption is a major requirement for circuit design of future technologies, such as in quantum circuits, the main features of several future technologies must include reversibility, and thus the new synthesis method, using reversible butterfly circuits, can play an important role in the synthesis of circuits that consume minimal power.
Many-valued quantum circuit synthesis of many-valued reversible expansions and fast transforms is introduced in this paper. Since the reduction of power consumption is a major requirement for the circuit design in future technologies, such as in quantum computing, the main features of several future technologies will include reversibility. Consequently, the new quantum circuits can play an important task in the design of future circuits that consume minimal power. In addition, the new quantum circuit methodology is general and can be used to realize any multiple-valued function in the quantum space. It is also shown that the structural generality advantage that results from the new design methodology is opposed with the increased quantum structural complexity disadvantage which may be considered in any further cost-benefit design analysis.
Reversible logic is a key technique for quantum computing so leading to low-power designs. However, current synthesis algorithms for reversible circuits are low efficiency and do not obtain optimized reversible circuits, so they are only applied to small logic functions. In this paper, we propose a new method based on positive Davio expansion to synthesize reversible circuits, which generates a positive Davio decision diagram for a logic function and transfers diagram nodes to reversible circuits. The algorithm has advantages of optimizing area and fast synthesis speed compared to BDD (Binary decision diagram) based and RM (Reed-Muller) based synthesis method, so it can be adapted for large functions.
Reversible logic is becoming more and more popular due to the fact that many novel technologies such as quantum computing, low power CMOS circuit design or quantum optical computing are becoming more and more realistic. In quantum computing, reversible computing is the main venue for the realization and design of classical functions and circuits. We present a new approach to synthesis of reversible circuits using Kronecker Functional Lattice Diagrams (KFLD). Unlike many of contemporary algorithms for synthesis of reversible functions that use n×n Toffoli gates, our method synthesizes functions using 3×3 Toffoli gates, Feynman gates and NOT gates. This reduces the quantum cost of the designed circuit but adds additional ancilla bits. The resulting circuits are always regular in a 4-neighbor model and all connections are predictable. Consequently resulting circuits can be directly mapped in to a quantum device such as quantum FPGA [14]. This is a significant advantage of our method, as it allows us to design optimum circuits for a given quantum technology.
The biggest problems in system design today, and in the future, are the high rate of power consumption, and the emergence of quantum effects for highly dense ICs. The real challenge is to design reliable systems that consume as little power as possible and in which the signals are processed and transmitted at very high speeds with very high signal integrity. Current tools are used to design ICs using only classical design methodologies that apply conventional synthesis constraints such as area, delay, and power. As it was proven, physical processes have to be logically reversible in order to reduce (theoretically eliminate) power consumption. Logical reversibility requires that one can obtain the vector of inputs from the vector of outputs (i.e., backward process) as well as the vector of outputs from the vector of inputs (i.e., forward process). Since only ad-hoc methods were used previously for the reversible synthesis of logic functions, and since systematic and efficient reversible logic synthesis methodologies were significantly missing from previous literature, this dissertation provides several original contributions to reversible logic synthesis by providing a set of tools that can be systematically used to synthesize and evaluate logic functions using reversible logic. This includes, among other new results, Reversible Lattice Structures, Reversible Modified Reconstructability Analysis, Reversible Nets, Reversible Decision Diagrams, and Reversible Cascades. To solve the problem of high signal integrity when processing (computing and transmitting) information using extremely high dense circuits one needs to incorporate the physical quantum mechanical effects that are unavoidable in the nano scales. Since quantum circuits are reversible, and since many of the underlying theorems and formalisms for multiple-valued quantum computing were significantly missing from previous literature, new fundamental foundations for such computations had to be established. These new results include, but not limited to, Quantum Chrestenson Operator, new types of Quantum Decision Trees and Quantum Decision Diagrams as efficient representations for quantum computing, new Composite Basis States, new multiple-valued Einstein-Podolsky-Rosen (EPR) Basis States, and new classes of quantum primitives. Initial evaluations and conclusions for the comparative advantages and disadvantages of the new reversible and quantum computing methodologies are also provided.
Reversible circuits play an important role in quantum computation. The realization of reversible quantum circuits is required for any quantum computer to be universal. This paper1studies the realization and synthesis problem of reversible circuits. Using group theory, we present two sets of new 3-bit reversible logic gates, which included g1g2g3and nine complex gates, respectively. It is shown that any 3-bit reversible logic circuit is realizable by cascading NOT and Feynman gates, and at most one instance of our proposed gates. Given any n-bit reversible function, there are N distinct input patterns different from their corresponding outputs, where N ≤ 2n, and the other (2n- N) input patterns will be the same as their outputs. According to a synthesis example, it demonstrates that the main results are constructive and practical than others.