EfficientSAM3 for Reversible Sequential Logic

• The paper introduces EfficientSAM3, a methodology leveraging the 3×3 SAM gate to significantly reduce quantum cost, delay, and garbage outputs in reversible sequential circuits.
• It details a quantum-efficient decomposition of the SAM gate using NOT, CNOT, Controlled-V, and Controlled-V+ primitives, achieving a minimal quantum cost of 4.
• The paper demonstrates how the approach optimizes the synthesis of SR, JK, and D flip-flops, outperforming previous designs with improvements of up to 62% in cost and 67% in garbage reduction.

EfficientSAM3 refers to the methodology and resulting family of quantum cost–optimized reversible sequential logic circuits based on the 3×3 SAM (Selim-Al-Mamun) gate. This design framework achieves significant reductions in quantum cost, logical depth (delay), and garbage outputs in fundamental memory elements including SR, JK, and D latches and flip-flops when compared to prior reversible logic constructions (Mamun et al., 2014).

1. Formal Specification of the SAM Gate

The SAM gate is a 3×3 reversible logic gate mapping three inputs $(A, B, C)$ to three outputs %%%%1%%%% as follows:

• $P = \lnot A$
• $Q = (\lnot A \land B) + (A \land \lnot C)$
• $R = (\lnot A \land C) \oplus (A \land B)$

This functionality is bijective, ensuring each input vector corresponds uniquely to an output vector. The gate’s truth table is as follows:

$A$	$B$	$C$	$P$	$Q$	$R$
0	0	0	1	0	0
0	0	1	1	0	1
0	1	0	1	1	0
0	1	1	1	1	1
1	0	0	0	1	0
1	0	1	0	0	0
1	1	0	0	1	1
1	1	1	0	0	1

The block symbol for the SAM gate reflects its mappings: $A \rightarrow P = \lnot A$, $$B \rightarrow Q = (\lnot A \land B) + (A \land \lnot C)$$, $$C \rightarrow R = (\lnot A \land C) \oplus (A \land B)$$ (Mamun et al., 2014).

2. Quantum Cost and Gate Decomposition

The SAM gate achieves minimal overhead in quantum technology metric by being decomposable into standard 1×1 and 2×2 primitives, each with quantum cost 1. A minimal quantum-cost decomposition follows:

• NOT gate on line $A$ ($\lnot A$): cost 1
• CNOT (control $A$, target $B$): cost 1
• Controlled-V (control $A$, on $C$): cost 1
• Controlled-V$^+$ (control $P = \lnot A$, on $B$): cost 1

Thus, the quantum cost of SAM is:

$$QC(\mathrm{SAM}) = 1~(\mathrm{NOT}) + 1~(\mathrm{CNOT}) + 1~(\mathrm{CV}) + 1~(\mathrm{CV^+}) = 4$$

This structure allows the SAM gate to serve as an efficient reversible primitive in more complex sequential circuits (Mamun et al., 2014).

3. Synthesis of Reversible Sequential Elements

The SAM gate, in conjunction with other reversible primitives such as the Feynman gate (FG, 2×2, $QC=1$), Double-Feynman gate (DFG, 3×3, $QC=2$), and Modified Peres Gate (MPG, 3×3, $QC=4$), enables the construction of SR, JK, and D latches and flip-flops:

• SR Latch: Implements $Q^+ = S + \lnot R \cdot Q$ using MPG, with $QC=5$, delay=5, garbage=1 (ungated).
• Gated SR: Uses MPG, SAM, and DFG, yielding $QC=10$, delay=10, garbage=2.
• Master–Slave SR: Cascades two gated stages (with inverted clocks), achieving $QC=14$, delay=14, garbage=4.
• JK Latch: Characteristic $Q^+ = J \cdot \lnot Q + \lnot K \cdot Q$, synthesizable using SAM and FG, $QC=5$, delay=5, garbage=1 (ungated).
• Gated JK: Utilizes two SAMs, DFG, and FG, with $QC=10$, delay=10, garbage=2.
• Master–Slave JK: Two cascaded gated JK units, $QC=15$, delay=15, garbage=4.
• D Latch/Flip-Flop: Characteristic $$Q^+ = \mathrm{CLK} \cdot D + \lnot \mathrm{CLK} \cdot Q$$, implemented as one SAM and one DFG ($QC=6$, delay=6, garbage=1 for gated; $QC=11$, delay=11, garbage=3 for master–slave).

Clocks are routed as controls to the appropriate reversible gates in gated and master–slave constructions. Each design minimizes the quantum cost, delay, and garbage outputs compared to previous methods (Mamun et al., 2014).

4. Analytical Delay and Garbage Output

The designs are analytically characterized by their quantum cost ($QC$), logical depth ($D$), and garbage count ($G$):

Element	QC	Delay	Garbage
SR Flip-flop	5	5	1
Gated SR	10	10	2
Master–Slave SR	14	14	4
JK Flip-flop	5	5	1
Gated JK	10	10	2
Master–Slave JK	15	15	4
Gated D	6	6	1
Master–Slave D	11	11	3

This optimization of delay and garbage is direct, as both metrics scale with the quantum cost due to the sequential nature of the gate arrangements.

5. Comparative Evaluation and Dominance

A comparative performance table for the gated D flip-flop is representative:

Design	Proposed	Thapliyal’10	Lafifa’12
Gated D-FF $(QC, D, G)$	(6, 6, 1)	(7, 7, 2)	(7, 7, 2)
Improvement (%)	–14%, –14%, –50%	–14%, –14%, –50%

Analogous trends hold for SR, JK, and master–slave designs: the EfficientSAM3 approach achieves 23–62% quantum cost reductions, delay reductions commensurate with cost, and a 33–67% reduction in garbage outputs relative to prior art. These quantitative improvements establish strict dominance of the SAM-based construction in reversible sequential-element design (Mamun et al., 2014).

6. Implementation Principles and Application Context

In all circuits, the clock signal (CLK) is applied to the control inputs of the associated reversible gates (SAM, MPG, Feynman type) to realize standard gating or master–slave stage separation. Master–slave synchronization is achieved by cascading two gated stages with oppositely phased clocks. The resulting circuits are directly implementable as memory blocks in prospective low-power computing devices, fulfilling stringent requirements for reversibility, minimal energy dissipation, and quantum efficiency.

The EfficientSAM3 methodology thus constitutes a foundational toolset for the synthesis of next-generation quantum and reversible sequential logic, providing improved performance in physical implementations and facilitating rigorous benchmarking of emerging reversible circuit paradigms (Mamun et al., 2014).