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Duplication-Aware Retiming and Cell Interface Redesign for Superconductor Circuit Minimization
Panagiotis Papanikolaou, Alex Vanasse, Haoran Jin, Georgios Tzimpragos, Jennifer Volk
TL;DR
Superconductor electronics remain limited in integration density, motivating efforts to deliver more functionality with fewer JJs. The paper combines duplication-aware retiming with redesigned SFQ cell interfaces. It reports lower JJ counts in single- and multi-cycle designs, with the latter establishing a new Pareto frontier against prior single-cycle implementations.
Problem
SCE integration density remains far below CMOS, motivating designs that deliver more functionality per unit area with fewer JJs.
Source: §1 · p. 1Method
The approach uses ILP-based retiming to co-optimize register placement and de-duplication, alongside an integrated xSFQ-amplifier cell library for interface costs.
Source: §1 · p. 2Results
63–71% fewer JJs in single-cycle and 41–66% fewer in multi-cycle implementations; the multi-cycle designs have shorter critical paths and lower JJ counts than prior single-cycle designs.
Source: §1 · p. 2Takeaways & Limitations
The retiming solution jointly optimizes sequential and combinational logic for monotonic circuits, beyond the SFQ application described here.
Source: §3.1 · p. 3
Abstract
from arXiv · showhide
Superconductor electronics have increasingly shifted away from RSFQ and its variants toward logic families that eliminate explicit gate-level clocking. While this transition enables simpler circuits and more efficient architectures, it also introduces an implicit reliance on dual-rail codes, resulting in inherent gate duplication. This work presents a duplication-aware retiming methodology for Josephson junction (JJ) count minimization, co-optimizing register placement and polarity assignment. The approach applies beyond SFQ to any monotonic circuit. We further identify cell interfaces---specifically, interconnect drivers, receivers, and fanout (FO) elements---as dominant contributors to JJ count in each cell. A new amplifier design is introduced to reduce these costs, integrated within existing SFQ cells, experimentally verified, and characterized to form a new SFQ cell library. Our results demonstrate a 63-71% JJ count reduction in single-cycle implementations and 41-66% reduction in multi-cycle implementations compared to the prior state-of-the-art. The latter establishes a new Pareto frontier, achieving both shorter critical paths and lower JJ counts than the best-to-date single-cycle implementations.
1 Introduction
Superconductor electronics remain far behind CMOS in integration density, motivating designs that deliver more functionality with fewer Josephson junctions. This work combines duplication-aware retiming and redesigned cell interfaces, reporting substantial JJ reductions in single- and multi-cycle implementations.
1 Introduction
- SCE reaches 1–10 million JJs per cm2, versus more than 10 billion transistors per cm2 in advanced CMOS, and planned fabrication advances are unlikely to close the gap.The paper proposes maximizing functionality per unit area—functional density—as a complementary path forward.Source: §1 · p. 1
- Clockless SFQ logic families share dual-rail signaling and passive transmission-line interconnection, and prior synthesis reduced dual-rail duplication by over 4× across combinational and sequential benchmarks.The prior method assigns a single polarity at synchronous interfaces and propagates choices backward [20].Source: §1 · p. 1
- The proposed ILP retiming method jointly balances register placement and gate de-duplication, while the integrated xSFQ-amplifier library reduces fanout and PTL-driving JJ costs.The library reduces these costs by 1.7–3.1× for FO2 and 2.8–6.0× for FO4.Source: §1 · p. 2
- 63–71% fewer JJs in single-cycle and 41–66% fewer in multi-cycle implementations establish a new Pareto frontier with shorter critical paths and lower JJ counts than prior single-cycle designs.The single-cycle comparison is against prior work [32]; the multi-cycle range reflects a tradeoff between throughput and JJ resources.Source: §1 · p. 2
2 Background
SFQ logic stores pulse-arrival state, while clockless monotonic families rely on dual-rail encoding because they lack native inversion. In xSFQ, polarity choices and DROC placement shape how much gate duplication can be removed.
2.1 SCE Digital Logic
- SFQ gates retain pulse-arrival state because picosecond-wide pulses cannot be practically synchronized for logic operations.A pulse arriving within a predefined interval represents 1 and triggers a transition; its absence represents 0 and leaves the gate state unchanged.Source: §2.1 · p. 2
- Clockless SFQ families support monotonic functions and use dual-rail encoding to implement arbitrary Boolean logic without native inversion, making some gate duplication unavoidable.Pulse encoding permits only 0 → 1 transitions within a cycle.Source: §2.1 · p. 2
2.2 xSFQ Logic Synthesis
- xSFQ uses First Arrival and Last Arrival cells as single-rail OR and AND equivalents, plus DROC storage; dual-rail OR and AND require FA-LA cell pairs.Because DROC cells provide complementary outputs, propagating only one polarity to a DROC input lets the logic producing the unused polarity be removed.Source: §2.2 · p. 2, §2.2.2 · p. 2
2.2.2 Polarity assignment.
- Polarity assignment and strategic DROC placement, including retiming, are the two factors identified for reducing duplication.Prior xSFQ synthesis addressed output polarity assignment [32]; this paper focuses on DROC placement.Source: §2.2.2 · p. 2
3 Duplication-Aware Retiming
In dual-rail circuits, register placement affects gate duplication, so minimizing register count alone need not minimize total cell or JJ count. The proposed ILP jointly chooses register placement and polarity assignment to minimize their combined cost.
3 Duplication-Aware Retiming
- Retiming changes duplication in dual-rail circuits, unlike traditional minimum-area retiming, which assumes register placement does not affect combinational logic.Figure 3 illustrates how moving a DROC can absorb an inverter and remove duplication, subject to timing constraints.Source: §3 · pp. 2–3
3.1 Proposed Method
- Joint register-placement and polarity optimization targets total JJ count because either minimizing registers or maximizing de-duplication alone can increase overall cost.Each DROC register costs more JJs than a single FA or LA cell.Source: §3.1 · p. 3
3.2 ILP Formulation
- The ILP takes an AIG and outputs register placements and polarity assignments that minimize the combined register and gate-duplication cost.Its input represents AND nodes, inverter-bearing edges, and the initial register count on each edge.Source: §3.2 · p. 3, §3.1 · p. 3
3.2.1 Variables.
- Node labels encode retiming direction, whether De Morgan’s law is applied, and whether a node must be duplicated; edge labels encode post-retiming inversion.Positive retiming values move registers backward, while negative values move them forward.Source: §3.2.1 · p. 3
3.2.2 Constraints.
- Constraints preserve non-negative edge register counts, require registers on paths exceeding the clock period, and prevent moving registers across primary inputs or outputs.They also update edge inversion under De Morgan transformations and identify duplication when an unregistered successor has an inverter or is itself duplicated.Source: §3.2.2 · pp. 3–4
3.2.3 Objective Function.
- The objective minimizes total area by balancing register cost against duplicated-gate cost, including interface overhead.The stated cost terms also account for register-sharing opportunities.Source: §3.2.3 · p. 4
4 Cell Interface Redesign
PTL interfaces and fanout splitters add substantial junction cost around xSFQ logic. The section quantifies those costs and motivates an amplifier with higher fanout and PTL-driving capability.
4.1.1 Gate interconnection.
- PTL receivers and drivers cost 3 and 2 JJs, respectively, producing an 8-JJ interface overhead for a two-input, one-output xSFQ gate.The proposed amplifier uses a variable-inductance path for higher fanout and PTL-driving capability than conventional splitters and PTL interfaces.Source: §4.1.1 · p. 4, Figure 4 · p. 4
4.1.2 Fanout.
- Each FO2 splitter costs 3 JJs, excluding its PTL drivers and receivers, and splitting a pulse without added energy can degrade operating margins or prevent loading cells from being driven.A 2-input, 1-output gate library needs one FO2 splitter at every cell output.Source: §4.1.2 · p. 4, §4.1.3 · p. 4
4.1.3 Cost summary.
- 13–18 JJs of interface cost per FO2 xSFQ cell exceeds the 4-JJ logic cost by more than 3×.The range depends on whether the gate and splitter are abutted; without abutment, the interface cost is 18 JJs.Source: §4.1.3 · p. 4
4.2 Proposed Interface
The proposed amplifier uses a varying-inductance path to drive fanout and PTLs, then integrates with xSFQ cells to replace conventional interface elements. Its operation also redirects current under near-overbias and near-underbias conditions, enabling operation at lower-than-typical bias points.
4.2.1 Key characteristics & functionality.
- The amplifier’s nonlinear, varying-inductance path lets an SFQ pulse propagate and drive fanout and PTLs at a cost of 4 JJs.As the pulse peaks, inductance increases and effectively turns the cell into two smaller JJs; bias and SFQ current are then redirected to the output.Source: §4.2.1 · pp. 4–5
- The varying-inductance path redirects current away from the most sensitive JJ near overbias and slows its bias decrease near underbias.These behaviors enable operation at lower-than-typical bias points.Source: §4.2.1 · p. 5
4.2.2 Integration with xSFQ.
- Integrating the amplifier into xSFQ cells replaces PTL receivers, PTL drivers, and FO2–FO4 splitters; fanouts beyond FO4 can use chained standalone SPLIT cells.With JJ sharing, FA and LA cells cost 7 JJs, compared with 17/22 JJs for FO2 and 27/42 JJs for FO4, with and without abutment, respectively.Source: §4.2.2 · p. 5
4.3 Cell Library
The proposed xSFQ-amplifier library supports fanout up to FO4 and reduces both junction counts and cell delays relative to xSFQ-DAC61.
4.3 Cell Library
- Every proposed cell natively supports up to FO4, while xSFQ-DAC61 FA, LA, and DROC cells support only FO1.The proposed SPLIT cell also provides additional FO4 fanout.Source: §4.3 · p. 5
- Proposed cells reduce JJ count by 1.7–3.1× for FO2 and 2.8–6.0× for FO4.The comparison is against xSFQ-DAC61.Source: §4.3 · p. 5
- Proposed cells cut cell delay by 2–4.5×, with the speedup attributed to fewer JJs on critical paths.The stated speedup is before accounting for fanout overhead in xSFQ-DAC61.Source: §4.3 · p. 5
5.1 Duplication-Aware Retiming
The evaluation compares duplication-aware retiming with an enhanced xSFQ-DAC61 baseline and measures JJ count and critical-path delay across ISCAS85 benchmarks. Across those circuits, retiming reduces JJ count, with gains that can accompany throughput improvement.
5.1.1 Baseline.
- The baseline is xSFQ-DAC61 augmented with minimum-register retiming and ILP-based polarity assignment.These enhancements provide a stronger comparison point than xSFQ-DAC61 with ABC’s default retiming.Source: §5.1.1 · p. 5
5.1.2 Methodology.
- The workflow synthesizes each benchmark to an AIG, computes fanout-dependent node delays, solves the retiming ILP, then maps and verifies netlists while extracting delay and JJ count.All ISCAS85 circuits completed within the one-hour solver timeout.Source: §5.1.2 · p. 5
5.1.3 Results.
- Experiments sweep pipeline depths of two, four, and six stages under strict and relaxed timing targets, using xSFQ-DAC61 to isolate retiming benefits.The two timing settings bound the practical clock-period range for each pipeline depth.Source: §5.1.3 · p. 6
- JJ count reduction across ISCAS85 circuits averages 7–15% and reaches 22% maximum.Gains are consistently higher under relaxed timing targets; c3540 uses 23,041 JJs after 4-stage pipelining and retiming, versus 35,146 JJs as a single-cycle design, a 34% reduction and Pareto improvement.Source: §5.1.3 · p. 6
5.2 Cell Interface Redesign
The redesigned library reduces JJ counts on single-cycle ISCAS85 and EPFL circuits, and its integrated amplifier is validated through simulation and a fabricated LA cell with FO2. The fabricated circuit’s measured transitions indicate correct operation.
5.2.1 Functional validation through simulation.
- Simulations using SFQ5ee device models verified global bias margins of at least ±15% across valid fanout and amplifier loading combinations.The cells were simulated while both driving and being loaded by fanouts from one to four.Source: §5.2.1 · p. 6
5.2.2 Functional validation through fabrication.
- A fabricated LA cell with FO2 was measured at 4.2 K, and all measured transitions demonstrated correct operation.Input current and output voltage transitions indicated logic 1.Source: §5.2.2 · p. 6
5.2.3 Results.
- The new library reduces JJ count by 2.7–3.4× compared to xSFQ-DAC61, with splitter counts dropping sharply and reaching zero for the priority benchmark.The single-cycle comparison includes PTL interface costs for both libraries.Source: §5.2.3 · p. 6
5.3 Combined Improvements
Combining duplication-aware retiming with redesigned cell interfaces reduces JJ counts by 41–66% in pipelined circuits. The results establish a Pareto frontier with shorter critical paths and lower JJ counts than prior single-cycle implementations.
5.3 Combined Improvements
- The designs achieve shorter critical paths and lower JJ counts than the best-to-date single-cycle implementations [32].Source: §5.3 · p. 6
- Introducing DROC cells, whose base designs use roughly 3× more JJs than FA/LA cells, may limit percentage savings from pipelining.Source: §5.3 · p. 6
- 41–66% JJ count reduction is achieved for pipelined circuits relative to the xSFQ-DAC61 cell library and enhanced baseline flow.Figure 6 covers 2, 4, and 6 pipeline stages.Source: Figure 6 · p. 6, §5.3 · p. 6
6 Conclusion
The paper combines duplication-aware retiming with redesigned SFQ cell interfaces, reporting 1.7–3.4× JJ count reduction over prior state-of-the-art dual-rail SFQ, depending on pipelining.
6 Conclusion
- 1.7–3.4× JJ count reduction over prior state-of-the-art dual-rail SFQ is achieved, depending on the degree of pipelining.Source: §6 · p. 6
- The work presents duplication-aware retiming combined with cell interface redesign for SFQ circuits.Source: §6 · p. 6
- The retiming solution jointly optimizes sequential and combinational logic for monotonic circuits, and the work presents the most compact SFQ cell library to date.Source: §6 · p. 6