Charge-Domain Computation Overview

• Charge-domain computation is defined as a family of techniques that treat stored electrical charge as the primary computational state in capacitive circuits, CiM architectures, or excited state analyses.
• Mixed-signal and compute-in-memory architectures leverage charge sampling, sharing, and accumulation to perform multiply-accumulate operations efficiently, reducing energy overhead.
• Wavefunction-based approaches use localized orbital domains to quantify electron excitation, offering detailed insights into directional charge transfer in molecular systems.

Charge-domain computation denotes a class of computational formalisms in which charge is treated as the primary computational state, either physically as stored or redistributed electrical charge on capacitive structures, or analytically as a domain-resolved descriptor of electronic excitation and transfer. In the cited literature, the term spans three closely related but distinct usages: a general signal-theoretic treatment of capacitive devices as mappings between voltage and charge in time and frequency domains; mixed-signal and compute-in-memory architectures that implement multiply-accumulate operations through charge sampling, charge sharing, and charge accumulation; and a wavefunction-based excited-state analysis in which localized orbital domains are used to quantify directional charge transfer between donor, bridge, and acceptor fragments (Allagui et al., 2022, Ghodrati et al., 2019, Yin et al., 2021, Pandey et al., 24 Nov 2025).

1. Scope and terminology

The phrase has a broader technical range than the textbook capacitor law alone suggests. In mixed-signal hardware, charge-domain computation refers to circuits in which values are represented by charge or voltage on capacitors, multiplication is realized through charge sampling and charge sharing, accumulation is realized by storing and combining charge on accumulation capacitors, and digitization is deferred until after multiple partial products have been integrated (Ghodrati et al., 2019). In charge-domain compute-in-memory, the same principle is specialized to array macros in which XNOR or MAC outcomes are encoded as charge contributions on a floating accumulation line rather than as summed currents (Yin et al., 2021). In quantum-chemical excited-state analysis, “charge-domain computation” is not a density partitioning scheme in the usual Mulliken, Löwdin, or Bader sense, but an excited-state wavefunction decomposition over localized orbital domains (Pandey et al., 24 Nov 2025).

Usage in the cited literature	Primary computational object	Representative relation
Capacitive system theory	System function between voltage and charge	$q(t)=\mathcal{C}_{\text{tt}}\{v(t)\}$
Switched-capacitor / CiM hardware	Stored or redistributed charge on capacitors	$Q=CV$, $V_{ScL}=V_{DD}\frac{M}{N}$
Domain-based excited-state analysis	Squared excitation-amplitude sum over domains	$$\%_{\mathrm{CT}(A \rightarrow B)}=\sum_{i\in A}\sum_{a\in B}|\mathrm{CI}_{i\rightarrow a}|^2$$

This terminological spread is not accidental. Across the cited work, the scalar view of capacitance or transfer is replaced by an explicitly structured description: kernel system functions in capacitive devices, grouped low-bit charge accumulation in mixed-signal arithmetic, or localized orbital-domain amplitudes in molecular excitation analysis.

2. Capacitive devices as charge-domain operators

A general theoretical foundation is provided by the formulation of capacitive devices as input/output systems with eight possible mapping functions: four for voltage-charged devices, where voltage is the input and charge is the output, and four inverse-capacitance functions for current-charged devices, where charge is the input and voltage is the output (Allagui et al., 2022). For voltage-charged devices, the mappings are

$$q(t)=\mathcal{C}_{\text{tt}}\{v(t)\},\qquad Q(f)=\mathcal{C}_{\text{ff}}\{V(f)\},$$

$$q(t)=\mathcal{C}_{\text{tf}}\{V(f)\},\qquad Q(f)=\mathcal{C}_{\text{ft}}\{v(t)\}.$$

For linear time-invariant devices, these become convolution or integral-kernel relations such as

$$q(t)=\int C_{\text{tt}}(t-x)\,v(x)\,dx, \qquad Q(f)=\int C_{\text{ff}}(f-y)\,V(y)\,dy.$$

This formulation is important because it states that the familiar law $q=cv$ is only a special limiting case. All eight system functions coincide with each other if and only if a constant time- and frequency-independent capacitance is considered (Allagui et al., 2022). In that ideal case,

$q(t)=Cv(t),\qquad Q(f)=CV(f),$

$v(t)=\frac{1}{C}q(t),\qquad V(f)=\frac{1}{C}Q(f).$

Outside that limit, the relevant object is a system function $Q=CV$0 or $Q=CV$1, not a scalar capacitance. The paper explicitly states that applying a step, ramp, or sinusoidal excitation can yield different capacitive or inverse-capacitive functions for the same physical device (Allagui et al., 2022).

The mapping functions are related by single or double Fourier transforms. For example, $Q=CV$2 and $Q=CV$3 form a Fourier-transform pair in the input variable, and $Q=CV$4 is linked to $Q=CV$5 by a double Fourier transform. In practical terms, this means that charge-domain primitives such as accumulation, redistribution, and readout may be non-instantaneous and nonlocal. A direct implication stated in the paper’s charge-domain-computing interpretation is that write and read stages cannot in general be modeled by the same scalar $Q=CV$6; voltage-to-charge and charge-to-voltage conversion may require separate identification through $Q=CV$7 and $Q=CV$8 (Allagui et al., 2022).

3. Switched-capacitor arithmetic and mixed-signal acceleration

In mixed-signal DNN acceleration, charge-domain computation is implemented as switched-capacitor arithmetic. The central idea is to reformulate a high-bitwidth dot product into many low-bitwidth grouped products, execute those low-bit products in parallel in analog hardware, and accumulate the results directly as electric charge on capacitors over multiple cycles before digitization (Ghodrati et al., 2019). This is the basis of the BIHIWE / ATLASS architecture, described as a 3D-stacked microarchitecture using wide, low-bitwidth multiply-accumulate units that share a single ADC (Ghodrati et al., 2019).

The primitive operator is a 3-bit sign-magnitude MACC with an input capacitive DAC $Q=CV$9, a weight capacitive DAC $V_{ScL}=V_{DD}\frac{M}{N}$0, and accumulation capacitors $V_{ScL}=V_{DD}\frac{M}{N}$1 and $V_{ScL}=V_{DD}\frac{M}{N}$2. The input magnitude is first sampled as charge,

$V_{ScL}=V_{DD}\frac{M}{N}$3

Charge sharing between $V_{ScL}=V_{DD}\frac{M}{N}$4 and $V_{ScL}=V_{DD}\frac{M}{N}$5 then produces

$V_{ScL}=V_{DD}\frac{M}{N}$6

and the charge stored on the weight DAC is

$V_{ScL}=V_{DD}\frac{M}{N}$7

Under the approximation $V_{ScL}=V_{DD}\frac{M}{N}$8,

$V_{ScL}=V_{DD}\frac{M}{N}$9

The product-proportional charge is then transferred to the accumulation capacitor, yielding the ideal single-step accumulation voltage

$$\%_{\mathrm{CT}(A \rightarrow B)}=\sum_{i\in A}\sum_{a\in B}|\mathrm{CI}_{i\rightarrow a}|^2$$0

Over $$\%_{\mathrm{CT}(A \rightarrow B)}=\sum_{i\in A}\sum_{a\in B}|\mathrm{CI}_{i\rightarrow a}|^2$$1 cycles, the ideal accumulated voltage becomes

$$\%_{\mathrm{CT}(A \rightarrow B)}=\sum_{i\in A}\sum_{a\in B}|\mathrm{CI}_{i\rightarrow a}|^2$$2

The architectural innovation is not limited to the circuit primitive. High-bit operands are decomposed into low-bit partitions,

$$\%_{\mathrm{CT}(A \rightarrow B)}=\sum_{i\in A}\sum_{a\in B}|\mathrm{CI}_{i\rightarrow a}|^2$$3

so that the dot product becomes

$$\%_{\mathrm{CT}(A \rightarrow B)}=\sum_{i\in A}\sum_{a\in B}|\mathrm{CI}_{i\rightarrow a}|^2$$4

The analog hardware computes the grouped low-bit inner products, while exact significance weighting is restored digitally by shifts and adds (Ghodrati et al., 2019). This bit-partitioned and interleaved organization reduces analog dynamic range, improves noise margin, and allows many operations to share one conversion event.

The reported implementation uses $$\%_{\mathrm{CT}(A \rightarrow B)}=\sum_{i\in A}\sum_{a\in B}|\mathrm{CI}_{i\rightarrow a}|^2$$5 parallel low-bit MACCs and $$\%_{\mathrm{CT}(A \rightarrow B)}=\sum_{i\in A}\sum_{a\in B}|\mathrm{CI}_{i\rightarrow a}|^2$$6 analog accumulation cycles before one ADC event, so one $$\%_{\mathrm{CT}(A \rightarrow B)}=\sum_{i\in A}\sum_{a\in B}|\mathrm{CI}_{i\rightarrow a}|^2$$7 performs $$\%_{\mathrm{CT}(A \rightarrow B)}=\sum_{i\in A}\sum_{a\in B}|\mathrm{CI}_{i\rightarrow a}|^2$$8 low-bit MACC operations before digitization (Ghodrati et al., 2019). The ADC is a 10-bit, 15 MS/s SAR ADC. For one $$\%_{\mathrm{CT}(A \rightarrow B)}=\sum_{i\in A}\sum_{a\in B}|\mathrm{CI}_{i\rightarrow a}|^2$$9 with 2-bit partitioning, the paper reports 5.1 fJ for one low-bit MACC, 1305.6 fJ for 256 MACCs, 1660.0 fJ for the SAR ADC over those 256 MACCs, 1956.6 fJ total energy, 11.6 fJ per 2b-2b MACC, and 185.3 fJ per 8b-8b MACC, which is approximately $$q(t)=\mathcal{C}_{\text{tt}}\{v(t)\},\qquad Q(f)=\mathcal{C}_{\text{ff}}\{V(f)\},$$0 lower than a digital 8-bit MACC at about 1 pJ (Ghodrati et al., 2019). The design is therefore charge-domain not merely because it uses capacitors, but because capacitive storage is the stateful accumulator that amortizes conversion overhead in both time and space.

4. Charge-domain compute-in-memory with FeFETs

A separate but related lineage is charge-domain compute-in-memory based on nonvolatile devices. In the FeFET-based 2T1C macro, binary multiply-accumulate for binary neural networks is implemented by storing complementary weight states in two n-type FeFETs $$q(t)=\mathcal{C}_{\text{tt}}\{v(t)\},\qquad Q(f)=\mathcal{C}_{\text{ff}}\{V(f)\},$$1 and $$q(t)=\mathcal{C}_{\text{tt}}\{v(t)\},\qquad Q(f)=\mathcal{C}_{\text{ff}}\{V(f)\},$$2, using the XNOR result to determine whether a local node $$q(t)=\mathcal{C}_{\text{tt}}\{v(t)\},\qquad Q(f)=\mathcal{C}_{\text{ff}}\{V(f)\},$$3 is pulled to $$q(t)=\mathcal{C}_{\text{tt}}\{v(t)\},\qquad Q(f)=\mathcal{C}_{\text{ff}}\{V(f)\},$$4 or remains at GND, and transferring the resulting charge contribution through a local capacitor $$q(t)=\mathcal{C}_{\text{tt}}\{v(t)\},\qquad Q(f)=\mathcal{C}_{\text{ff}}\{V(f)\},$$5 onto a shared floating source line $$q(t)=\mathcal{C}_{\text{tt}}\{v(t)\},\qquad Q(f)=\mathcal{C}_{\text{ff}}\{V(f)\},$$6 (Yin et al., 2021). The paper presents this as the first concept and analysis of charge-domain CiM using nonvolatile memory devices (Yin et al., 2021).

The cell stores one logical 1 and one logical 0 through complementary FeFET threshold states: positive polarization corresponds to lower $$q(t)=\mathcal{C}_{\text{tt}}\{v(t)\},\qquad Q(f)=\mathcal{C}_{\text{ff}}\{V(f)\},$$7, and negative polarization corresponds to higher $$q(t)=\mathcal{C}_{\text{tt}}\{v(t)\},\qquad Q(f)=\mathcal{C}_{\text{ff}}\{V(f)\},$$8. Inputs are applied as complementary voltages on WL/WLB. If the activated branch corresponds to the ON-state FeFET, the internal node $$q(t)=\mathcal{C}_{\text{tt}}\{v(t)\},\qquad Q(f)=\mathcal{C}_{\text{ff}}\{V(f)\},$$9 is pulled to $$q(t)=\mathcal{C}_{\text{tf}}\{V(f)\},\qquad Q(f)=\mathcal{C}_{\text{ft}}\{v(t)\}.$$0; otherwise it remains at GND. Because $$q(t)=\mathcal{C}_{\text{tf}}\{V(f)\},\qquad Q(f)=\mathcal{C}_{\text{ft}}\{v(t)\}.$$1 is floating, the change in $$q(t)=\mathcal{C}_{\text{tf}}\{V(f)\},\qquad Q(f)=\mathcal{C}_{\text{ft}}\{v(t)\}.$$2 is transferred through $$q(t)=\mathcal{C}_{\text{tf}}\{V(f)\},\qquad Q(f)=\mathcal{C}_{\text{ft}}\{v(t)\}.$$3 to the column accumulation line. For a column of $$q(t)=\mathcal{C}_{\text{tf}}\{V(f)\},\qquad Q(f)=\mathcal{C}_{\text{ft}}\{v(t)\}.$$4 participating cells, if $$q(t)=\mathcal{C}_{\text{tf}}\{V(f)\},\qquad Q(f)=\mathcal{C}_{\text{ft}}\{v(t)\}.$$5 cells produce XNOR = 1, the ideal accumulation result is

$$q(t)=\mathcal{C}_{\text{tf}}\{V(f)\},\qquad Q(f)=\mathcal{C}_{\text{ft}}\{v(t)\}.$$6

The column therefore performs a population count in the charge domain (Yin et al., 2021).

The effective charging capacitance for a column with $$q(t)=\mathcal{C}_{\text{tf}}\{V(f)\},\qquad Q(f)=\mathcal{C}_{\text{ft}}\{v(t)\}.$$7 cells is

$$q(t)=\mathcal{C}_{\text{tf}}\{V(f)\},\qquad Q(f)=\mathcal{C}_{\text{ft}}\{v(t)\}.$$8

and more generally

$$q(t)=\mathcal{C}_{\text{tf}}\{V(f)\},\qquad Q(f)=\mathcal{C}_{\text{ft}}\{v(t)\}.$$9

This relation explains a major energy claim: if all cells output 0 or all output 1, then $$q(t)=\int C_{\text{tt}}(t-x)\,v(x)\,dx, \qquad Q(f)=\int C_{\text{ff}}(f-y)\,V(y)\,dy.$$0; the worst case occurs near $$q(t)=\int C_{\text{tt}}(t-x)\,v(x)\,dx, \qquad Q(f)=\int C_{\text{ff}}(f-y)\,V(y)\,dy.$$1 (Yin et al., 2021). The nonideal MAC voltage is given by

$$q(t)=\int C_{\text{tt}}(t-x)\,v(x)\,dx, \qquad Q(f)=\int C_{\text{ff}}(f-y)\,V(y)\,dy.$$2

which reduces to the ideal $$q(t)=\int C_{\text{tt}}(t-x)\,v(x)\,dx, \qquad Q(f)=\int C_{\text{ff}}(f-y)\,V(y)\,dy.$$3 limit when $$q(t)=\int C_{\text{tt}}(t-x)\,v(x)\,dx, \qquad Q(f)=\int C_{\text{ff}}(f-y)\,V(y)\,dy.$$4 and $$q(t)=\int C_{\text{tt}}(t-x)\,v(x)\,dx, \qquad Q(f)=\int C_{\text{ff}}(f-y)\,V(y)\,dy.$$5 (Yin et al., 2021).

The FeFET choice is central. The paper emphasizes nonvolatility, zero standby storage leakage, compact 2T1C organization, and ON/OFF ratio $$q(t)=\int C_{\text{tt}}(t-x)\,v(x)\,dx, \qquad Q(f)=\int C_{\text{ff}}(f-y)\,V(y)\,dy.$$6, which is much higher than MRAM and typical RRAM (Yin et al., 2021). The simulated operating range is 0.45 V to 0.90 V and 100 MHz to 1.0 GHz. For binary neural network application simulations, the reported energy reduction is over 47% versus SRAM-based charge-domain CiM, 60% versus SRAM-based current-domain CiM, and 64% versus RRAM-based current-domain CiM. The reported application accuracies are over 95% on MNIST and over 80% on CIFAR-10 (Yin et al., 2021). The paper therefore frames charge-domain CiM not as precise analog current summation, but as charge redistribution governed mainly by capacitor ratios and high-ratio switching.

5. Domain-based charge-transfer computation in excited states

In quantum chemistry, charge-domain computation acquires a different meaning. The pCCD-based framework for carbazole dyes defines a domain-based charge-transfer analysis built on localized pair Coupled Cluster Doubles orbitals, where the first excited state is computed with EOM-pCCD+S and its CI-like excitation amplitudes are decomposed over three chemically defined fragments: donor $$q(t)=\int C_{\text{tt}}(t-x)\,v(x)\,dx, \qquad Q(f)=\int C_{\text{ff}}(f-y)\,V(y)\,dy.$$7, bridge $$q(t)=\int C_{\text{tt}}(t-x)\,v(x)\,dx, \qquad Q(f)=\int C_{\text{ff}}(f-y)\,V(y)\,dy.$$8, and acceptor $$q(t)=\int C_{\text{tt}}(t-x)\,v(x)\,dx, \qquad Q(f)=\int C_{\text{ff}}(f-y)\,V(y)\,dy.$$9 (Pandey et al., 24 Nov 2025). The donor is the carbazole moiety, the bridge is the three fused five-membered rings constituting the $q=cv$0-linker, and the acceptor is the cyanoacrylic acid group (Pandey et al., 24 Nov 2025).

The key descriptor is

$q=cv$1

where $q=cv$2 is an occupied orbital assigned to domain $q=cv$3 and $q=cv$4 is a virtual orbital assigned to domain $q=cv$5. The method resolves nine domain contributions, including $q=cv$6, $q=cv$7, $q=cv$8, local channels such as $q=cv$9, and reverse channels such as $q(t)=Cv(t),\qquad Q(f)=CV(f),$0 and $q(t)=Cv(t),\qquad Q(f)=CV(f),$1 (Pandey et al., 24 Nov 2025). Forward transfer is defined as

$q(t)=Cv(t),\qquad Q(f)=CV(f),$2

reverse transfer as

$q(t)=Cv(t),\qquad Q(f)=CV(f),$3

and local excitation as the sum of $q(t)=Cv(t),\qquad Q(f)=CV(f),$4, $q(t)=Cv(t),\qquad Q(f)=CV(f),$5, and $q(t)=Cv(t),\qquad Q(f)=CV(f),$6 (Pandey et al., 24 Nov 2025).

The computational workflow is explicit: compute the ground state with orbital-optimized pCCD, use the resulting localized pCCD orbitals as the orbital basis for domain assignment, compute the first excited state with EOM-pCCD+S, read off the excited-state $q(t)=Cv(t),\qquad Q(f)=CV(f),$7 components, assign occupied and virtual orbitals to $q(t)=Cv(t),\qquad Q(f)=CV(f),$8, $q(t)=Cv(t),\qquad Q(f)=CV(f),$9, or $v(t)=\frac{1}{C}q(t),\qquad V(f)=\frac{1}{C}Q(f).$0, and sum squared excitation amplitudes over each domain pair (Pandey et al., 24 Nov 2025). The localized nature of orbital-optimized pCCD orbitals is crucial because canonical HF or delocalized KS orbitals can obscure fragment assignments.

Applied to 33 carbazole-based dyes with systematically N/O/S-doped $v(t)=\frac{1}{C}q(t),\qquad V(f)=\frac{1}{C}Q(f).$1-bridges, the method shows that nitrogen doping most strongly enhances directional forward transfer (Pandey et al., 24 Nov 2025). Among mono-doped systems, CCN gives the largest forward $v(t)=\frac{1}{C}q(t),\qquad V(f)=\frac{1}{C}Q(f).$2 value at 31.5%. Among homogeneous di-doped nitrogen systems, NCN reaches 39.2%. The headline result is the fully nitrogen-doped bridge NNN, for which the first excited state has 42.6% forward donor-to-acceptor transfer character, dominated by $v(t)=\frac{1}{C}q(t),\qquad V(f)=\frac{1}{C}Q(f).$3, with $v(t)=\frac{1}{C}q(t),\qquad V(f)=\frac{1}{C}Q(f).$4 and $v(t)=\frac{1}{C}q(t),\qquad V(f)=\frac{1}{C}Q(f).$5; the reverse transfer is 4.8% (Pandey et al., 24 Nov 2025). The paper explicitly states that this 42.6% is not a net transfer and not a density difference, but the total forward domain-transfer character for the first excited state. The same analysis also leads to a major conceptual result: all systems exhibit weak charge separation, because excitation remains strongly bridge-centered and the dominant transfer pathway is bridge-to-acceptor rather than a clean donor-to-acceptor separation (Pandey et al., 24 Nov 2025).

6. Misconceptions, limits, and technical implications

Several recurrent simplifications are explicitly rejected by the cited work. First, charge-domain computation is not synonymous with current summation. The FeFET CiM paper contrasts charge-domain operation with current-domain CiM by emphasizing no static or DC compute current during evaluation, less sensitivity to device current variation, and voltage or charge-based linear accumulation on a shared source line rather than Kirchhoff current summation (Yin et al., 2021). Second, $v(t)=\frac{1}{C}q(t),\qquad V(f)=\frac{1}{C}Q(f).$6 is not a generally valid model of charge-domain hardware. The capacitive-system formulation states that all eight direct and inverse system functions coincide only for constant, time- and frequency-independent capacitance, and that different excitations can produce different capacitive or inverse-capacitive functions for the same device (Allagui et al., 2022). This suggests that write, accumulate, and read stages in realistic charge-domain circuits may require kernel or operator descriptions rather than a single scalar weight.

A third misconception is that analog conversion cost disappears in charge-domain accelerators. In the bit-partitioned switched-capacitor design, the ADC is not eliminated; it is amortized. The reported energy numbers show 1660.0 fJ of SAR ADC energy for 256 low-bit MACCs out of a total of 1956.6 fJ, so the success of the scheme depends on sharing one conversion event across many low-bit operations (Ghodrati et al., 2019). The same paper also makes clear that its strongest applicability is 8-bit inference with 2-bit partitioning, $v(t)=\frac{1}{C}q(t),\qquad V(f)=\frac{1}{C}Q(f).$7, and $v(t)=\frac{1}{C}q(t),\qquad V(f)=\frac{1}{C}Q(f).$8, rather than arbitrary-precision analog arithmetic (Ghodrati et al., 2019). Likewise, the FeFET macro is specifically designed for binary neural networks, and its application-level accuracy evaluation assumes ideal peripherals such as reference generators and sense amplifiers (Yin et al., 2021).

In the wavefunction context, the largest numerical transfer character should not be conflated with static charge separation. The pCCD domain-analysis paper explicitly warns that the NNN value of 42.6% means total forward domain-transfer character in the first excited state, not that 42.6% of an electron is statically transferred, and not that the state is fully donor-hole/acceptor-electron separated (Pandey et al., 24 Nov 2025). Weak charge separation can coexist with appreciable charge transfer because the excitation may be dominated by $v(t)=\frac{1}{C}q(t),\qquad V(f)=\frac{1}{C}Q(f).$9 and $Q=CV$00 channels. More broadly, the cited literature indicates that charge-domain computation is best understood not as a single device class or single equation, but as a family of methods in which the physically or chemically relevant charge pathway is made explicit—through system kernels, switched-capacitor accumulation, floating-node redistribution, or localized orbital-domain amplitudes.