Non-Energy Preserving Gates (NEPG)

• Non-Energy Preserving Gates (NEPG) are defined as operations that do not conserve the supplied energy—through dissipation in classical systems or non-commutation with the Hamiltonian in quantum systems.
• Research shows that NEPG implementations challenge traditional views on Landauer’s principle by decoupling logical irreversibility from physical dissipation under quasi-static or controlled conditions.
• Advances in NEPG designs leverage techniques like slow actuation, smooth pulse shaping, and reversible logic embeddings to mitigate energy loss and enhance operational fidelity.

Searching arXiv for papers on non-energy preserving gates and related implementations/theory. Non-Energy Preserving Gates (NEPG) denote gate operations for which energy preservation fails in a physically relevant sense, but the precise sense is domain-dependent. In classical thermodynamics and low-dissipation logic, an NEPG is a logic operation whose physical implementation does not conserve the energy supplied during switching, so that some energy is dissipated to the environment through friction, damping, resistive currents, capacitive charging, leakage, or related non-conservative processes. In quantum information and resource-theoretic settings, the term instead usually denotes a gate $U$ that does not commute with the system Hamiltonian, $[U,H_S]\neq 0$, and therefore changes the system’s energy distribution for some input states. Across these literatures, NEPGs are central to questions about Landauer’s principle, logical versus physical reversibility, control under conservation laws, and the resource costs of high-fidelity gate implementation (Lopez-Suarez et al., 2015, Castellano et al., 1 Sep 2025).

1. Terminology and domain-specific meanings

The expression “non-energy preserving gate” is not fully uniform across subfields. In the micro-electromechanical logic literature, standard irreversible Boolean gates such as AND, OR, and XOR are treated as NEPGs because their practical implementations dissipate energy during switching and because their many-to-one mappings are commonly associated with information reduction. In quantum resource theory, the defining criterion is commutation failure with the relevant Hamiltonian. In bosonic and encoded-oscillator platforms, the same language is used for gates that do not conserve oscillator energy or that transiently couple protected manifolds to excited states (Cattaneo et al., 2017, Xu et al., 2021, Rojkov et al., 2023).

Setting	NEPG criterion	Representative consequence
Classical irreversible logic	Physical implementation does not conserve the energy supplied during switching	Dissipation through friction, damping, resistive currents, capacitive charging, or leakage
Quantum system with Hamiltonian $H_S$	$[U,H_S]\neq 0$	The gate changes the energy distribution of $S$ for some input states
Bosonic two-mode gates	$[U,\hat{H}_{\text{tot}}]\neq 0$ or $[U,\hat{N}_{\text{tot}}]\neq 0$	The gate can broaden finite-energy code states and couple logical information to energy or mode degrees of freedom

A recurring source of confusion is that “energy-preserving” can itself mean different things. In the Landauer-centered classical discussion, “energy-preserving” often means that no information erasure occurs and hence no fundamental dissipation floor due to erasure applies. In the quantum resource-theoretic discussion, “energy-preserving” refers to commutation with the Hamiltonian or to total-energy-preserving interactions. These notions are related to different questions and should not be conflated.

2. Logical irreversibility, physical irreversibility, and Landauer’s principle

The classical debate around NEPGs turns on the distinction between logical irreversibility and physical irreversibility. A Boolean gate is logically irreversible when the output does not determine the input uniquely; OR is the canonical example because $01$, $10$, and $11$ all map to the same logical output. It has therefore been widely argued that any gate with an input-output information reduction must dissipate at least a finite minimum energy. The micro-electromechanical OR-gate experiment directly challenges that inference by showing that logical irreversibility alone does not impose a nonzero lower bound on dissipation when the device is operated quasi-statically and frictional losses are properly managed (Lopez-Suarez et al., 2015).

The relevant thermodynamic benchmark is Landauer’s principle. In the form used throughout this literature, bit erasure has a minimum energy cost

$[U,H_S]\neq 0$0

This lower bound applies to resetting a memory to a known state, or more generally to processes with a net decrease of physical entropy due to erasing information. The central conceptual claim of the 2015 experiment is that performing a logically irreversible combinational gate, such as OR, is not the same operation as erasing a stored bit. The experiment therefore separates logical irreversibility from physical dissipation: a combinational gate can be logically irreversible yet dissipate arbitrarily little heat if it is implemented slowly enough and without a reset step (Lopez-Suarez et al., 2015).

A later discussion sharpened the same distinction in a different way. It argued that “irreversibility is information erasure,” and that if the full physical system is modeled as a reversible mapping with ancilla and garbage retained, then the Landauer erasure bound does not apply to the gate realization itself. In that view, NEPG behavior arises only when output interpretation or measurement coarsening merges distinct states, or when garbage bits are discarded without reversible cleanup (Cattaneo et al., 2017).

3. Micro-electromechanical realization of a sub-$[U,H_S]\neq 0$1 irreversible gate

A prototypical classical NEPG is the micro-electromechanical OR gate realized with a single Si$[U,H_S]\neq 0$2N$[U,H_S]\neq 0$3 cantilever actuated electrostatically by two nearby electrodes. The cantilever is V-shaped, has length $[U,H_S]\neq 0$4, stiffness $[U,H_S]\neq 0$5, first-mode resonance $[U,H_S]\neq 0$6, quality factor $[U,H_S]\neq 0$7, and relaxation time $[U,H_S]\neq 0$8. The device operates in vacuum at $[U,H_S]\neq 0$9 and room temperature $H_S$0. Two tungsten tips of $H_S$1 radius are positioned at gap $H_S$2 from the cantilever tip, and the logical output is the tip position $H_S$3, read by an AFM-like optical lever with conversion $H_S$4 and $H_S$5; the signals are digitized at $H_S$6. Logical inputs $H_S$7 and $H_S$8 are encoded as voltages on the probes, and a single position threshold separates logical states $H_S$9 and $[U,H_S]\neq 0$0 (Lopez-Suarez et al., 2015).

The electrostatic force is approximated as

$[U,H_S]\neq 0$1

with coefficients $[U,H_S]\neq 0$2 and $[U,H_S]\neq 0$3 dependent on the input configuration. Because the force produces larger deflection for “11” than for “01” or “10,” a single threshold yields the OR truth table: “00” maps to $[U,H_S]\neq 0$4, while “01,” “10,” and “11” map to $[U,H_S]\neq 0$5. Raising the threshold allows the same hardware to implement an AND gate. The gate is therefore logically irreversible in the standard 2-input/1-output sense, since output alone does not reconstruct the input pair (Lopez-Suarez et al., 2015).

Energy accounting is performed at the device level through a Hamiltonian

$[U,H_S]\neq 0$6

and over a full cycle of duration $[U,H_S]\neq 0$7 that returns the device to its initial state on average, $[U,H_S]\neq 0$8, so the dissipated heat equals the work done by the inputs, $[U,H_S]\neq 0$9. The work is computed with the Stratonovich convention,

$S$0

All input combinations are exercised, ramp amplitudes $S$1 and durations $S$2 are varied, and approximately Gaussian heat distributions are estimated from about $S$3 repetitions per condition. The mean heat is positive, but negative tails occur, consistent with equilibrium fluctuations and fluctuation theorems (Lopez-Suarez et al., 2015).

The key operational principle is slow actuation. Each logical input transition is a linear voltage ramp over $S$4, the output is read during $S$5, and protocol durations are varied from roughly $S$6 to $S$7, which is tens to hundreds of mechanical periods while still shorter than $S$8. Dissipation is modeled with a loss angle

$S$9

where structural damping is frequency independent, thermo-elastic and viscous contributions scale $[U,\hat{H}_{\text{tot}}]\neq 0$0 at low $[U,\hat{H}_{\text{tot}}]\neq 0$1, and clamp recoil scales $[U,\hat{H}_{\text{tot}}]\neq 0$2. The dissipated heat obeys

$[U,\hat{H}_{\text{tot}}]\neq 0$3

so increasing $[U,\hat{H}_{\text{tot}}]\neq 0$4 lowers the effective frequency $[U,\hat{H}_{\text{tot}}]\neq 0$5 and suppresses the frequency-dependent loss channels (Lopez-Suarez et al., 2015).

Quantitatively, for $[U,\hat{H}_{\text{tot}}]\neq 0$6, the mean dissipated heat $[U,\hat{H}_{\text{tot}}]\neq 0$7 for inputs “01,” “10,” and “11” decreases as $[U,\hat{H}_{\text{tot}}]\neq 0$8 increases, and at longer $[U,\hat{H}_{\text{tot}}]\neq 0$9 of approximately $[U,\hat{N}_{\text{tot}}]\neq 0$0–$[U,\hat{N}_{\text{tot}}]\neq 0$1 it drops below $[U,\hat{N}_{\text{tot}}]\neq 0$2 for all single- and double-input cases. The normalized quantity $[U,\hat{N}_{\text{tot}}]\neq 0$3 collapses across different inputs and voltages from $[U,\hat{N}_{\text{tot}}]\neq 0$4 to $[U,\hat{N}_{\text{tot}}]\neq 0$5, supporting the Zener-based power-law fit. By electrostatically coupling a second biased cantilever to the OR output, the same work realized a NOR gate with average dissipated energy $[U,\hat{N}_{\text{tot}}]\neq 0$6, or approximately $[U,\hat{N}_{\text{tot}}]\neq 0$7 at $[U,\hat{N}_{\text{tot}}]\neq 0$8, in adiabatic conditions (Lopez-Suarez et al., 2015).

The scope of this energy accounting is explicitly limited. What is counted is the mechanical work on the cantilever, identified with dissipated heat per cycle. What is not counted includes external control electronics, signal generation, and optical readout. Moreover, because the device is combinational, removing the inputs returns it to $[U,\hat{N}_{\text{tot}}]\neq 0$9; persisting the output would require coupling to a sequential element, and a later reset could then incur the Landauer cost.

4. Reversible reinterpretation of the cantilever gate

The 2017 discussion of the LNG experiment proposed a different theoretical description of the same physical device. Rather than treating it as a 2-input/1-output irreversible OR gate, it modeled the full system “probes + cantilever” as a 3in/3out self-reversible gate, with the first two outputs corresponding to the probe states and the third output obtained from a normalization function applied to the cantilever angle. In this interpretation, the apparent irreversibility of OR arises only after coarse-graining, namely when the outcomes for inputs $01$0 and $01$1 are treated as indistinguishable and the first two outputs are ignored (Cattaneo et al., 2017).

The central reversible primitive is the self-reversible CL gate

$01$2

which satisfies $01$3. Fixing the ancilla $01$4 yields $01$5, with $01$6 and $01$7 treated as garbage. In the same framework, the Toffoli gate

$01$8

with $01$9 yields AND on the third output, while an X-gate

$10$0

with $10$1 yields XOR. By fixing $10$2 and using negated normalization functions, the same scheme yields NOR, NAND, and NXOR (Cattaneo et al., 2017).

A crucial element of this reinterpretation is the family of angle normalization functions. For example,

$10$3

allow OR- and AND-type readouts, while $10$4 and its negation support XOR and NXOR. A discrete mapping $10$5 is introduced for the case in which the outputs generated by $10$6 and $10$7 are distinguished, using angles near $10$8, $10$9, and $11$0. The paper’s claim is that, with suitable normalization and with ancilla and garbage handled reversibly, the LNG device realizes a reversible logic embedding rather than an intrinsically dissipative irreversible gate (Cattaneo et al., 2017).

This reinterpretation does not deny finite-time thermodynamic dissipation. It instead claims that no mandatory Landauer lower bound greater than zero applies as long as no information is erased. The broader significance is terminological: within this framework, a device behaves as a NEPG only when state merging, coarse-graining, or garbage disposal renders the mapping non-bijective. A plausible implication is that part of the classical NEPG debate is actually a debate about where the computational boundary is drawn.

5. Quantum resource-theoretic NEPGs and the cost of asymmetry

In the quantum resource-theoretic formulation, the setting is a finite-dimensional system $11$1 with Hamiltonian $11$2 and an auxiliary battery $11$3 with Hamiltonian $11$4, initially prepared in state $11$5. The gate implementation is constrained to a total-energy-preserving unitary $11$6 satisfying

$11$7

which induces a channel

$11$8

The target gate is a unitary $11$9 on $[U,H_S]\neq 0$00, written as $[U,H_S]\neq 0$01. In this language, a gate is a NEPG precisely when $[U,H_S]\neq 0$02; equivalently, the gate changes the energy distribution of $[U,H_S]\neq 0$03 for some input states. The problem is then to approximate $[U,H_S]\neq 0$04 by some $[U,H_S]\neq 0$05 with prescribed accuracy (Castellano et al., 1 Sep 2025).

The necessary battery resource is entropic coherence relative to the battery Hamiltonian. For the dephasing map in the energy basis,

$[U,H_S]\neq 0$06

the entropic coherence is

$[U,H_S]\neq 0$07

Because $[U,H_S]\neq 0$08 cannot increase under energy-preserving operations and partial trace, any time-translation asymmetry created on $[U,H_S]\neq 0$09 by implementing a NEPG must be supplied by the battery. Precision is quantified by worst-case infidelity between channels,

$[U,H_S]\neq 0$10

with

$[U,H_S]\neq 0$11

The main lower bounds depend on gate-dependent quantities $[U,H_S]\neq 0$12 and $[U,H_S]\neq 0$13, from which the paper defines $[U,H_S]\neq 0$14 and $[U,H_S]\neq 0$15. If $[U,H_S]\neq 0$16 is any suitable family of batteries able to implement $[U,H_S]\neq 0$17 with $[U,H_S]\neq 0$18, then

$[U,H_S]\neq 0$19

For a “proportionate” battery family satisfying

$[U,H_S]\neq 0$20

a stronger bound holds:

$[U,H_S]\neq 0$21

For qubits, $[U,H_S]\neq 0$22, and the constants are determined by $[U,H_S]\neq 0$23 in the energy basis (Castellano et al., 1 Sep 2025).

One immediate corollary is a finite-dimensional no-go statement. Since $[U,H_S]\neq 0$24 for a $[U,H_S]\neq 0$25-dimensional battery, any fixed finite-dimensional battery is doomed to a nonzero minimal error in the implementation of a NEPG, and perfect implementation requires $[U,H_S]\neq 0$26. For qubits, the resulting exponent implies battery levels scaling as $[U,H_S]\neq 0$27, matching prior explicit constructions. Under additional level-density assumptions,

$[U,H_S]\neq 0$28

the same entropic-coherence lower bounds imply lower bounds on average battery energy and quantum Fisher information. For harmonic-oscillator batteries,

$[U,H_S]\neq 0$29

and

$[U,H_S]\neq 0$30

When $[U,H_S]\neq 0$31 or $[U,H_S]\neq 0$32, these exponents can be stronger than earlier universal $[U,H_S]\neq 0$33 energy and $[U,H_S]\neq 0$34 QFI bounds. The conceptual conclusion is that NEPG implementation under exact energy conservation requires time-translation asymmetry supplied by the battery, quantified here by entropic coherence.

6. Platform-specific quantum NEPGs: stabilized cat qubits and finite-energy GKP codes

In stabilized Kerr-cat qubits, NEPGs are control operations that do not commute with the stabilizing Hamiltonian of the cat manifold and therefore transiently couple the logical ground manifold to excited states during the gate. They are nevertheless designed to be bias-preserving: the dominant noise remains $[U,H_S]\neq 0$35-type phase flips, while $[U,H_S]\neq 0$36-type bit flips stay exponentially suppressed with cat size. Useful logical gates such as $[U,H_S]\neq 0$37 rotations, $[U,H_S]\neq 0$38, and $[U,H_S]\neq 0$39 require control terms, including single-photon drives and two-mode squeezing, that do not commute with the stabilizer; in this architecture, useful bias-preserving gates are therefore NEPGs rather than energy-preserving gates. Their error budget decomposes into non-adiabatic and loss-induced terms,

$[U,H_S]\neq 0$40

with $[U,H_S]\neq 0$41 for $[U,H_S]\neq 0$42 rotations and $[U,H_S]\neq 0$43 for $[U,H_S]\neq 0$44 and $[U,H_S]\neq 0$45. To suppress non-adiabatic leakage, the work introduced a derivative-based leakage suppression scheme using smooth Gaussian-like envelopes and corrective quadratures. For the single-qubit $[U,H_S]\neq 0$46 rotation, the added quadrature is

$[U,H_S]\neq 0$47

For $[U,H_S]\neq 0$48, the DBC-controlled Kerr gates significantly reduce both $[U,H_S]\neq 0$49 and $[U,H_S]\neq 0$50, permit operation in the fast-gate regime $[U,H_S]\neq 0$51, and yield the favorable optimal scaling

$[U,H_S]\neq 0$52

compared with $[U,H_S]\neq 0$53 for hard-pulse Kerr gates and $[U,H_S]\neq 0$54 for dissipative gates (Xu et al., 2021).

In finite-energy Gottesman-Kitaev-Preskill encodings, NEPG behavior appears when ideal bosonic entanglers for infinite-energy codewords are applied to realistic finite-energy states. The standard controlled-$[U,H_S]\neq 0$55 gate,

$[U,H_S]\neq 0$56

is a NEPG because

$[U,H_S]\neq 0$57

Applied to finite-energy GKP states $[U,H_S]\neq 0$58 with $[U,H_S]\neq 0$59, the gate broadens only the $[U,H_S]\neq 0$60 quadrature and entangles logical information with oscillator energy or mode degrees of freedom. After tracing out the partner mode, the marginal state retains finite-energy GKP form but with

$[U,H_S]\neq 0$61

so both the peak width and the envelope width increase by $[U,H_S]\neq 0$62. For input $[U,H_S]\neq 0$63, the physical overlap fidelity tends to

$[U,H_S]\neq 0$64

as $[U,H_S]\neq 0$65, showing that finite-energy distortion persists even in the strong-squeezing limit. Local stabilization rounds before and after the NEPG can correct much of this distortion; with nine rounds before and after, the stabilized logical infidelity stays below $[U,H_S]\neq 0$66 for $[U,H_S]\neq 0$67. The same work also proposed energy-conserving finite-energy alternatives, especially a qutrit-mediated entangler using one auxiliary three-level system, two $[U,H_S]\neq 0$68 beam splitters, conditional finite-energy unitaries, and one reset; this protocol achieves logical infidelity below $[U,H_S]\neq 0$69 for $[U,H_S]\neq 0$70 and largely avoids the NEPG-induced broadening (Rojkov et al., 2023).

These two platforms illustrate a general point. In cat qubits, NEPGs are necessary because nontrivial bias-preserving control does not commute with the stabilizer. In finite-energy GKP codes, NEPG behavior arises because a mathematically correct logical entangler for ideal codewords becomes energy-distorting on approximate states. In both cases, the central challenge is to realize the desired logical action while suppressing leakage, broadening, or asymmetry costs.

7. Conceptual synthesis, misconceptions, and design implications

Several persistent misconceptions are resolved by the combined literature. First, logical irreversibility does not by itself imply a fixed heat cost for performing a gate; the micro-electromechanical OR experiment shows that a logically irreversible combinational gate can dissipate well below $[U,H_S]\neq 0$71 when operated quasi-statically and without erasing a stored bit (Lopez-Suarez et al., 2015). Second, this does not invalidate Landauer’s principle, because Landauer’s bound applies to erasure or reset, not to every logically irreversible computation. Third, the term “energy-preserving” is not uniform: in the 2017 reinterpretation it means “no dispersion/dissipation due to information erasure” in the Landauer sense, whereas in the battery-based quantum setting it means exact conservation under Hamiltonian commutation or total-energy-preserving interaction (Cattaneo et al., 2017, Castellano et al., 1 Sep 2025).

Across domains, two broad design principles recur. One is quasi-static or spectrally selective control: slow voltage ramps in the cantilever gate, smooth derivative-shaped pulses in Kerr-cat gates, and finite-energy-aware couplings in GKP architectures all reduce unwanted dissipation or leakage. The other is careful accounting of hidden resources: ancillas and garbage bits in reversible classical embeddings, battery coherence in quantum implementations under conservation laws, and stabilization rounds or auxiliary qutrits in finite-energy bosonic codes.

A plausible implication is that NEPGs are best understood not as a single phenomenon but as a family of obstructions to ideal gate implementation under physical constraints. In classical thermodynamics the obstruction is friction, damping, and erasure; in quantum resource theory it is time-translation asymmetry under exact conservation laws; in protected bosonic encodings it is leakage or state deformation induced by control Hamiltonians that do not preserve the relevant energy structure. The common theme is that non-energy preservation is not merely a label on a truth table or a unitary. It is a statement about how logical action, physical implementation, and resource accounting interact.