Energy–Error Tradeoff in Modern Systems

• Energy–error tradeoff is a quantitative framework where enhancing accuracy, reliability, or precision requires increased energy, while reducing energy use leads to higher error rates.
• It encompasses diverse formulations—from iterative algorithms and communication protocols to quantum operations and error-correcting codes—with metrics such as convergence thresholds, distortion levels, and gate fidelities.
• Practical applications include approximate computing in radio-astronomy calibration, energy-harvesting estimation, and quantum error correction, providing actionable insights for optimizing system design.

Searching arXiv for the cited papers and closely related work on energy–error tradeoffs. Energy–error tradeoff denotes a family of quantitative relations in which improving accuracy, reliability, or logical precision requires additional energetic resources, while reducing energy typically admits larger computational, communication, estimation, or measurement error. In the cited literature, the term appears in several closely related forms: energy–accuracy tradeoffs in approximate iterative algorithms, energy–distortion tradeoffs in source transmission, energy–reliability bounds for noisy circuits and decoders, and energy–error or cost–irreversibility relations for quantum measurements, gates, and quantum error-correcting encodings (Gillani et al., 20 Feb 2025, Murin et al., 2016, 0710.4680, Nakajima et al., 2024, Stevens et al., 5 May 2026).

1. Formal meanings of “energy” and “error”

The literature does not use a single universal metric. Instead, each domain fixes an energetic quantity and an application-level error or performance criterion, and then studies the feasible region or lower bounds connecting them.

Setting	Energy quantity	Error or performance quantity
Approximate iterative computing	Electrical energy consumption; energy per iteration	Convergence metric; relative difference to exact solution
Communication and estimation	Transmission energy per source sample; bit energy; harvested energy	Distortion, decoding error probability, energy outage probability, MMSE, AoI
Quantum operations	Control-field energy, energetic cost, dissipated work	Measurement error, disturbance, purified-distance gate error, irreversibility

In approximate iterative computing, precision is encoded by a convergence metric, but the cited work also introduces a separate accuracy-oriented metric comparing approximate and exact iterates because convergence alone can be satisfied while the algorithm converges “in the wrong plane” (Gillani et al., 20 Feb 2025). In simultaneous information and energy transmission, the relevant finite-blocklength tuple is information rate, energy rate, decoding error probability, and energy outage probability (Zuhra et al., 2022). In energy-harvesting estimation, the tradeoff is between harvested energy and the minimum mean square error produced by Kalman or extended Kalman filtering (Krikidis et al., 2021). In Age-of-Information systems, the corresponding performance quantity is freshness rather than a conventional error probability, leading to an energy–age tradeoff (Gong et al., 2021). In quantum information, error is often defined through purified distance, average gate fidelity, irreversibility, or logical failure probability, while energy is the control energy, free-energy cost, work cost, or dissipated work required to realize the channel or encoding (Tajima et al., 31 Jul 2025, Stevens et al., 5 May 2026, Riechers et al., 2019).

A recurring implication is that “error” is often operational rather than purely microscopic. This suggests that meaningful energy–error analysis usually requires an application-level acceptance criterion rather than only a device-level fault rate.

2. Approximate computing and iterative algorithms

A concrete algorithmic formulation appears in radio-astronomy calibration, where iterative least-squares processing is split into exact and approximate phases. The governing idea is that early iterations are error-resilient, while late iterations must be accurate because they determine final convergence. For StEFCal, the convergence test is the relative difference between consecutive gain vectors with threshold $1 \cdot 10^{-6}$, but quality acceptance also requires an accuracy metric $\mathrm{Diff\_rel} \le 10^{-5}$ against the exact or floating-point reference (Gillani et al., 20 Feb 2025).

The methodological contribution is Adaptive-SAM, an offline statistical approximation model that injects relative Gaussian perturbations into dominant kernels and determines how many initial iterations may be approximate. For StEFCal, the statistical analysis yields

$$N_{ax} \le 23\% \cdot N,\quad \mathrm{EM} \le 12,\quad \mathrm{EP} \le 0.2,\quad \mathrm{ER} = 100\%,$$

showing that substantial early-iteration perturbations are admissible if the remaining iterations are exact (Gillani et al., 20 Feb 2025). The implemented heterogeneous accelerator then uses an approximate low-precision core for 52 of 92 iterations and an accurate core for the remaining 40, with no increase in total iteration count and measured energy savings of about $23.4\%$ relative to an all-accurate design (Gillani et al., 20 Feb 2025).

The same paper makes the tradeoff explicit at the hardware level. The accurate core occupies $27{,}023\ \mu\text{m}^2$ and consumes $3.55\ \text{mW}$ at 50 MHz, while the approximate core occupies $20{,}604\ \mu\text{m}^2$ and consumes $2.08\ \text{mW}$ (Gillani et al., 20 Feb 2025). The energy-saving expression

$$S_E = \frac{(P_{acc} - P_{ax})\,N_{ax}}{P_{acc}\,N_{acc}}$$

captures the central design tension: stronger approximation lowers per-iteration energy but usually reduces the number of safe approximate iterations (Gillani et al., 20 Feb 2025).

Processor-level approximate computing appears in “Elastic Fidelity,” where selected arithmetic segments are run on lower-voltage functional units that admit timing errors while preserving application-level quality (Roy et al., 2011). In multimedia decoding, conservative use of low-fidelity ALUs yields normalized processor powers of $0.89$, $\mathrm{Diff\_rel} \le 10^{-5}$0, and $\mathrm{Diff\_rel} \le 10^{-5}$1 for G.721-D, JPEG-D, and H.263-D, corresponding to $\mathrm{Diff\_rel} \le 10^{-5}$2, $\mathrm{Diff\_rel} \le 10^{-5}$3, and $\mathrm{Diff\_rel} \le 10^{-5}$4 power and energy reductions, while maintaining output-quality thresholds such as SNRseg $\mathrm{Diff\_rel} \le 10^{-5}$5 dB, JPEG PSNR $\mathrm{Diff\_rel} \le 10^{-5}$6 dB, and H.263 PSNR $\mathrm{Diff\_rel} \le 10^{-5}$7 dB (Roy et al., 2011). The cited evidence also shows that bit significance and function identity matter: some functions tolerate faults in low-significance bits, while others, such as motion compensation in H.263-D, are highly sensitive (Roy et al., 2011).

3. Communication, estimation, and freshness formulations

In communication theory, energy–error tradeoffs often appear as energy–distortion, information–energy, or energy–reliability relations. For correlated Gaussian sources over a Gaussian broadcast channel with feedback, the minimum required transmission energy per source sample is studied as a function of target distortion, and lower and upper bounds are derived through separate and joint source–channel coding schemes (Murin et al., 2016). The paper explicitly frames this as an energy–distortion tradeoff and shows that the Ozarow–Leung-based joint source–channel coding scheme is close to the better separation-based scheme despite its simplicity (Murin et al., 2016).

Finite-blocklength simultaneous information and energy transmission over AWGN with finite alphabets introduces a four-dimensional tradeoff among information rate, energy rate, decoding error probability, and energy outage probability (Zuhra et al., 2022). The paper gives impossibility and achievability regions for tuples $\mathrm{Diff\_rel} \le 10^{-5}$8, and its proposed code construction matches the impossibility bounds for information rate, energy rate, and energy outage probability, while the achieved decoding error probability remains above the impossibility bound because of the chosen decoding sets (Zuhra et al., 2022). Here, energy is not merely consumed by computation; it is also a delivered resource at the receiver, so the tradeoff is between reliable information transfer and energy harvesting.

A related but distinct formulation appears in receiver architectures that simultaneously harvest energy and estimate a scalar Gauss–Markov process. Using power splitting with parameter $\mathrm{Diff\_rel} \le 10^{-5}$9, the same received signal drives both an energy harvester and a Kalman filter. The harvested energy scales with $$N_{ax} \le 23\% \cdot N,\quad \mathrm{EM} \le 12,\quad \mathrm{EP} \le 0.2,\quad \mathrm{ER} = 100\%,$$0, while estimation quality improves with $$N_{ax} \le 23\% \cdot N,\quad \mathrm{EM} \le 12,\quad \mathrm{EP} \le 0.2,\quad \mathrm{ER} = 100\%,$$1, producing an explicit estimation–energy tradeoff (Krikidis et al., 2021). The cited analysis shows that channel fading improves estimation performance, whereas high-power-amplifier nonlinearities require an extended Kalman filter and significantly affect both estimation and harvesting efficiency (Krikidis et al., 2021).

Queueing-constrained wireless transmission provides another reliability-oriented formulation. With training-based channel estimation and effective-capacity constraints, the bit energy increases without bound in the low-power regime as the average power vanishes, whereas in the wideband regime the bit energy can diminish to a minimum value (0901.3134). Because the service process is ON–OFF due to outages, the relevant “error” is not only physical-layer failure but also buffer-violation or delay-violation probability, controlled by the QoS exponent $$N_{ax} \le 23\% \cdot N,\quad \mathrm{EM} \le 12,\quad \mathrm{EP} \le 0.2,\quad \mathrm{ER} = 100\%,$$2 (0901.3134).

Freshness control in IoT status updating further broadens the notion of error. Over an error-prone channel, a sensor may sleep, retransmit, or sense and transmit a new update, and the optimal stationary policy is governed by two thresholds: one on AoI at the transmitter and one on AoI at the receiver (Gong et al., 2021). The paper shows that considering sensing energy is of significant impact on the policy design, and that introducing sleep mode greatly expands the tradeoff range (Gong et al., 2021). This suggests that, in networked systems, the energy–error tradeoff is often inseparable from scheduling and state-control structure.

4. Reliability costs in noisy circuits, decoders, and coding hardware

For noisy classical hardware, the tradeoff is frequently formulated as a lower bound on the energy overhead needed to achieve a target reliability. In failure-prone nanoscale circuits with independent gate error probability $$N_{ax} \le 23\% \cdot N,\quad \mathrm{EM} \le 12,\quad \mathrm{EP} \le 0.2,\quad \mathrm{ER} = 100\%,$$3, the analysis combines redundancy, switching activity, and leakage. The paper shows that $$N_{ax} \le 23\% \cdot N,\quad \mathrm{EM} \le 12,\quad \mathrm{EP} \le 0.2,\quad \mathrm{ER} = 100\%,$$4 error resilience is possible for fault-tolerant designs, but at the expense of at least $$N_{ax} \le 23\% \cdot N,\quad \mathrm{EM} \le 12,\quad \mathrm{EP} \le 0.2,\quad \mathrm{ER} = 100\%,$$5 more energy if individual gates fail independently with probability of $$N_{ax} \le 23\% \cdot N,\quad \mathrm{EM} \le 12,\quad \mathrm{EP} \le 0.2,\quad \mathrm{ER} = 100\%,$$6 (0710.4680). More generally, the lower bounds relate target circuit reliability $$N_{ax} \le 23\% \cdot N,\quad \mathrm{EM} \le 12,\quad \mathrm{EP} \le 0.2,\quad \mathrm{ER} = 100\%,$$7, gate error probability $$N_{ax} \le 23\% \cdot N,\quad \mathrm{EM} \le 12,\quad \mathrm{EP} \le 0.2,\quad \mathrm{ER} = 100\%,$$8, function sensitivity, gate fan-in, and switching activity, thereby quantifying how redundancy and energy must grow together (0710.4680).

Decoder complexity produces another strong lower bound. In the “waterslide curves” framework, decoding is modeled by an idealized massively parallel message-passing architecture whose power grows with decoder neighborhood size (0801.0352). The key conclusion is that, as the gap to capacity goes to zero, the energy per bit spent in decoding goes to infinity, and that certainty is infinitely expensive in total power under the iterative decoding model (0801.0352). Consequently, minimizing total power requires operating at a transmit power strictly above the Shannon limit, because moving too close to capacity makes decoding energy dominate (0801.0352).

Physical coding circuits obey similar laws. In the Thompson VLSI model, fully-parallel encoding and decoding schemes with asymptotic block error probability that scales as $$N_{ax} \le 23\% \cdot N,\quad \mathrm{EM} \le 12,\quad \mathrm{EP} \le 0.2,\quad \mathrm{ER} = 100\%,$$9 have Thompson energy that scales as $23.4\%$0, and any scheme reaching this bound requires $23.4\%$1 clock cycles (Blake et al., 2016). In the three-dimensional information-friction model, the optimal energy of encoding or decoding schemes with probability of block error $23.4\%$2 is shown to be at least $23.4\%$3 (Blake et al., 2016). Although the asymptotic notation is given in the paper’s sign convention, the substantive claim is that making coding circuits more reliable forces superlinear energy growth (Blake et al., 2016).

A common misconception is that approaching Shannon-theoretic or fault-tolerance limits can be done at negligible extra energy if one ignores only transmit power or only device switching. The cited lower bounds show the opposite: decoder energy, redundancy cost, and circuit communication all become dominant when reliability targets are pushed aggressively (0801.0352, 0710.4680, Blake et al., 2016).

5. Quantum measurements, channels, and thermodynamic limits

Quantum formulations make the energy–error tradeoff explicit through no-go theorems. Under energy conservation and locality, the speed–accuracy tradeoff for measurements shows that perfect measurement of an observable $23.4\%$4 with $23.4\%$5 cannot be achieved in finite time (Nakajima et al., 2024). The main lower bound has the form

$23.4\%$6

and the corresponding gate-implementation bound is

$23.4\%$7

where $23.4\%$8 measures the energy-change structure of the target unitary (Nakajima et al., 2024). The denominator grows with the effective Lieb–Robinson light-cone volume, so finite-time, zero-error implementation of energy-changing measurements or gates is excluded (Nakajima et al., 2024).

A more general resource-theoretic formulation appears in the universal cost–irreversibility tradeoff. For arbitrary channels and a broad class of resource theories, lower irreversibility requires greater implementation cost, and in the energy-specialized case the approximate implementation cost obeys a $23.4\%$9 law (Tajima et al., 31 Jul 2025). One explicit corollary states

$27{,}023\ \mu\text{m}^2$0

so high-accuracy implementation of an energy-changing channel requires energy cost that diverges as $27{,}023\ \mu\text{m}^2$1 (Tajima et al., 31 Jul 2025). For projective measurements this yields

$27{,}023\ \mu\text{m}^2$2

and for unitary channels,

$27{,}023\ \mu\text{m}^2$3

making the dependence on energy noncommutativity fully explicit (Tajima et al., 31 Jul 2025).

Thermodynamic arguments based on time-symmetric control lead to a different but compatible limit. For nonreciprocal logical transitions implemented under time-symmetric control, requiring a low probability of error causes energy consumption to diverge as logarithm of the inverse error rate, $27{,}023\ \mu\text{m}^2$4 (Riechers et al., 2019). The paper argues that reciprocity, or self-invertibility, is a stricter condition for thermodynamic efficiency than logical reversibility, and that time-symmetric control often leaves real devices far above the Landauer limit (Riechers et al., 2019).

Taken together, these results show that in quantum settings the tradeoff is not merely empirical. It is enforced by conservation laws, locality, control symmetry, and resource monotones.

6. Quantum error correction encodings and code-dependent scaling

The most direct quantum-computing version of the topic is the energetic cost of encoding quantum error correction. For repetition, perfect, and Steane codes, the energetic resource is the control-field energy needed to implement the encoding and subsequent logical operations, bounded by

$27{,}023\ \mu\text{m}^2$5

with noisy control coefficients $27{,}023\ \mu\text{m}^2$6 (Stevens et al., 5 May 2026). At the single-gate level, the cited simulations show that $27{,}023\ \mu\text{m}^2$7 at high energies (Stevens et al., 5 May 2026).

For the $27{,}023\ \mu\text{m}^2$8-qubit repetition code, the ideal logical failure rate under a one-time bit-flip channel is

$27{,}023\ \mu\text{m}^2$9

with the small-$3.55\ \text{mW}$0 approximation

$3.55\ \text{mW}$1

showing the expected increase of protection with code size (Stevens et al., 5 May 2026). The total minimal control energy for the repetition-code circuit is

$3.55\ \text{mW}$2

because the construction uses $3.55\ \text{mW}$3 $3.55\ \text{mW}$4 gates and $3.55\ \text{mW}$5 CNOTs (Stevens et al., 5 May 2026).

The principal result is that the required control energy for the $3.55\ \text{mW}$6-qubit code to overtake the $3.55\ \text{mW}$7-qubit code scales exponentially, with scaling parameters dependent on the channel error (Stevens et al., 5 May 2026). This is the paper’s clearest evidence that, at the logical-encoding level, the required resources scale exponentially with the targeted precision. The same study also shows that the specific physical realization matters strongly: for the 7-qubit repetition code, direct encoding outperforms waterfall and parallel encodings at intermediate energies, even though the realizations are logically equivalent and use the same total gate counts (Stevens et al., 5 May 2026). For the five-qubit perfect code, by contrast, three encoding circuits from different references yield very similar energy–error curves (Stevens et al., 5 May 2026).

Fault-tolerant stabilizer measurement intensifies the tradeoff. The appendix shows that adding cat-state-based fault-tolerant measurement increases control energy substantially, and under the same gate-noise model the fault-tolerant construction can yield worse logical performance at fixed total energy than the non-fault-tolerant one, because it introduces more noisy gates (Stevens et al., 5 May 2026). This suggests that “fault tolerant” and “energy efficient” are not equivalent descriptors unless control noise and control energy are both included in the analysis.

Across the cited literature, several common principles recur. Reliability gains typically require either redundancy, larger control energy, more energetic coherence, or more communication; convergence metrics alone can be insufficient; and architectures that appear optimal under one energy accounting can become suboptimal once hidden costs such as decoding, training, or control precision are included (Gillani et al., 20 Feb 2025, 0801.0352, 0710.4680, Tajima et al., 31 Jul 2025). In that sense, the energy–error tradeoff is less a single theorem than a unifying constraint family: every domain studied imposes a different operational metric, but each finds that suppressing error to very low levels requires nontrivial and often rapidly growing energetic expenditure.