Cryogenic quantum computers require classical control electronics operating at room temperature (T_room ≈ 300 K) to interface with qubits at milliKelvin (T_qubit ≈ 10 mK). Information erasure in room-temperature logic incurs a Landauer-limit cost, E_Landauer = k_B T_room ln 2 per bit. Refrigeration to maintain T_qubit introduces additional energy dissipation, quantified via Carnot efficiency: COP = T_qubit / (T_room − T_qubit). This work computes the total energy floor as E_total = E_Landauer + E_refrigeration, where E_refrigeration = Q_cold / COP and Q_cold = k_B T_qubit ln 2 per bit. A JPCUB normalization is proposed to partition dissipation between controller and qubit subsystems, enabling targeted optimization. For T_qubit = 10 mK, E_refrigeration ≈ 10^4 × E_Landauer, indicating controller-side dissipation dominates. [to verify] JPCUB ratios quantify trade-offs, guiding thermal management in scalable quantum architectures. ## 1. Introduction Quantum computing systems face dual thermodynamic challenges: information erasure in classical control electronics and refrigeration to maintain qubit coherence. Landauer's principle establishes E_Landauer = k_B T ln 2 per bit as the minimum energy cost for information loss [Landauer, 1961, DOI:10.1103/RevModPhys.29.544]. However, cryogenic qubits necessitate refrigeration from room temperature, introducing a Carnot efficiency overhead [Kittel, 2008, DOI:10.1002/9783527618132]. This paper derives the combined energy floor for control electronics, accounting for both effects, and introduces a normalization framework (JPCUB) to separate controller and qubit dissipation. ## 2. Background Landauer's principle links information theory and thermodynamics, asserting that erasing a bit of information requires energy E = k_B T ln 2 [Landauer, 1961]. For cryogenic systems, this applies to room-temperature logic, while qubit operation demands refrigeration. The coefficient of performance (COP) for a Carnot refrigerator is COP =
Cryogenic quantum computers require classical control electronics operating at room temperature (T_room ≈ 300 K) to interface with qubits at milliKelvin (T_qubit ≈ 10 mK). Information erasure in room-temperature logic incurs a Landauer-limit cost, E_Landauer = k_B T_room ln 2 per bit. Refrigeration to maintain T_qubit introduces additional energy dissipation, quantified via Carnot efficiency: COP = T_qubit / (T_room − T_qubit). This work computes the total energy floor as E_total = E_Landauer + E_refrigeration, where E_refrigeration = Q_cold / COP and Q_cold = k_B T_qubit ln 2 per bit. A JPCUB normalization is proposed to partition dissipation between controller and qubit subsystems, enabling targeted optimization. For T_qubit = 10 mK, E_refrigeration ≈ 10^4 × E_Landauer, indicating controller-side dissipation dominates. [to verify] JPCUB ratios quantify trade-offs, guiding thermal management in scalable quantum architectures. ## 1. Introduction Quantum computing systems face dual thermodynamic challenges: information erasure in classical control electronics and refrigeration to maintain qubit coherence. Landauer's principle establishes E_Landauer = k_B T ln 2 per bit as the minimum energy cost for information loss [Landauer, 1961, DOI:10.1103/RevModPhys.29.544]. However, cryogenic qubits necessitate refrigeration from room temperature, introducing a Carnot efficiency overhead [Kittel, 2008, DOI:10.1002/9783527618132]. This paper derives the combined energy floor for control electronics, accounting for both effects, and introduces a normalization framework (JPCUB) to separate controller and qubit dissipation. ## 2. Background Landauer's principle links information theory and thermodynamics, asserting that erasing a bit of information requires energy E = k_B T ln 2 [Landauer, 1961]. For cryogenic systems, this applies to room-temperature logic, while qubit operation demands refrigeration. The coefficient of performance (COP) for a Carnot refrigerator is COP =
Quantum error correction is priced in combinatorial currency — rate, distance, threshold — while the physical cost of performing correction goes uncounted. Every cycle of active (measurement-based) error correction erases information: syndrome extraction, majority voting, and ancilla re-initialization each reset redundancy, and Landauer's principle prices each erasure at no less than kT·ln2 per bit. This paper derives the resulting thermodynamic floor for correction overhead — (n−k)/k · kT·ln2 per logical qubit per round for an [[n,k]] code — verifies its scaling across code families (linear for repetition, quadratic for surface, constant for constant-rate), and applies the JPCUB joules-per-correct-answer metric to rank correction architectures. A two-level nested (tree-structured) code is compared with flat Hamming codes of equal rate on a clustered NAND-like channel: hierarchical decoding cuts the per-round erasure count by a factor of 1.6 to 3, while the naive construction pays a residual-error penalty — a partial null that leaves the floor claim intact and defines the protocol for an industrial flash-memory test. Biological systems that run robust quantum processes without correction (photosynthetic energy transfer, radical-pair magnetoreception) show the floor is an architecture choice: structure prepays in capital energy what correction pays in operating erasures. Autonomous and dissipative correction is identified as the boundary of the claim. A closing section documents the 2026 quantum-algorithms-and-QEC curriculum snapshot: the energy bill the paper prices is absent from the field's own teaching materials as of 2026-08-26.
Current joules-per-compute (JPCUB) benchmarks normalize energy consumption per physical gate, conflating logical operations with error-correction overhead. This obscures trade-offs between qubit connectivity, gate fidelity, and fault-tolerant design, hindering equitable comparisons of quantum processors. We propose a revised JPCUB metric that normalizes energy use per logical-qubit operation, isolating logical computation from error-correction infrastructure. The protocol involves executing standardized logical circuits (e.g., Clifford operations) on surface-code qubits, measuring total energy, and decomposing it into logical gate costs and error-correction overhead. By standardizing benchmark circuits and energy measurement techniques, this approach enables reproducible evaluations of quantum architectures. Simulations suggest [to verify] a 2–5× reduction in JPCUB for fault-tolerant systems compared to uncorrected benchmarks, highlighting the interplay between logical efficiency and error-correction demands. ## 1. Introduction Quantum computing architectures face escalating energy demands as they scale, necessitating rigorous metrics to evaluate efficiency. The existing JPCUB benchmark, which quantifies energy per physical gate [arXiv:2006.07065], fails to distinguish between the energy required for logical operations and that consumed by error correction. Logical-qubit operations inherently involve multiple physical gates and repeated error-correction cycles, yet current metrics conflate these components, leading to inconsistent evaluations. This limitation impedes the development of architectures optimized for both computational speed and energy efficiency. Here, we address this gap by redefining JPCUB to normalize energy consumption per logical-qubit operation, enabling fair comparisons across diverse quantum processor designs. ## 2. Background Quantum processors execute logical operations through sequences of physical gates, with error correction ensuri
Quantum error correction is priced in combinatorial currency — rate, distance, threshold — while the physical cost of performing correction goes uncounted. Every cycle of active (measurement-based) error correction erases information: syndrome extraction, majority voting, and ancilla re-initialization each reset redundancy, and Landauer's principle prices each erasure at no less than kT·ln2 per bit. This paper derives the resulting thermodynamic floor for correction overhead — (n−k)/k · kT·ln2 per logical qubit per round for an [[n,k]] code — verifies its scaling across code families (linear for repetition, quadratic for surface, constant for constant-rate), and applies the JPCUB joules-per-correct-answer metric to rank correction architectures. A two-level nested (tree-structured) code is compared with flat Hamming codes of equal rate on a clustered NAND-like channel: hierarchical decoding cuts the per-round erasure count by a factor of 1.6 to 3, while the naive construction pays a residual-error penalty — a partial null that leaves the floor claim intact and defines the protocol for an industrial flash-memory test. Biological systems that run robust quantum processes without correction (photosynthetic energy transfer, radical-pair magnetoreception) show the floor is an architecture choice: structure prepays in capital energy what correction pays in operating erasures. Autonomous and dissipative correction is identified as the boundary of the claim.
Irreversibility has a fundamental thermodynamic cost, erasing information inevitably generates heat. This connection is quantified by the Landauer bound, which gives the minimum dissipation needed to erase a single bit of information. While this bound applies in both classical and quantum settings, it is saturated only in idealised limits of infinite resources. Here, we provide a unified first principles description of finite-resource erasure in both classical and quantum systems. We begin by proving the communal folklore that in the idealised regime the erasure cost of a bit encoded in a quantum or classical system is the same. Despite this, we show that their practical implementation differs substantially: achieving comparable erasure quality in quantum systems requires more control, larger accessible energy gaps and longer operation times. Classical protocols can achieve the erasure of a comparable quantum protocol under far weaker constraints which we expose in trade-off relations. Our results explain why practical erasure schemes fall short of Landauer's bound and show that classical systems enjoy several fundamental thermodynamic advantages.
The standard Landauer bound W≥kBTln2 sets the fundamental thermodynamic cost for information erasure under ideal conditions: weak system-bath coupling, quasistatic operation, and equilibrium reservoirs. However, realistic quantum error correction (QEC) operates in a profoundly different regime-finite-time syndrome extraction, strong coupling to cryogenic environments, and non-equilibrium dynamics. Here, we develop a unified thermodynamic framework for fault-tolerant quantum computing that incorporates corrections beyond the ideal Landauer limit. We derive a generalized bound on the heat dissipation per QEC cycle: Qmin≥kBTln2+kBTΔISB+ℏτ, and scaling this result to large-scale quantum computers, we find that the total heat load grows polynomially with code distance but remains in the nanowatt range for million-qubit systems-well within the cooling power of modern dilution refrigerators. Applying our model to superconducting qubit architectures, we show that while strong coupling can add up to ∼20% to the ideal cost, finite-time effects contribute approximately 0.55% at 100 ns and 5.5% at 10 ns reset operations. Our results establish that the true thermodynamic cost of fault tolerance, while exceeding the naive Landauer estimate, does not pose a fundamental obstacle to scalability; the dominant engineering challenges lie in the heat load of control electronics and wiring, not in the fundamental dissipation of qubit reset.
Quantum information reset is an intrinsically irreversible process constrained by the second law of thermodynamics. In contemporary superconducting processors, dissipation during qubit reset is frequently dominated not only by quantum limits but by a classical energy floor on the order of 10^{-21} J. In contrast, the entropic bound T \Delta S at dilution refrigerator temperatures (\sim 15 mK) corresponds to a lower energy scale of approximately 10^{-23} J, establishing entropy production as the organizing physical principle that limits both the efficiency and the speed of reset operations. We introduce an entropy-limited efficiency estimator, \eta_S = 1 - (T \Delta S / W_{\rm diss}), which quantifies the thermodynamic distance from the entropic (Landauer) limit. By construction, \eta_S = 0 corresponds to the saturation of the entropy production limit, while \eta_S \to 1 indicates dissipation dominated by non-entropic (classical) contributions. Maximizing \eta_S is equivalent to a constrained variational problem in which dissipated work is minimized subject to the second law of thermodynamics. For typical superconducting qubits (\omega_q/2\pi \sim 5 GHz, T_1 \sim 50–100 \mus, T \sim 15 mK), the entropic bound T \Delta S \sim 10^{-23} J lies two orders of magnitude below the experimentally observed classical dissipation floor W_{\rm floor} \sim 10^{-21} J. Figures 1 and 2 were generated using the provided scripts_generate_figures.py script, using parameters: \omega_q/2\pi = 5 GHz, T_1 = 80 \mus, and T = 15 mK.