his paper proves that exhaustive reconstruction attacks on product-quantized (PQ) embeddings transmitted in federated learning settings are physically impossible for any attacker conforming to the laws of physics. The proof establishes two independent physical barriers targeting distinct attacker classes. The first barrier, derived from Landauer's principle (Landauer, 1961), shows that irreversible classical attackers must expend energy exceeding 10 to the power of 38,778 joules to perform exhaustive reconstruction. This surpasses the total energy of the observable universe by 10 to the power of 38,709 orders of magnitude, making the attack thermodynamically impossible. The second barrier, derived from the Margolus-Levitin theorem (Margolus and Levitin, 1998), closes the reversible computation loophole identified by Bennett (1973, 1982). It shows that reversible classical and quantum attackers -- including fault-tolerant quantum computers using the optimal Grover algorithm -- require more operations than the entire observable universe can perform in its lifetime. Even with Grover acceleration, the required operations exceed the universal computational ceiling by at least 10 to the power of 19,277 orders of magnitude. Advances in quantum error correction, including recent below-threshold demonstrations, do not affect this conclusion: the Margolus-Levitin bound constrains operation rate for any physical system regardless of hardware architecture or error rates. The two barriers are physically independent. The first rests on the second law of thermodynamics; the second rests on the time-energy uncertainty relation of quantum mechanics. Neither can be invalidated by any computational or engineering advance. The information loss quantity driving both barriers -- 128,886 bits -- is derived from first principles from the mutual information upper bound of the PQ quantization mapping, accounting for side-channel leakage and encryption key entropy. A conditional security certificate (T14-C) integrates both barriers with an explicit premise dependency matrix. The principal open problem is the MC4b structural conjecture, which asserts that the PQ codebook presents as a black box to the attacker; all conclusions are conditional on this premise, which is under active investigation by the authors. The framework applies directly to value caches in transformer architectures. Application to RoPE-modulated key caches requires additional treatment and is declared out of scope. This work represents, to the authors' knowledge, the first application of a dual thermodynamic and quantum mechanical barrier to privacy guarantees for product quantization in federated learning, and the first systematic closure of the reversible computation loophole in this security context. The closest prior work, Coiteux-Roy (2022), applies Landauer's principle to general cryptographic protocols but does not address reversible computation, quantum attackers, or the specific information-loss structure of PQ quantization. This preprint is part of the Mnemosyne Project, a post-quantum distributed computing architecture grounding cryptographic security in physical laws rather than mathematical complexity assumptions.
his paper proves that exhaustive reconstruction attacks on product-quantized (PQ) embeddings transmitted in federated learning settings are physically impossible for any attacker conforming to the laws of physics. The proof establishes two independent physical barriers targeting distinct attacker classes. The first barrier, derived from Landauer's principle (Landauer, 1961), shows that irreversible classical attackers must expend energy exceeding 10 to the power of 38,778 joules to perform exhaustive reconstruction. This surpasses the total energy of the observable universe by 10 to the power of 38,709 orders of magnitude, making the attack thermodynamically impossible. The second barrier, derived from the Margolus-Levitin theorem (Margolus and Levitin, 1998), closes the reversible computation loophole identified by Bennett (1973, 1982). It shows that reversible classical and quantum attackers -- including fault-tolerant quantum computers using the optimal Grover algorithm -- require more operations than the entire observable universe can perform in its lifetime. Even with Grover acceleration, the required operations exceed the universal computational ceiling by at least 10 to the power of 19,277 orders of magnitude. Advances in quantum error correction, including recent below-threshold demonstrations, do not affect this conclusion: the Margolus-Levitin bound constrains operation rate for any physical system regardless of hardware architecture or error rates. The two barriers are physically independent. The first rests on the second law of thermodynamics; the second rests on the time-energy uncertainty relation of quantum mechanics. Neither can be invalidated by any computational or engineering advance. The information loss quantity driving both barriers -- 128,886 bits -- is derived from first principles from the mutual information upper bound of the PQ quantization mapping, accounting for side-channel leakage and encryption key entropy. A conditional security certificate (T14-C) integrates both barriers with an explicit premise dependency matrix. The principal open problem is the MC4b structural conjecture, which asserts that the PQ codebook presents as a black box to the attacker; all conclusions are conditional on this premise, which is under active investigation by the authors. The framework applies directly to value caches in transformer architectures. Application to RoPE-modulated key caches requires additional treatment and is declared out of scope. This work represents, to the authors' knowledge, the first application of a dual thermodynamic and quantum mechanical barrier to privacy guarantees for product quantization in federated learning, and the first systematic closure of the reversible computation loophole in this security context. The closest prior work, Coiteux-Roy (2022), applies Landauer's principle to general cryptographic protocols but does not address reversible computation, quantum attackers, or the specific information-loss structure of PQ quantization. This preprint is part of the Mnemosyne Project, a post-quantum distributed computing architecture grounding cryptographic security in physical laws rather than mathematical complexity assumptions.
This upload contains the full preprint "Han's Structure-Induced Quantum Blindness Conjecture (Han-SIQB)" (version v6), a technical report from the Mnemosyne research project. The paper formally introduces and systematically studies Han's Structure-Induced Quantum Blindness Conjecture (Han-SIQB), a central open problem in product quantization security theory. The core statement of the conjecture is that, for any product quantization codebook C trained by k-means or LBG and a publicly known Soft-ZCA whitening transformation W, the composite function PQ_C composed with W admits no quantum algorithm that achieves a search speedup beyond Grover's algorithm (that is, no acceleration factor alpha_Q greater than 2). The conjecture currently lacks a complete proof and is positioned as a structural working hypothesis supported by the convergence of multiple independent lines of evidence from quantum query complexity, computational geometry, statistical physics, cryptography, and neuroscience. The work organizes the conjecture into five concrete sub-obligations (exclusion of known quantum paradigms, statistical structure testing, reduction to known hard problems, formal reduction of quantum mode-finding to unstructured search, and experimental verification of near-uniform conditional entropy), honestly assesses the formalization level of each piece of supporting evidence, and identifies the critical open problems on the path to a full proof. For the Mnemosyne post-quantum distributed computing architecture, Han-SIQB constitutes the sole remaining condition required to fully close its dual physical security barriers (Landauer thermodynamic bound and Margolus-Levitin operation-count bound) in the time dimension, potentially enabling a security system grounded solely in physical laws rather than mathematical hardness assumptions. The paper also provides a detailed epistemological positioning of the conjecture, discusses historical precedents, and explicitly outlines fallback mechanisms (hybrid post-quantum cryptography mode) should the conjecture eventually be falsified. It invites collaboration from the quantum query complexity, algebraic geometry, and machine learning theory communities to advance its formalization or disproof. This upload includes the complete manuscript with all appendices, tables, and references. Keywords: product quantization, quantum blindness, structural conjecture, post-quantum security, Grover lower bound, whitening transformation, physical security, Mnemosyne
Abstract Computation requires energy. Landauer’s principle establishes that erasing one bit of information irreversibly dissipates at least kBT ln 2 of thermal energy. Modern computers operate 10^6 to 10^9 times above this fundamental bound. In 1973, Bennett demonstrated that any computation can be made thermodynamically reversible through procedures that preserve information, thereby approaching the Landauer limit. However, a critical question remained unanswered for fifty years: how quickly does reversible compilation converge to the theoretical minimum, and what ensemble size achieves practical energy savings? We resolve this problem through the Fluctuation-Dissipation Compilation Theorem, which provides the first explicit convergence rate for thermodynamically optimal computing. Reversible compilation of any algorithm dissipates average energy equal to the thermal energy scale multiplied by the logical entropy change, plus corrections that decrease as the inverse square root of the ensemble size. This N^(−1/2) convergence rate, derived by combining fluctuation theorems from statistical mechanics with the central limit theorem, is provably optimal—no compilation strategy can achieve faster convergence. The proof unifies three historically independent disciplines: statistical mechanics through the Jarzynski equality and Crooks fluctuation theorem, information theory through Shannon entropy bounds, and compilation theory through Bennett’s reversible computing framework. By formulating thermodynamic compilation as a rigorous resource theory, we establish that reversible transformations exactly conserve entropy while irreversible erasure operations incur fundamental energy costs bounded by information-theoretic limits. Critically, we provide finite-ensemble performance guarantees essential for practical implementations, characterizing not just average energy dissipation but complete statistical distributions including fluctuations and tail probabilities. Numerical experiments on Grover search and quantum Fourier transforms validate our predictions. Simulations confirm the predicted N^(−1/2) convergence rate within 3% error and demonstrate energy reductions exceeding 99.9% compared to irreversible implementations. These results establish that thermodynamically optimal compilation is achievable with realistic ensemble sizes for near-term quantum hardware operating under thermal constraints. This framework bridges fundamental physics and practical computing. Applications span quantum processors cooled to millikelvin temperatures where every picojoule of dissipation limits performance, battery-powered Internet-of-Things sensors requiring decade-long autonomous operation, data centers consuming one percent of global electricity where incremental efficiency improvements yield billion-dollar savings, and deep-space missions operating on hundred-watt radioisotope generators where every watt conserved enables additional science. By establishing thermodynamic compilation as a rigorous engineering discipline with explicit finite-size guarantees, this work transforms Bennett’s asymptotic theoretical insight into a practical methodology for approaching fundamental physical limits in real computing systems
Quantum error correction (QEC) is widely regarded as the indispensable cornerstone for achieving scalable, fault-tolerant quantum computation. However, conventional fault-tolerance theorems traditionally rely on abstract circuit models that frequently abstract away fundamental physical laws, treating noise channels and error syndromes as purely mathematical parameters. In this paper, we establish the Physical Limit of Quantum Error Correction (PLQEC) framework, investigating whether there exists an absolute, nature-imposed lower bound on the residual error probability, denoted as p_min, below which reliable quantum information preservation becomes physically unattainable due to deeply rooted thermodynamic, informational, and spacetime constraints. By systematically integrating Landauer's principle, von Neumann entropy bounds, the intrinsic thermodynamics of quantum measurement, and holographic gravity principles, we derive universal fundamental limits on quantum error correction thresholds. We demonstrate that physical constraints such as minimal energy dissipation per syndrome extraction cycle, quantum speed limits, decoherence velocities, and spacetime curvature cutoffs impose strict, unyielding boundaries on fault-tolerant quantum architectures. This work bridges the profound gap between abstract quantum information theory and fundamental physical law, opening a new frontier in understanding the ultimate operational boundaries of quantum information processing in our physical universe.
The Margolus-Levitin (ML) bound establishes a fundamental limit on the speed of quantum evolution, stating that a system with mean energy ⟨H⟩ above its ground state requires a minimum time τ≥πℏ/(2⟨H⟩) to evolve between orthogonal states. This preprint rigorously derives the ML bound and critically examines its applicability as a universal per-operation energy-time limit for quantum computation at the Landauer-relevant energy scale. We show that while the ML bound constrains the intrinsic dynamics of the qubit register, it does not account for the dominant energy dissipation from cryogenic control hardware. Consequently, the ML bound alone is insufficient to characterize the energy cost of a logical operation. To address this, we connect the theoretical limit to the QNFO joules-per-compute (JPCUB) benchmark program and propose a concrete normalization of computational cost. This normalization quantifies energy per logical qubit operation by incorporating both the intrinsic qubit energy and the overhead of cryogenic controllers, providing a practical metric for quantum computation efficiency. ## 1. Introduction Quantum speed limits (QSLs) bound the minimum time required for a quantum state to evolve to an orthogonal state. The Margolus-Levitin (ML) bound is a prominent QSL, relating this minimum time to the mean energy of the system. As quantum computing hardware matures, understanding the fundamental energy limits of computation becomes critical. The Landauer limit sets a lower bound on the energy dissipated by irreversible operations, while the ML bound sets a lower bound on the time required for state evolution. This preprint derives the ML bound and assesses its role as a universal per-operation energy-time bound for quantum computation. We then connect this theoretical limit to the QNFO joules-per-compute (JPCUB) benchmark program, proposing a concrete normalization for the energy cost of logical qubit operations in cr
We establish a unified framework for understanding the fundamental physical limits of computation by developing a geometric theory of resource trade-offs. Our approach treats computation as a physical process constrained by thermodynamics, quantum mechanics, and relativity, leading to a multidimensional capacity region in resource space. We prove that any algorithm achieving error ε on problem class P must satisfy joint constraints across five dimensions: space S, time T, bandwidth H, energy E, and coherence C. The capacity region is characterized by fundamental physical constants: Landauer’s principle (kBT ln 2 per bit erasure), the Margolus-Levitin quantum speed limit (2E/(π~) operations per second), the Bekenstein bound (maximum information density), and statistical sampling limits. We provide explicit constructions showing these bounds are achievable within constant factors for canonical problems including matrix multiplication, covariance estimation, and Gaussian process inference. This framework unifies classical complexity theory with quantum information and thermodynamics, providing the first general theory of multi-resource computational limits with applications to algorithm design, hardware optimization, and fundamental physics.
Phase III of the Qubit Delusion series. Examines fundamental physical limits on computation: Landauer bound, Margolus-Levitin theorem, Bremermann limit, Bekenstein bound. Shows that quantum error-correction overhead (1e2-1e3 physical operations per logical operation) multiplies thermodynamic cost beyond what any polynomial algorithmic speedup can overcome. Proposes a falsifiable criterion: joules per solution on commercially relevant problems. Assesses all major quantum platforms against this criterion. Identifies reversible computing as the only paradigm that can approach the true physical limits. Full text and updates: https://papers.qnfo.org/papers/paper-physics-of-computation/