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DOE OSTI2024

Group-theoretic error mitigation enabled by classical shadows and symmetries

Abstract Estimating expectation values is a key subroutine in quantum algorithms. Near-term implementations face two major challenges: a limited number of samples required to learn a large collection of observables, and the accumulation of errors in devices without quantum error correction. To address these challenges simultaneously, we develop a quantum error-mitigation strategy called symmetry-adjusted classical shadows , by adjusting classical-shadow tomography according to how symmetries are corrupted by device errors. As a concrete example, we highlight global U(1) symmetry, which manifests in fermions as particle number and in spins as total magnetization, and illustrate their group-theoretic unification with respective classical-shadow protocols. We establish rigorous sampling bounds under readout errors obeying minimal assumptions, and perform numerical experiments with a more comprehensive model of gate-level errors derived from existing quantum processors. Our results reveal symmetry-adjusted classical shadows as a low-cost strategy to mitigate errors from noisy quantum experiments in the ubiquitous presence of symmetry.

Zhao, Andrew (ORCID:0000000202990277)↗

DOE OSTI2021

Test Points for Online Monitoring of Quantum Circuits

Noisy Intermediate-Scale Quantum (NISQ) computers consisting of tens of inherently noisy quantum bits (qubits) suffer from reliability problems. Qubits and their gates are susceptible to various types of errors. Due to limited numbers of qubits and high error rates, quantum error correction cannot be applied. Physical constraints of quantum hardware including the error rates are used to guide the design and the layout of quantum circuits. The error rates determine the selection of qubits and their operations. The resulting circuit is executed on the quantum computer. This study explores the risk of unexpected changes in the error rates of NISQ computers post-calibration. We show that unexpected changes in error rates can alter the output state of a quantum circuit. To detect these changes, we propose the insertion of test points into the quantum circuit to enable online monitoring of the physical qubit behavior. We utilize classical, superposition, and uncompute test points. Furthermore, we use a gate error coverage metric to assess the quality of the tests. We verify the effectiveness of the proposed scheme on different IBM quantum computers (IBM Q), in addition to a noisy simulation that shows the scalability of the proposed approach.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

RISE: robust iterative surface extension for sub-nanometer X-ray mirror fabrication

Precision optics have been widely required in many advanced technological applications. X-ray mirrors, as an example, serve as the key optical components at synchrotron radiation and free electron laser facilities. They are rectangular silicon or glass substrates where a rectangular Clear Aperture (CA) needs to be polished to sub-nanometer Root Mean Squared (RMS) to keep the imaging capability of the incoming X-ray wavefront at the diffraction limit. The convolutional polishing model requires a CA to be extended with extra data, from which the dwell time is calculated via deconvolution. However, since deconvolution is very sensitive to boundary errors and noise, the existing surface extension methods can hardly fulfill the sub-nanometer requirement. On one hand, the figure errors in a CA were improperly modeled during the extension, leading to continuity issues along the boundary. On the other hand, uncorrectable high-frequency errors and noise were also extended. In this study, we propose a novel Robust Iterative Surface Extension (RISE) method that resolves these problems with a data fitting strategy. RISE models the figure errors in a CA with orthogonal polynomials and ensures that only correctable errors are fit and extended. Combined with boundary conditions, an iterative refinement of dwell time is then proposed to compensate the errors brought by the extension and deconvolution, which drastically reduces the estimated figure error residuals in a CA while the increase of total dwell time is negligible. To our best knowledge, RISE is the first data fitting-based surface extension method and is the first to optimize dwell time based on iterative extension. An experimental verification of RISE is given by fabricating two elliptic cylinders (10 mm × 80 mm CAs) starting from a sphere with a radius of curvature around 173 m using ion beam figuring. The figure errors in the two CAs greatly improved from 204.96 nm RMS and 190.28 nm RMS to 0.62 nm RMS and 0.71 nm RMS, respectively, which proves that RISE is an effective method for sub-nanometer level X-ray mirror fabrication.

36 MATERIALS SCIENCE↗

DOE OSTI2022

System and method for error detection and correction in virtual reality and augmented reality environments

Embodiments of the present disclosure are related to training one or more of machine learning algorithms in a virtual reality environment for error detection and correction and/or for employing one or more trained machine learning models in an augmented reality environment to detect and/or correct user errors associated the performance of one or more tasks.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Author Correction: Genome-guided isolation of the hyperthermophilic aerobe Fervidibacter sacchari reveals conserved polysaccharide metabolism in the Armatimonadota

Correction to: Nature Communicationshttps://doi.org/10.1038/s41467-024-53784-3, published online 4 November 2024 In the version of this article initially published, Table 1 did not include the properties of the taxa being proposed or refer directly to another location in the main manuscript describing the properties. As such, the original manuscript did not comply with Rule 27 (2)(c) of the ICNP. Also, Table 1 listed the order Fervidibacterales as the nomenclatural type for the class Fervidibacteria, which violates latest emended version of Rule 15 stating that the nomenclatural type for a class must be a genus. Below we provide a modification of Table 1 containing protologues with these errors corrected. We have also changed the order of the taxa in the table to meet the most common ordering. (Table presented.) Taxon names proposed under the ICNP Proposed taxon Etymology Description Genus Fervidibacter Fer.vi.di.bac’ter. L. masc. adj. fervidus, hot, steaming; N.L. masc. n. bacter, a rod; N.L. masc. n. Fervidibacter, a hot rod Thermophilic or hyperthermophilic inhabitants of freshwater thermal environments. All members are likely polysaccharide-degrading chemoheterotrophs with numerous carbohydrate-active enzymes encoded in their genomes. Aerobic, with high-affinity and/or low-affinity terminal oxidases present in the genomes. The oxidative pentose phosphate pathway and the tricarboxylic acid cycle are complete in genomes belonging to the genus. Gram-stain-negative and diderm cell envelope structure. Ovoid- to rod-shaped morphology. Spores are not formed. The genus is a distinct phylogenetic lineage in the family Fervidibacteraceae, the order Fervidibacterales, and the class Fervidibacteria in the phylum Armatimonadota. The type species is Fervidibacter sacchariT. Species Fervidibacter sacchari sac’cha.ri. N.L. gen. n. sacchari, of sugar Hyperthermophilic, microaerophilic, facultatively anaerobic, and grows chemoheterotrophically on monosaccharides and polysaccharides. Cells are ovoid- to rod-shaped, Gram-stain negative, and are 0.9–1.3 µm in width and 1.6–3.6 µm in length. Grows between 65 and 87.5 °C and an optimum temperature of 80 °C, and a pH range of 6.5–8.6 with an optimum pH of 7.5. Grows at an optimum O2 concentration of 5–10%. Grows on D-arabinose, D-galactose, D-glucose, D-rhamnose, D-ribose, D-xylose, chondroitin sulfate, colloidal chitin, galactan, gellan gum, guar gum, karaya gum, locust bean gum, xantham gum, xyloglucan, β-glucan, glycogen, starch, AFEX-pretreated corn stover, miscanthus, sugarcane bagasse, acetate and casamino acids. Grows weakly on xyloglucan under fermentation conditions. The major fatty acids (>10%) are C16:0, C18:0 and/or cyclo-C17:0, and iso-C16:0. The major respiratory quinones (>10%) are MK-8 and MK-9. The isolate and genomes of the species have been recovered from geothermal springs in the Great Basin, Nevada, USA. GC content of genomes range between 51–52%. Subunits for both the high-affinity and low-affinity terminal oxidases are encoded in the genomes. Genomes also encode a Group 3d [NiFe] hydrogenase, which produces hydrogen as an electron sink for NAD+ regeneration. The type strain PD1T (= JCM 39283T = DSM 113467T) was isolated from Great Boiling Spring in Nevada, USA. Family Fervidibacteraceae Fer.vi.di.bac.te.ra’ce.ae. N.L. masc. n. Fervidibacter type genus of the family; L. suff. -aceae ending to denote a family; N.L. fem. pl. n. Fervidibacteraceae the family of the genus Fervidibacter Thermophilic or hyperthermophilic inhabitants of freshwater thermal environments. All members are likely polysaccharide-degrading chemoheterotrophs with numerous carbohydrate-active enzymes encoded in their genomes. Aerobic, with high-affinity and/or low-affinity terminal oxidases present in the genomes. The oxidative pentose phosphate pathway and the tricarboxylic acid cycle are complete in genomes belonging to the family. The family is a distinct phylogenetic lineage in the order Fervidibacterales and the class Fervidibacteria in the phylum Armatimonadota. The type genus is Fervidibacter. Order Fervidibacterales Fer.vi.di.bac.te.ra’les. N.L. masc. n. Fervidibacter type genus of the order; L. suff. -ales ending to denote an order; N.L. fem. pl. n. Fervidibacterales the order of the genus Fervidibacter Thermophilic or hyperthermophilic inhabitants of freshwater thermal environments. All members are likely polysaccharide-degrading chemoheterotrophs with numerous carbohydrate-active enzymes encoded in their genomes. Aerobic or strictly anaerobic. Phylogenomic placement of this lineage within the Fervidibacteria and relative evolutionary divergence supports delineation of this lineage as an order within the class Fervidibacteria and phylum Armatimonadota. The type genus is Fervidibacter. Class Fervidibacteria Fer.vi.di.bac.te’ri.a. N.L. masc. n. Fervidibacter type genus of the type order of the class; L. suff. -ia ending to denote a class; N.L. neut. pl. n. Fervidibacteria the class of the order Fervidibacterales Thermophilic or hyperthermophilic inhabitants of freshwater thermal environments. All members are likely polysaccharide-degrading chemoheterotrophs with numerous carbohydrate-active enzymes encoded in their genomes. Aerobic or strictly anaerobic. Phylogenomic placement of this lineage within the Armatimonadota and relative evolutionary divergence supports delineation of this lineage as a class within the Armatimonadota. The type genus is Fervidibacter. The error has not been corrected in the PDF or HTML versions of the Article.

Nou, Nancy O↗

DOE OSTI2022

Characterizing Midcircuit Measurements on a Superconducting Qubit Using Gate Set Tomography

Measurements that occur within the internal layers of a quantum circuit—midcircuit measurements—are a useful quantum-computing primitive, most notably for quantum error correction. Midcircuit measurements have both classical and quantum outputs, so they can be subject to error modes that do not exist for measurements that terminate quantum circuits. In this work, we show how to characterize midcircuit measurements, modeled by quantum instruments, using a technique that we call quantum instrument linear gate set tomography (QILGST). We then apply this technique to characterize a dispersive measurement on a superconducting transmon qubit within a multiqubit system. By varying the delay time between the measurement pulse and subsequent gates, we explore the impact of residual cavity photon population on measurement error. QILGST can resolve different error modes and quantify the total error from a measurement; in our experiment, for delay times above 1000 ns we measure a total error rate (i.e., half diamond distance) of ϵ$\diamond$ = 8.1 ± 1.4%, a readout fidelity of 97.0 ± 0.3%, and output quantum-state fidelities of 96.7 ± 0.6% and 93.7 ± 0.7 % when measuring 0 and 1, respectively.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2026

ORBIT CORRECTION STUDIES ON THE ELECTRON TRANSPORT LINE FROM RCS TO ESR

A dedicated electron transfer line from Rapid Cycling Synchrotron (RCS) to Electron Storage Ring (ESR), referred to as the RTE line has been designed for the Electron-Ion Collider (EIC). The beamline follows a straight-line geometry, with a length of 133 m, and is consists with two matching sections and a FODO section for beam diagnostics. Imperfections with magnet alignments introduce orbit distortions, making orbit correction scheme a critical component in the design. To facilitate orbit correction, each quadrupole magnet is equipped with a pair of beam position monitors (BPMs) and kickers. The Singular Value Decomposition (SVD) algorithm is used for orbit correction and tolerance studies. This paper presents the ongoing progress in the optics design and error correction scheme of the RTE line.

Neththikumara, I. [Thomas Jefferson National Accel↗

DOE OSTI2017

RedThreads: An Interface for Application-Level Fault Detection/Correction Through Adaptive Redundant Multithreading

In the presence of accelerated fault rates, which are projected to be the norm on future exascale systems, it will become increasingly difficult for high-performance computing (HPC) applications to accomplish useful computation. Due to the fault-oblivious nature of current HPC programming paradigms and execution environments, HPC applications are insufficiently equipped to deal with errors. We believe that HPC applications should be enabled with capabilities to actively search for and correct errors in their computations. The redundant multithreading (RMT) approach offers lightweight replicated execution streams of program instructions within the context of a single application process. Furthermore, the use of complete redundancy incurs significant overhead to the application performance.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2016

AMiBA: cluster Sunyaev-Zel'dovich effect observations with the expanded 13-element array

The Yuan-Tseh Lee Array for Microwave Background Anisotropy (AMiBA) is a co-planar interferometer array operating at a wavelength of 3 mm to measure the Sunyaev–Zel’dovich effect (SZE) of galaxy clusters at arcminute scales. The first phase of operation—with a compact 7-element array with 0.6 m antennas (AMiBA-7)—observed six clusters at angular scales from 5{sup ′} to 23{sup ′}. Here, we describe the expansion of AMiBA to a 13-element array with 1.2 m antennas (AMiBA-13), its subsequent commissioning, and cluster SZE observing program. The most noticeable changes compared to AMiBA-7 are (1) array re-configuration with baselines ranging from 1.4 m to 4.8 m, allowing us to sample structures between 2{sup ′} and 10{sup ′}, (2) 13 new lightweight carbon-fiber-reinforced plastic (CFRP) 1.2 m reflectors, and (3) additional correlators and six new receivers. Since the reflectors are co-mounted on and distributed over the entire six-meter CFRP platform, a refined hexapod pointing error model and phase error correction scheme have been developed for AMiBA-13. These effects—entirely negligible for the earlier central close-packed AMiBA-7 configuration—can lead to additional geometrical delays during observations. Our correction scheme recovers at least 80 ± 5% of the point-source fluxes. We, therefore, apply an upward correcting factor of 1.25 to our visibilities to correct for phase decoherence, and a ±5% systematic uncertainty is added in quadrature with our statistical errors. We demonstrate the absence of further systematics with a noise level consistent with zero in stacked uv-visibilities. From the AMiBA-13 SZE observing program, we present here maps of a subset of 12 clusters with signal-to-noise ratios above five. We demonstrate combining AMiBA-7 with AMiBA-13 observations on Abell 1689, by jointly fitting their data to a generalized Navarro–Frenk–White model. Our cylindrically integrated Compton-y values for five radii are consistent with results from the Berkeley-Illinois-Maryland Array, the Owens Valley Radio Observatory, the Sunyaev–Zel’dovich Array, and the Planck Observatory. We also report the first targeted SZE detection toward the optically selected cluster RCS J1447+0828, and we demonstrate the ability of AMiBA SZE data to serve as a proxy for the total cluster mass. Finally, we show that our AMiBA-SZE derived cluster masses are consistent with recent lensing mass measurements in the literature.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2020

3D Coded SUMMA: Communication-Efficient and Robust Parallel Matrix Multiplication

In this paper, we propose a novel fault-tolerant parallel matrix multiplication algorithm called 3D Coded SUMMA that achieves higher failure-tolerance than replication-based schemes for the same amount of redundancy. This work bridges the gap between recent developments in coded computing and fault-tolerance in high-performance computing (HPC). The core idea of coded computing is the same as algorithm-based fault-tolerance (ABFT), which is weaving redundancy in the computation using error-correcting codes. In particular, we show that MatDot codes, an innovative code construction for parallel matrix multiplications, can be integrated into three-dimensional SUMMA (Scalable Universal Matrix Multiplication Algorithm [30]) in a communication-avoiding manner. To tolerate any two node failures, the proposed 3D Coded SUMMA requires ~50% less redundancy than replication, while the overhead in execution time is only about 5–10%.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2019

Quantum information in the Posner model of quantum cognition

Matthew Fisher recently postulated a mechanism by which quantum phenomena could influence cognition: Phosphorus nuclear spins may resist decoherence for long times. The spins would serve as biological qubits. The qubits may resist decoherence longer when in Posner molecules. We imagine that Fisher postulates correctly. How adroitly could biological systems process quantum information (QI)? We establish a framework for answering. Additionally, we construct applications of biological qubits to quantum error correction, quantum communication, and quantum computation. First, we posit how the QI encoded by the spins transforms as Posner molecules form. The transformation points to a natural computational basis for qubits in Posner molecules. From the basis, we construct a quantum code that detects arbitrary single-qubit errors. Each molecule encodes one qutrit. Shifting from information storage to computation, we define the model of Posner quantum computation. To illustrate the model’s quantum-communication ability, we show how it can teleport information incoherently: A state’s weights are teleported. Dephasing results from the entangling operation’s simulation of a coarse-grained Bell measurement. Whether Posner quantum computation is universal remains an open question. However, the model’s operations can efficiently prepare a Posner state usable as a resource in universal measurement-based quantum computation. The state results from deforming the Affleck–Kennedy–Lieb–Tasaki (AKLT) state and is a projected entangled-pair state (PEPS). Finally, we show that entanglement can affect molecular-binding rates, boosting a binding probability from 33.6% to 100% in an example. This work opens the door for the QI-theoretic analysis of biological qubits and Posner molecules.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2024

An Assessment of Receiver Algorithms for Distributed Massive MIMO Systems: Investigating Design Solutions and Performance

This study investigates receiver design solutions for distributed Massive Multiple Input Multiple Output (D-m MIMO) systems, taking into account parameters such as number of access points as well as concerns related to channel estimates that use single-carrier frequency-domain equalization (SC-FDE). A significant contribution of this research is the integration of Low-Density Parity-Check (LDPC) codes to simplify coding complexity and enhance communication efficiency. The research examines different receiver designs, such as spatial antenna correlation and sophisticated channel estimation methods. The authors propose integrating LDPC codes into the receiver architecture to simplify computations and enhance error correction and decoding. Moreover, the paper examines performance evaluation measures and approaches, highlighting the trade-offs among complexity, spectral efficiency, and error performance. The comparative analysis indicates the benefits, in terms of performance, of incorporating LDPC codes and improving system throughput and dependability. We examine four distinct receiver algorithms: zero-forcing (ZF), minimum mean square error (MMSE), maximum ratio combining (MRC), and equal gain combining (EGC). The study shows that MRC and EGC receivers work well in D-m MIMO because they make the receiver system less computationally demanding.

Gashtasbi, Ali↗

DOE OSTI2023

Mitigation of Cosmic Rays-Induced Errors in Superconducting Quantum Processors

Environmental radioactivity and cosmic-rays have recently been identified as a source of decoherence in super-conducting quantum bits (qubits). In particular, the absorption of cosmic-ray muons and gamma rays emitted by naturally occurring radioactive isotopes in the qubit substrate leads to correlated errors in superconducting quantum processors, posing significant challenges to quantum error correction. To enable quantum computing to scale, it is therefore necessary the devel-opment of mitigation strategies to prevent, or keep under control, error bursts due to particle impacts in the chip. While most environmental radioactive sources can be effectively suppressed using dedicated shielding, cosmic-ray muons, with their high penetration capability, can only be mitigated by moving the entire facility in a deep underground laboratory. This work explores the potential for developing a novel class of quantum processors equipped with an active veto system to protect superconducting-based quantum computers from the detrimental effects of atmospheric muons. Such a device would enable the identification of an atmospheric muon interaction within the processor and veto all operations performed during the occurrence of such an interaction. By demonstrating high detection efficiency and negligible dead time, we aim to establish that the future of quantum processors can be envisioned in above-around facilities.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

DOE OSTI2024

Single-Shot Decoding of Good Quantum LDPC Codes

Abstract Quantum Tanner codes constitute a family of quantum low-density parity-check codes with good parameters, i.e., constant encoding rate and relative distance. In this article, we prove that quantum Tanner codes also facilitate single-shot quantum error correction (QEC) of adversarial noise, where one measurement round (consisting of constant-weight parity checks) suffices to perform reliable QEC even in the presence of measurement errors. We establish this result for both the sequential and parallel decoding algorithms introduced by Leverrier and Zémor. Furthermore, we show that in order to suppress errors over multiple repeated rounds of QEC, it suffices to run the parallel decoding algorithm for constant time in each round. Combined with good code parameters, the resulting constant-time overhead of QEC and robustness to (possibly time-correlated) adversarial noise make quantum Tanner codes alluring from the perspective of quantum fault-tolerant protocols.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Efficient Multigrid Reduction-in-Time for Method-of-Lines Discretizations of Linear Advection

Parallel-in-time methods for partial differential equations (PDEs) have been the subject of intense development over recent decades, particularly for diffusion-dominated problems. It has been widely reported in the literature, however, that many of these methods perform quite poorly for advection-dominated problems. In this report we analyze the particular iterative parallel-in-time algorithm of multigrid reduction-in-time (MGRIT) for discretizations of constant-wave-speed linear advection problems. We focus on common method-of-lines discretizations that employ upwind finite differences in space and Runge-Kutta methods in time. Using a convergence framework we developed in previous work, we prove for a subclass of these discretizations that, if using the standard approach of rediscretizing the fine-grid problem on the coarse grid, robust MGRIT convergence with respect to CFL number and coarsening factor is not possible. This poor convergence and non-robustness is caused, at least in part, by an inadequate coarse-grid correction for smooth Fourier modes in space-time known as characteristic components. We propose an alternative coarse-grid operator that provides a better correction of these modes. This coarse-grid operator is related to previous work and uses a semi-Lagrangian discretization combined with an implicitly treated truncation error correction. Theory and numerical experiments show the proposed coarse-grid operator yields fast MGRIT convergence for many of the method-of-lines discretizations considered, including for both implicit and explicit discretizations of high order. Parallel results demonstrate speed-up over sequential time-stepping.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Structural basis for DNA proofreading

DNA polymerase (DNAP) can correct errors in DNA during replication by proofreading, a process critical for cell viability. However, the mechanism by which an erroneously incorporated base translocates from the polymerase to the exonuclease site and the corrected DNA terminus returns has remained elusive. Here, we present an ensemble of nine high-resolution structures representing human mitochondrial DNA polymerase Gamma, Polγ, captured during consecutive proofreading steps. The structures reveal key events, including mismatched base recognition, its dissociation from the polymerase site, forward translocation of DNAP, alterations in DNA trajectory, repositioning and refolding of elements for primer separation, DNAP backtracking, and displacement of the mismatched base into the exonuclease site. Altogether, our findings suggest a conserved ‘bolt-action’ mechanism of proofreading based on iterative cycles of DNAP translocation without dissociation from the DNA, facilitating primer transfer between catalytic sites. Functional assays and mutagenesis corroborate this mechanism, connecting pathogenic mutations to crucial structural elements in proofreading steps.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2021

Mitigating Depolarizing Noise on Quantum Computers with Noise-Estimation Circuits

A significant problem for current quantum computers is noise. While there are many distinct noise channels, the depolarizing noise model often appropriately describes average noise for large circuits involving many qubits and gates. We present a method to mitigate the depolarizing noise by first estimating its rate with a noise-estimation circuit and then correcting the output of the target circuit using the estimated rate. The method is experimentally validated on the simulation of the Heisenberg model. We find that our approach in combination with readout-error correction, randomized compiling, and zero-noise extrapolation produces results close to exact results even for circuits containing hundreds of CNOT gates.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2022

Masked fault detection for reliable low voltage cache operation

Systems, apparatuses, and methods for implementing masked fault detection for reliable low voltage cache operation are disclosed. A processor includes a cache that can operate at a relatively low voltage level to conserve power. However, at low voltage levels, the cache is more likely to suffer from bit errors. To mitigate the bit errors occurring in cache lines at low voltage levels, the cache employs a strategy to uncover masked faults during runtime accesses to data by actual software applications. For example, on the first read of a given cache line, the data of the given cache line is inverted and written back to the same data array entry. Also, the error correction bits are regenerated for the inverted data. On a second read of the given cache line, if the fault population of the given cache line changes, then the given cache line's error protection level is updated.

Ganapathy, Shrikanth↗