Cambridge Quantum (CQ) is a global leader in quantum computing

Cambridge Quantum

The race for quantum computing dominance is on with four protagonists: IBM, Microsoft, Alibaba and Google. Whereas most of the attention in the quantum world is focused on the new generation of super computers that are breaking computation records regularly, we shouldn’t forget about the developers that are going to be building the first wave of quantum applications. Microsoft took the initiative to attract developers to the quantum world by releasing the first version of its which even includes a new programming language called Q#. Yesterday, at the (QSML), Google decided to enter the race by releasing the first public alpha of Cirq, an open source framework that allow developers to create algorithms without needing a background in quantum physics. Quantum Developer Kit First International Workshop on Quantum Software and Quantum Machine Learning Building applications for the quantum computers is full of challenges. For starters, developers need to get accustomed to the properties of “qubits” that differ from traditional bits in more than one annoying way 😉. For instance, qubits can be in both binary states [1,0] at the same time which is a phenomenon known as superposition. Additionally, qubits can influence each other even if they are not physically connected and their state changes incredibly rapidly. As a result, building quantum applications requires new types of programming models like Q# and now Cirq. The idea behind Cirq is to provide a simple programming model that abstracts the fundamental building blocks of quantum applications. The initial version of Cirq has been implemented in Python and is available on . Architecturally, Cirq models a quantum application using a series of base constructs. Github Circuits Any quantum programin Cirq starts is abstracted by one of two fundamental classes: a Ciruit or a Schedule. In Cirq, a Cirquit[capital C] represents the most basic form of a quantum circuit[lowercase c]. A Cirq Circuit is represented as a collection of Moments which include operations that can be executed on Qubits during some abstract slide of time. The syntax for combining Circuit, Moments and Operations in Cirq is fairly simple as shown in the following code. qubits = [cirq.GridQubit(x, y) x range(3) y range(3)] for in for in (qubits[0]) print # prints "(0, 0)" cz01 = cirq.CZ(qubits[0], qubits[1])x2 = cirq.X(qubits[2])cz12 = cirq.CZ(qubits[1], qubits[2])moment0 = cirq.Moment([cz01, x2])moment1 = cirq.Moment([cz12])circuit = cirq.Circuit((moment0, moment1)) (circuit) print # prints the text diagram for the circuit: # (0, 0): ───@─────── #            │ # (0, 1): ───@───@─── #                │ # (0, 2): ───X───@─── Devices The Device construct in Cirq represents the constraints of a specific type of quantum hardware. For instance, most hardware only allows certain gates to be enacted on qubits. Or, as another example, some gates may be constrained to not be able to run at the same time as neighboring gates Schedules & Devices A Schedule is another form of quantum circuit that includes more detailed information about the timing and duration of the gates. Conceptually, a Schedule is made up of a set of ScheduledOperations as well as a description of the Device on which the schedule is intended to be run. Each ScheduledOperation is made up of a timewhen the operation starts and a duration describing how long the operation takes, in addition to the Operation itself. In the following code, we take a Circuit on a specific type of Device and convert it to a Schedule using a single call: ExpWGatecircuit = cirq.Circuit()CZ = Exp11Gate(half_turns=1.0)X = ExpWGate(half_turns=1.0)circuit.append([CZ(device.qubits[0], device.qubits[1]), X(device.qubits[0])]) (circuit) from cirq.google.xmon_gates import print schedule = cirq.moment_by_moment_schedule(device, circuit) Gates In Cirq, Gates abstract operations on collections of qubits. To apply a Gate on a qubit, we simply call its On method as shown in the following code: CNOT GridQubitq0, q1 = (GridQubit(0, 0), GridQubit(0, 1)) (CNOT.on(q0, q1)) (CNOT(q0, q1)) from cirq.ops import from cirq.devices import print print # prints # CNOT((0, 0), (0, 1)) # CNOT((0, 0), (0, 1)) Simulators Cirq includes a Python simulator that can be used to run Circuits and Schedules. The Simulator architecture can scale across multiple threads and CPUs which allows it to run fairly sophisticated Circuits. The following code uses a Simulator modeled after the Google’s Xmom architecture: XmonSimulatorsimulator = XmonSimulator()result = simulator.run(circuit) from cirq.google import (result) print # prints something like # q0=1 q1=1 Early Adopters Even though Cirq was just announced yesterday, it already includes an all-star group of early adopters: · : simulation of a ( , ) Zapata Computing quantum autoencoder example code video tutorial · : QAOA implementation and integration into QC Ware’s AQUA platform ( , ) QC Ware example code video tutorial · : integration of True-Q software tools for assessing and extending hardware capabilities ( ) Quantum Benchmark video tutorial · : simulating the Heisenberg Quantum Simulations Anderson Model · : integration of proprietary quantum compiler t|ket> ( ) Cambridge Quantum Computing video tutorial · : architecture-aware compiler based on ( ) and simulator of quantum computers ( ) NASA temporal-planning for QAOA slides slides Additionally, Google announced the release of . The framework uses a quantum chemistry platform and adds an open source library that compiles quantum simulation algorithms to Cirq. OpenFermion-Cirq OpenFermion Quantum computing requires a new type of programming language and efforts like Cirq and Microsoft’s Q# are the first steps to bring quantum computing to mainstream developers. The next version of Cirq is likely to include better tooling and libraries that abstract even more important constructs to build the next generation of quantum applications.

Google Cirq and the New World of Quantum Programming

It’s Here, and Not Here at the Same Time Photo by on Daniel Falcão Unsplash In the past few weeks, I’ve ended up a little about quantum computing. I’ve enjoyed discussions with leaders from Microsoft’s quantum team, IBM Q, Ctrl-Q, IonQ and the Cambridge Quantum Computing centre. I even saw a . The processor needs to be cooled to close to absolute zero to operate. (For more amazing pics of how liquid-helium-punk these rigs are, .) This essay is dedicated to questioning: what’s holding back investors from putting more money into quantum startups and general R&D. learning model of the distillation refrigerator of an IBM quantum processor check this out This piece was originally published in the latest issue of my weekly newsletter Exponential View. Subscribe here The promise of quantum computing Quantum computing promises to be a totally novel form of computation, bringing a mastery to the physics unveiled by De Broglie, Schrödinger, Heisenberg, Bohr and their chums less than 100 years ago. I won’t dig too much into what quantum computing is and what opportunities it holds, but if you need a primer watch . Julie Love’s accessible 20-minute presentation The critical thing to understand about quantum computing is that it could deliver a tremendous amount of affordable computing power, compute that we need to solve key problems in the world. Assuming we can get enough stable “logical” qubits — the fundamental building blocks of quantum calculations — to build a working computer, the technology could turn vital calculations that are currently intractable into soluble ones. With the ditillation refrigerator for an IBM quantum chip. One example of a quantum speedup is in search problems. These grow exponentially complex in classical computers, but sub-exponentially using Grover’s algorithms on a quantum computer. For a 1 million item path, a quantum computer takes 500 times fewer steps than a traditional computer. To give another practical example, consider trying to understand how FeMoco, the chemical cofactor of bacterial nitrogenase, helps make those helpful prokaryotes so darn energy-shrewd at nitrogen fixation. The Haber-Bosch process consumes about 3% of the world energy; learning from the bacterial approach to nitrogen fixation would have a huge green dividend. There are other wins at the new frontier of novel biochemistries, including carbon capture, identifying new materials and exponential improvements in drug discovery that could be unleashed by quantum computing. Other applications include machine learning and logistics. The physics has long said we can build these machines. It was in the early 1980s that, most famously, Feynman & Deutsch, initiated the field of quantum computing. Some experts argue that in the past couple of years we’ve made significant progress in tackling the key issues of building quantum computers, and their components, qubits. Currently, nearly half a dozen different technologies vie for supremacy in building qubits: superconducting qubits (as found in IBM’s computer); topological qubits (which Microsoft is trying to use); (IonQ’s approach); quantum dots (which Intel took a hedge on when it backed QuTech, a Dutch University spinout); and, Of these approaches, superconducting qubits & trapped ions appear to have the most progress. trapped ions potentially, diamond-defect qubits. When experts argue that we’re past the science stage. In some senses they are right. The biggest technical issues relate to error-correction, coherence and scaling the number of qubits to usable. But I hear time and again from quantum experts that these problems are now increasingly in the realm of engineering, rather than science. (Watch Oxford’s Simon Benjamin .) discuss this point Today we actually have working quantum computers which are accessible via web services. The best known is IBM’s Quantum Experience, a 20-qubit quantum computer. The more technically might like this brief write-up of using a Rigetti quantum computer . au courant to calculate the binding energy of a deuteron There is a twist If quantum computing promises so many multi-trillion dollars opportunities, in machine learning, drug discovery and search, as well as in the not-so-distant-future quantum computing giving humanity its biggest dividend ever — surviving or mitigating the impact of climate change — then shouldn’t we be doubling down on it? This would lead to expectations of huge levels of investment. A multi-trillion dollar payout in the foreseeable is worth something today, even discounting for risk. future What we actually see is a very nascent field. While the investment levels are slowly rising, the field remains tiny. We looked at three vectors: what had governments announced, what we found from public announcements of investments in quantum computing companies, the amounts the large companies invested in the quantum computing domain. What we found was surprising. Given the large potential of quantum computing, the actual investment levels are low (with one exception, see at the end of this). We reckon, from a rough LinkedIn count, that fewer than 2,000 people are involved in companies working in any part of the quantum stack (and that includes all the marketing and PR types attached to these groups in large companies). IBM Q, which runs a developer ecosystem via the IBM Quantum Experience API, should have the deepest team. My scan on LinkedIn (far from perfect) shows only 300-or-so names attached to quantum computing in all of IBM. The startups are of similar scale, Rigetti, numbering less than 150. Some estimates go beyond this. There are 7,000 researchers working on quantum computing around the world — a more healthy, but still small number — . according to the European Commission The amount of venture dollars flowing intro quantum computing is small, with D-Wave ($175m), Rigetti ($70m), Cambridge Quantum Computing ($50m) and IonQ ($20m) leading the pack. The European Commission further estimates that total global annual investment in quantum research is some €1.5bn per annum. (Deloitte has further estimates: .) suggesting that there is about $2.2bn investments globally by governments in quantum computing So quantum computing is Schrödinger’s opportunity, simultaneously here and not here at the same time. On the one hand, quantum computing is getting all the accoutrements of a technology close to maturity (press briefings, Gartner reports, analysis from and management consultancies) and the large tech firms are trumpeting working systems within 5 years. investment banks On the other, investment levels are tiny by the standards of what large companies can put to work, or what VCs invest (Wag, a dog-walking app, .) This suggests that these smart investors are discounting the potential upside very heavily, i.e. there are many hurdles, which these investors cannot easily enumerate, to overcome or the time frame to realise is very long. recently raised $300m Which is it? Close to maturity or facing a long journey? The three things holding us back: The science has been hard. This hasn’t been metaphorically quantum physics, it has really been quantum physics. However, as many of the players in the field argue, lots of the problems are now engineering rather than science problems. And there are lots more engineers than theoretical physicists out there. Venture investors need returns within a time frame and with less uncertainty. The fact that there are only three venture-backed startups building the hardware (D-wave, with their application-specific system, Rigetti & IonQ) suggests that most venture investors are more cagey about whether this really is the realm of engineering vs. research, and just how much capital it will take to make these things work. Governments haven’t stepped up to the plate as the “investor of first resort” in basic technologies, in recent years. (I have a podcast coming up on exactly this topic with venture capitalist & economist, Bill Janeway, in a couple of weeks.) In fact, it seems that today the second and third reasons are the main attenuators. There is still too much uncertainty about when the technical barriers will be overcome and we can actually have useful quantum computers accessible. And many governments are chary about really backing core research, despite the ample long-term evidence of their positive impact in fundamental domains. In , which I recommend reading, the consultants’ estimates for the size of the quantum computing market by 2035 vary by a factor of 28, from $57bn (in their high case) or $2bn (in their base case)!! this excellent BCG report The projections suggest: A substantial market for quantum computing, but the timing could vary widely depending on when the critical technical milestones are reached that unlock actual computing capacity. In other words, the mid-term potential for this market is rather binary, dependent on when we actually get it all to work. Of course, there is an exception to all this. And that exception is China where a National Laboratory for Quantum Information Sciences will attract between $10–16bn of investment when it opens in two years. This is pretty significant, likely enough to put China in the lead in developing this technology. China’s Alibaba also has a public-cloud implementation of quantum computing, an 11-qubit machine, at this point smaller than IBM’s cloud-accessible device. Only two years ago, I was pretty sceptical about where quantum computing was in its cycle. And as Jerry Neumann , so quantum could be one of those technologies, like controllable fusion, that is always just around the corner. But something feels a bit different this time… I’d just love to see higher levels of investments in this exciting field. points out, quantum computing was only “five years away” back in 2000 I’m intrigued about speaking to founders tackling the problems in the quantum computing domain, including associated toolsets or applications. Early stage, pre-A-round, preferably, but not exclusively, based in the UK. I’m also happy to look at more mature teams. If this fits you or someone you know, email me at azeem.azhar@exponentialview.co.

Quantum Computing is Schrödinger’s Opportunity

Authors: (1) Eugene Y. S. Chua, Division of the Humanities and Social Sciences, California Institute of Technology; (2) Eddy Keming Chen, ‡Department of Philosophy, University of California. Table of Links Abstract & Introduction Decoherence and Branching The Born Rule Discussion Conclusion and References 5 Conclusion We suggest that the Evereian understanding of decoherence and branching, as well as the justifications for the Born rule, apply to both WFRE and DMRE. Hence, the theoretical benefits of DMR are available on EQM. Another consequence is that Everettians face a choice between two types of theories, one allowing only pure states for the multiverse and the other allowing mixed states also. The choice will not be based on different understandings of the branching structure or the Born rule, as the Everettian justifications equally apply in both theories, but must involve some other theoretical considerations. In any case, the availability of different versions of EQM is an interesting example of empirical underdetermination. Its implications and possible resolutions are questions we leave for future work. Acknowledgements For helpful feedback, we thank Jefferey Barrett, Charles Sebens, Kelvin McQueen, Katie Robertson, Simon Saunders, Tony Short, Karim Thebault, David Wallace, and the ´ participants at the 2023 Workshop on Relational Clocks, Decoherence, and the Arrow of Time at the University of Bristol, and the 2022 California Quantum Interpretation Network Conference at Chapman University. References Albert, David. “Elementaryantum Metaphysics”. In: Bohmian Mechanics and Quantum theory: An Appraisal. Ed. by J. T. Cushing, Arthur Fine, and Sheldon Goldstein. Kluwer Academic Publishers, 1996, pp. 277–284. — Time and chance. Cambridge: Harvard University Press, 2000. Allori, Valia et al. “Predictions and primitive ontology in quantum foundations: a study of examples”. In: The British Journal for the Philosophy of Science 65.2 (2013), pp. 323–352. Baker, David. “Measurement outcomes and probability in Everettian quantum mechanics”. In: Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 38.1 (2007), pp. 153–169. Barrett, Jeffrey. “Everettian Quantum Mechanics”. In: The Stanford Encyclopedia of Philosophy. Ed. by Edward N. Zalta and Uri Nodelman. Summer 2023. Metaphysics Research Lab, Stanford University, 2023. Chen, Eddy Keming. “From time asymmetry to quantum entanglement: The Humean unification”. In: Nousˆ 56 (2022), pp. 227–255. — “Fundamental Nomic Vagueness”. In: The Philosophical Review 131(1) (2022). — “Quantum Mechanics in a Time-Asymmetric Universe: On the Nature of the Initial Quantum State”. In: The British Journal for the Philosophy of Science 72.4 (2021), pp. 1155–1183. — “Realism about the wave function”. In: Philosophy Compass 14.7 (2019). — “Strong Determinism”. In: Philosophers’ Imprint forthcoming (2022). — “Time’s Arrow in a Quantum Universe: On the Status of Statistical Mechanical Probabilities”. In: Statistical Mechanics and Scientific Explanation: Determinism, Indeterminism and Laws of Nature. Ed. by Valid Allori. Singapore: World Scientific, 2020. Deutsch, David. “Quantum Theory of probability and decisions”. In: Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 455.1988 (1999), pp. 3129–3137. Durr, Detlef et al. “On the role of density matrices in Bohmian m ¨ echanics”. In: Foundations of Physics 35.3 (2005), pp. 449–467. Kent, Adrian. “307One World Versus Many: e Inadequacy of Everettian Accounts of Evolution, Probability, and Scientific Confirmation”. In: Many Worlds?: Everett, Quantum Theory, and Reality. Oxford University Press, June 2010. Maroney, Owen. “The Density Matrix in the de Broglie–Bohm Approach”. In: Foundations of Physics 35.3 (2005), pp. 493–510. McQueen, Kelvin J. and Lev Vaidman. “In defence of the self-location uncertainty account of probability in the many-worlds interpretation”. In: Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 66 (2018), pp. 14–23. Ney, Alyssa. The world in the wave function: a metaphysics for quantum physics. Oxford University Press, 2021. Nielsen, Michael A and Isaac L Chuang. Quantum computation and quantum information. Cambridge university press, 2010. Robertson, Katie. “In search of the holy grail: How to reduce the second law of thermodynamics”. In: The British Journal for the Philosophy of Science 73.4 (2022), pp. 987– 1020. Saunders, Simon. “Branch-counting in the Everett interpretation of quantum mechanics”. In: Proceedings of the Royal Society A 477.2255 (2021), p. 20210600. Schlosshauer, Maximilian A.Decoherence: and the quantum-to-classical transition. Springer Science & Business Media, 2007. Sebens, Charles T. and Sean M. Carroll. “Self-Locating Uncertainty and the Origin of Probability in Everettian Quantum Mechanics”. In: British Journal for the Philosophy of Science 1 (2018), axw004. Tappenden, Paul. “Evidence and Uncertainty in Everett?s Multiverse”. In: British Journal for the Philosophy of Science 62.1 (2011), pp. 99–123. Vaidman, Lev. “Derivations of the Born Rule”. In: antum, Probability, Logic: The Work and Influence of Itamar Pitowsky. Ed. by Meir Hemmo and Orly Shenker. Cham: Springer International Publishing, 2020, pp. 567–584. — “Many-Worlds Interpretation of Quantum Mechanics”. In: The Stanford Encyclopedia of Philosophy. Ed. by Edward N. Zalta. Fall 2021. Metaphysics Research Lab, Stanford University, 2021. — “On schizophrenic experiences of the neutron or why we should believe in the many-worlds interpretation of quantum theory”. In: International Studies in the Philosophy of Science 12.3 (1998), pp. 245–261. Wallace, David. e Emergent Multiverse: Quantum Theory According to the Everett Interpretation. Oxford University Press, 2012. — “The sky is blue, and other reasons quantum mechanics is not underdetermined by evidence”. In: arXiv preprint: 2205.00568 (2022). This paper is under CC 4.0 license. available on arxiv

Decoherence, Branching, and the Born Rule in a Mixed-State Everettian Multiverse: Conclusion

This paper is available on arxiv under CC 4.0 license. Authors: (1) Xiaoyu Ai, School of Electrical Engineering & Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia; (2) Robert Malaney, School of Electrical Engineering & Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia. Table of Links Abstract & Introduction System Overview Overcoming the limitation of key reconciliation Final Key Rate Experimental Results Discussion Conclusion & References Appendix VII. CONCLUSION In this work, we have carried out a full-blown analysis and experimental implementation of a Slice Reconciliation scheme applied to a specific CV QKD protocol (with post-selection) under simulated channel conditions anticipated for satelliteto-Earth channels. We have provided the optimal solution for the classical reconciliation process for this CV-QKD protocol in the context of massive parallelisation under the finite key regime. More specifically, we have identified the optimal block length when a large-code block is to be subdivided so as to improve the final secure key rate in bits per second. Although our results were based on a specific CV-QKD protocol and a specific GPU architecture, the type of analysis we have introduced here will apply in general terms a large suite of CV-QKD protocols run over any form of architecture that offers massive parallelisation. Our results, therefore, pave the way to optimal reconciliation system design for a wide range of practical CV-QKD systems that operate in the finite key regime. As the demand on the finite key size grows (better security thresholds), and technology advances lead to larger quantum signalling rates, the importance of optimised multithreaded CV-QKD reconciliation will grow. REFERENCES [1] N. Hosseinidehaj, Z. Babar, R. Malaney, S. X. Ng, and L. Hanzo, “Satellite-Based Continuous-Variable Quantum Communications: Stateof-the-Art and a Predictive Outlook,” IEEE Communications Surveys & Tutorials, vol. 21, no. 1, 881–919, 2018. [2] C. H. Bennett and G. Brassard, “Quantum Cryptography: Public Key Distribution and Coin Tossing,” in Proceedings of IEEE International Conference on Computers, Systems and Signal Processing,, vol. 175, 8, 1984. [3] H. Weinfurter, “Quantum Communication Experiments with Discrete Variables,” Quantum Information: From Foundations to Quantum Technology Applications, 369–381, 2016. [4] L. Gy ¨ongy ¨osi, L. Bacsardi, and S. Imre, “A Survey on Quantum Key Distribution,” Infocommunications Journal, vol. 11, no. 2, 14–21, 2019. [5] S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, et al., “Advances in Quantum Cryptography,” Advances in Optics and Photonics, vol. 12, no. 4, 1012–1236, 2020. [6] B. Korzh, C. C. W. Lim, R. Houlmann, N. Gisin, M. J. Li, D. Nolan, B. Sanguinetti, R. Thew, and H. Zbinden, “Provably Secure and Practical Quantum Key Distribution over 307 km of Optical Fibre,” Nature Photonics, vol. 9, no. 3, 163–168, 2015. [7] T. A. Eriksson, T. Hirano, B. J. Puttnam, G. Rademacher, R. S. Lu´ıs, M. Fujiwara, R. Namiki, Y. Awaji, M. Takeoka, N. Wada, et al., “Wavelength Division Multiplexing of Continuous Variable Quantum Key Distribution and 18.3 Tbit/s Data Channels,” Communications Physics, vol. 2, no. 1, 1–8, 2019. [8] S.-Y. Shen, M.-W. Dai, X.-T. Zheng, Q.-Y. Sun, G.-C. Guo, and Z.- F. Han, “Free-Space Continuous-Variable Quantum Key Distribution of Unidimensional Gaussian Modulation Using Polarized Coherent States in an Urban Environment,” Physical Review A, vol. 100, 012325, 2019. [9] L. Gyongyosi and S. Imre, “Secret Key Rates of Free-Space Optical Continuous-Variable Quantum Key Distribution,” International Journal of Communication Systems, vol. 32, no. 18, e4152, 2019. [10] P. Jouguet, S. Kunz-Jacques, and A. Leverrier, “Long-Distance Continuous-Variable Quantum Key Distribution with a Gaussian Modulation,” Physical Review A, vol. 84, no. 6, 062317, 2011. [11] P. Jouguet, S. Kunz-Jacques, T. Debuisschert, S. Fossier, E. Diamanti, R. All´eaume, R. Tualle-Brouri, P. Grangier, A. Leverrier, P. Pache, et al., “Field Test of Classical Symmetric Encryption with Continuous Variables Quantum Key Distribution,” Optics Express, vol. 20, no. 13, 14030–14041, 2012. [12] Q. Liao, Y. Guo, D. Huang, P. Huang, and G. Zeng, “Long-Distance Continuous-Variable Quantum Key Distribution Using Non-Gaussian State-Discrimination Detection,” New Journal of Physics, vol. 20, no. 2, 023015, 2018. [13] D. Guo, C. He, T. Guo, Z. Xue, Q. Feng, and J. Mu, “Comprehensive High-Speed Reconciliation for Continuous-Variable Quantum Key Distribution,” Quantum Information Processing, vol. 19, no. 9, 1–19, 2020. [14] Y. Zhang, Z. Chen, S. Pirandola, X. Wang, C. Zhou, B. Chu, Y. Zhao, B. Xu, S. Yu, and H. Guo, “Long-Distance Continuous-Variable Quantum Key Distribution over 202.81 km of Fiber,” Physical Review Letters, vol. 125, no. 1, 010502, 2020. [15] A. Leverrier, F. Grosshans, and P. Grangier, “Finite-Size Analysis of a Continuous-Variable Quantum Key Distribution,” Physical Review A, vol. 81, no. 6, 062343, 2010. [16] F. Furrer, T. Franz, M. Berta, A. Leverrier, V. B. Scholz, M. Tomamichel, and R. F. Werner, “Continuous Variable Quantum Key Distribution: Finite-Key Analysis of Composable Security against Coherent Attacks,” Physical Review Letters, vol. 109, no. 10, 100502, 2012. [17] S. Kish, E. Villase˜nor, R. Malaney, K. Mudge, and K. Grant, “Feasibility Assessment for Practical Continuous Variable Quantum Key Distribution over the Satellite-to-Earth Channel,” Quantum Engineering, vol. 2, no. 3, e50, 2020. [18] M. Tomamichel, C. C. W. Lim, N. Gisin, and R. Renner, “Tight FiniteKey Analysis for Quantum Cryptography,” Nature Communications, vol. 3, no. 1, 1–6, 2012. [19] Z. Yuan, A. Plews, R. Takahashi, K. Doi, W. Tam, A. Sharpe, A. Dixon, E. Lavelle, J. Dynes, A. Murakami, et al., “10-Mb/s Quantum Key Distribution,” Journal of Lightwave Technology, vol. 36, no. 16, 3427– 3433, 2018. [20] S.-K. Liao, W.-Q. Cai, W.-Y. Liu, L. Zhang, Y. Li, J.-G. Ren, J. Yin, Q. Shen, Y. Cao, Z.-P. Li, et al., “Satellite-to-Ground Quantum Key Distribution,” Nature, vol. 549, no. 7670, 43–47, 2017. [21] D. Lin, D. Huang, P. Huang, J. Peng, and G. Zeng, “High Performance Reconciliation for Continuous-Variable Quantum Key Distribution with LDPC Code,” International Journal of Quantum Information, vol. 13, no. 02, 1550010, 2015. [22] X. Wang, Y. Zhang, S. Yu, B. Xu, Z. Li, and H. Guo, “Efficient Rate-Adaptive Reconciliation for Continuous-Variable Quantum Key Distribution,” Quantum Information & Computation, vol. 17, no. 13– 14, 1123–1134, 2017. [23] C. Zhou, X. Wang, Y. Zhang, Z. Zhang, S. Yu, and H. Guo, “ContinuousVariable Quantum Key Distribution with Rateless Reconciliation Protocol,” Physical Review Applied, vol. 12, no. 5, 054013, 2019. [24] M. Milicevic, C. Feng, L. M. Zhang, and P. G. Gulak, “Quasi-Cyclic Multi-Edge LDPC Codes for Long-Distance Quantum Cryptography,” NPJ Quantum Information, vol. 4, no. 1, 1–9, 2018. [25] S.-S. Yang, Z.-G. Lu, and Y.-M. Li, “High-Speed Post-Processing in Continuous-Variable Quantum Key Distribution Based on FPGA Implementation,” Journal of Lightwave Technology, vol. 38, no. 15, 3935–3941, 2020. [26] H. Li and Y. Pang, “FPGA-Accelerated Quantum Computing Emulation and Quantum Key Distillation,” IEEE Micro, vol. 41, no. 4, 49–57, 2021. [27] C. Weedbrook, A. M. Lance, W. P. Bowen, T. Symul, T. C. Ralph, and P. K. Lam, “Quantum Cryptography without Switching,” Physical Review Letters, vol. 93, no. 17, 170504, 2004. [28] N. Hosseinidehaj, A. M. Lance, T. Symul, N. Walk, and T. C. Ralph, “Finite-Size Effects in Continuous-Variable Quantum Key Distribution with Gaussian Postselection,” Physical Review A, vol. 101, no. 5, 052335, 2020. [29] D. Dequal, L. T. Vidarte, V. R. Rodriguez, G. Vallone, P. Villoresi, A. Leverrier, and E. Diamanti, “Feasibility of Satellite-to-Ground Continuous-Variable Quantum Key Distribution,” NPJ Quantum Information, vol. 7, no. 1, 1–10, 2020. [30] M. Bloch, A. Thangaraj, and S. W. McLaughlin, “Efficient Reconciliation of Correlated Continuous Random Variables Using LDPC Codes,” arXiv preprint cs/0509041, 2005. [31] P. Jouguet, D. Elkouss, and S. Kunz-Jacques, “High-Bit-Rate Continuous-Variable Quantum Key Distribution,” Physical Review A, vol. 90, no. 4, 042329, 2014. [32] H. Mani, T. Gehring, P. Grabenweger, B. Omer, C. Pacher, and ¨ U. L. Andersen, “Multiedge-Type Low-Density Parity-Check Codes for Continuous-Variable Quantum Key Distribution,” Physical Review A, vol. 103, no. 6, 062419, 2021. [33] H. Imai and S. Hirakawa, “A New Multilevel Coding Method Using Error-Correcting Codes,” IEEE Transactions on Information Theory, vol. 23, no. 3, 371–377, 1977. [34] U. Wachsmann, R. F. Fischer, and J. B. Huber, “Multilevel codes: Theoretical concepts and practical design rules,” IEEE Transactions on Information Theory, vol. 45, no. 5, 1361–1391, 1999. [35] R. D. Wesel, X. Liu, J. M. Cioffi, and C. Komninakis, “Constellation Labeling for Linear Encoders,” IEEE Transactions on Information Theory, vol. 47, no. 6, 2417–2431, 2001. [36] M. Bloch, A. Thangaraj, S. W. McLaughlin, and J.-M. Merolla, “LDPCBased Gaussian Key Reconciliation,” in IEEE Information Theory Workshop-ITW’06 Punta del Este, 116–120, IEEE, 2006. [37] A. Leverrier, R. All´eaume, J. Boutros, G. Z´emor, and P. Grangier, “Multidimensional Reconciliation for a Continuous-Variable Quantum Key Distribution,” Physical Review A, vol. 77, no. 4, 042325, 2008. [38] J. Fiur´aˇsek and N. J. Cerf, “Gaussian Postselection and Virtual Noiseless Amplification in Continuous-Variable Quantum Key Distribution,” Physical Review A, vol. 86, no. 6, 060302, 2012. [39] F. Darema, “The SPMD Model: Past, Present and Future,” in European Parallel Virtual Machine/Message Passing Interface Users’ Group Meeting, 1–1, Springer, 2001. [40] M. Pharr and W. R. Mark, “A SPMD Compiler for High-Performance CPU Programming,” in Innovative Parallel Computing (InPar), 1–13, IEEE, 2012. [41] J. Lodewyck, M. Bloch, R. Garc´ıa-Patr ´on, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N. J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, et al., “Quantum Key Distribution over 25 km with an AllFiber Continuous-Variable System,” Physical Review A, vol. 76, no. 4, 042305, 2007. [42] F. Laudenbach, C. Pacher, C.-H. F. Fung, A. Poppe, M. Peev, B. Schrenk, M. Hentschel, P. Walther, and H. H ¨ubel, “Continuous-Variable Quantum Key Distribution with Gaussian Modulation—The Theory of Practical Implementations,” Advanced Quantum Technologies, vol. 1, no. 1, 1800011, 2018. [43] Y. Polyanskiy, H. V. Poor, and S. Verd ´u, “Channel Coding Rate in the Finite Blocklength Regime,” IEEE Transactions on Information Theory, vol. 56, no. 5, 2307–2359, 2010. [44] D. Slepian and J. Wolf, “Noiseless Coding of Correlated Information Sources,” IEEE Transactions on Information Theory, vol. 19, no. 4, 471–480, 1973. [45] T. Richardson and R. Urbanke, Modern Coding Theory. Cambridge University Press, 2008. [46] X. Ai, R. Malaney, and S. X. Ng, “A Reconciliation Strategy for RealTime Satellite-Based QKD,” IEEE Communications Letters, vol. 24, no. 5, 1062–1066, 2020. [47] S.-Y. Chung, T. J. Richardson, and R. L. Urbanke, “Analysis of SumProduct Decoding of Low-Density Parity-Check Codes Using a Gaussian Approximation,” IEEE Transactions on Information Theory, vol. 47, no. 2, 657–670, 2001. [48] V. A. Chandrasetty and S. M. Aziz, “FPGA Implementation of an LDPC Decoder Using a Reduced Complexity Message Passing Algorithm,” Journal of Networks, vol. 6, no. 1, 36, 2011. [49] A. Leverrier, “Composable Security Proof for Continuous-Variable Quantum Key Distribution with Coherent States,” Physical Review Letters, vol. 114, no. 7, 070501, 2015. [50] C. Lupo, C. Ottaviani, P. Papanastasiou, and S. Pirandola, “ContinuousVariable Measurement-Device-Independent Quantum Key Distribution: Composable Security against Coherent Attacks,” Physical Review A, vol. 97, no. 5, 052327, 2018. [51] N. Hosseinidehaj, N. Walk, and T. C. Ralph, “Optimal Realistic Attacks in Continuous-Variable Quantum Key Distribution,” Physical Review A, vol. 99, no. 5, 052336, 2019. [52] S. Pirandola, “Limits and Security of Free-Space Quantum Communications,” Physical Review Research, vol. 3, no. 1, 013279, 2021. [53] S. Pirandola, “Satellite Quantum Communications: Fundamental Bounds and Practical Security,” Physical Review Research, vol. 3, no. 2, 023130, 2021. [54] J. H. Mathews and K. D. Fink, Numerical Methods Using MATLAB, vol. 4. Pearson Prentice Hall Upper Saddle River, NJ, 2004. [55] A. Leverrier, “Security of Continuous-Variable Quantum Key Distribution via a Gaussian de Finetti Reduction,” Physical Review Letters, vol. 118, no. 20, 200501, 2017. [56] E. Villase˜nor, R. Malaney, K. A. Mudge, and K. J. Grant, “Atmospheric Effects on Satellite-to-Ground Quantum Key Distribution Using Coherent States,” in GLOBECOM 2020-2020 IEEE Global Communications Conference, 1–6, IEEE, 2020. [57] J. Mateo, Efficient Information Reconciliation for Quantum Key Distribution. PhD thesis, Universidad Politecnica de Madrid, 2011. [58] T. Tian, C. R. Jones, J. D. Villasenor, and R. D. Wesel, “Selective Avoidance of Cycles in Irregular LDPC Code Construction,” IEEE Transactions on Communications, vol. 52, no. 8, 1242–1247, 2004. [59] D. V. Nguyen, S. K. Chilappagari, M. W. Marcellin, and B. Vasic, “On the Construction of Structured LDPC Codes Free of Small Trapping Sets,” IEEE Transactions on Information Theory, vol. 58, no. 4, 2280– 2302, 2012. [60] S. Landner and O. Milenkovic, “Algorithmic and Combinatorial Analysis of Trapping Sets in Structured LDPC Codes,” in International Conference on Wireless Networks, Communications and Mobile Computing, vol. 1, 630–635, IEEE, 2005. [61] X. Wang, Y. Zhang, S. Yu, and H. Guo, “High Speed Error Correction for Continuous-Variable Quantum Key Distribution with Multi-Edge Type LDPC Code,” Scientific Reports, vol. 8, no. 1, 1–7, 2018. [62] Y. Li, X. Zhang, Y. Li, B. Xu, L. Ma, J. Yang, and W. Huang, “HighThroughput GPU Layered Decoder of Quasi-Cyclic Multi-Edge Type Low Density Parity Check Codes in Continuous-Variable Quantum Key Distribution Systems,” Scientific Reports, vol. 10, no. 1, 1–11, 2020. [63] F. Grosshans, “Collective Attacks and Unconditional Security in Continuous Variable Quantum Key Distribution,” Physical Review Letters, vol. 94, 020504, 2005. [64] M. Navascu´es, F. Grosshans, and A. Acin, “Optimality of Gaussian Attacks in Continuous-Variable Quantum Cryptography,” Physical Review Letters, vol. 97, no. 19, 190502, 2006. [65] S. Fossier, E. Diamanti, T. Debuisschert, R. Tualle-Brouri, and P. Grangier, “Improvement of Continuous-Variable Quantum Key Distribution Systems by Using Optical Preamplifiers,” Journal of Physics B: Atomic, Molecular and Optical Physics, vol. 42, no. 11, 114014, 2009. [66] P. Jouguet, S. Kunz-Jacques, E. Diamanti, and A. Leverrier, “Analysis of Imperfections in Practical Continuous-Variable Quantum Key Distribution,” Physical Review A, vol. 86, no. 3, 032309, 2012.

Optimised Multithreaded CV-QKD Reconciliation for Global Quantum Networks: Conclusion & References

This paper is available on arxiv under CC 4.0 license. Authors: (1) Amit Kumar Chatterjee, Yukawa Institute for Theoretical Physics, Kyoto University & Department of Physics, Ramakrishna Mission Vidyamandira; (2) Satoshi Takada, Department of Mechanical Systems Engineering and Institute of Engineering; (3) Hisao Hayakawa, Yukawa Institute for Theoretical Physics, Kyoto University. Table of Links Introduction Model and Protocol Different Regimes in d˜− Γ˜ Plain and Multiple QMPE QMPE in Region (d): Second-order Ep QMPE in Region (a1): Oscillations Summary Appendix A: Eigenvectors of L in region (d) Appendix B: Eigenvectors of L in region (a1) Appendix C: QMPE in region (b): purely exponential relaxation Appendix D: QMPE in region (e): 3rd order EP References References [1] N. C. Keim, J. D. Paulsen, Z. Zeravcic, S. Sastry and S. R. Nagel, Rev. Mod. Phys. 91, 035002 (2019).. [2] J. Gill, Solid State Commun. 39, 1203 (1981). [3] R. M. Fleming and L. F. Schneemeyer, Phys. Rev. B 28, 6996 (1983). [4] J. Kaiser, An investigation into the occurrence of noises in tensile tests or a study of acoustic phenomena, Ph.D. thesis (PhD thesis, Technical University, Munich, Germany) (1950). [5] A. Zang, B. Oye, P. Jousset, N. Deichmann, R. Gritto, A. McGarr, E. Majer and D, Bruhn, Geothermics 52, 6 (2014). [6] E. R. Heimisson, P. Einarsson, F. Sigmundsson and B. Brandsd´ottir, Geophys. Res. Lett., 42, 7958 (2015). [7] L. Mullins, Rubber Chem. Technol. 42, 339 (1969). [8] A. Dorfmann and R. W. Ogden, Int. J. Solids Struct. 41, 1855 (2004). [9] J. Diani, B. Fayolle, and P. Gilormini , European Polymer Journal 45, 601 (2009). [10] A. J. Kovacs, Adv. Polym. Sci 3, 394 (1963). [11] A. J. Kovacs, J. J. Aklonis, J. M. Hutchinson and A. R. Ramos, J. Polym. Sci. B 17, 1097 (1979). [12] E. M. Bertin, J. P. Bouchaud, J. M. Drouffe, and C. Godr`eche, J. Phys. A: Math. Gen. 36, 10701 (2003). [13] A. Prados and E. Trizac, Phys. Rev. Lett. 112, 198001 (2014). [14] A. Militaru, A. Lasanta, M. Frimmer, L. L. Bonilla, L. Novotny, and R. A. Rica, Phys. Rev. Lett. 127, 130603 (2021). [15] K. Jonason, E. Vincent, J. Hammann, J. P. Bouchaud and P. Nordblad, Phys. Rev. Lett. 81, 3243 (1998). [16] G. F. Rodriguez, G. G. Kenning and R. Orbach, Phys. Rev. Lett. 91, 037203 (2003). [17] M. Baity Jesi, Spin Glasses: Criticality and Energy Landscapes(Springer, 2018). [18] Y. Zheng and G. B. McKenna, Macromolecules 36, 2387 (2003). [19] X. Di, K. Z. Win, G. B. McKenna, T. Narita, F. Lequeux, S. R. Pullela and Z. Cheng, Phys. Rev. Lett. 106, 095701 (2011). [20] G. B. McKenna and S. L. Simon, Macromolecules 50, 6833 (2017). [21] C. Scalliet and L. Berthier, Phys. Rev. Lett. 122, 255502 (2019). [22] C. Josserand, A.V. Tkachenko, D.M. Mueth and H.M. Jaeger, Phys. Rev. Lett. 85, 3632 (2000). [23] N. Kumar and S. Luding, Granul. Matter 18, 58 (2016). [24] M. Otsuki and H. Hayakawa, Phys. Rev. E 101, 032905 (2020). [25] T. Kawasaki and K. Miyazaki, arXiv:2003.10716. [26] Pradipto and H. Hayakawa, Soft Matter 16, 945 (2020). [27] L. M. C. Janssen, A. Kaiser and H. L¨owen, Sci Rep. 7, 5667 (2017). [28] Aristotle and W. D. Ross, Aristotle’s Metaphysics (Clarendon, Oxford, UK, 1981). [29] E. B. Mpemba and D. G. Osborne, Phys. Educ. 4, 172 (1969). [30] A. Kumar and J. Bechhoefer, Nature 584, 64 (2020). [31] A. Kumar, R. Che´etrite and J. Bechhoefer, Proc. Natl. Acad. Sci. U.S.A. 119, e2118484119 (2022). [32] A. Lasanta, F. Vega Reyes, A. Prados, and A. Santos, Phys. Rev. Lett. 119, 148001 (2017). [33] A. Torrente, M. A. L´opez-Casta˜no, A. Lasanta, F. Vega Reyes, A. Prados, and A. Santos, Phys. Rev. E 99, 060901(R) (2019). [34] A. Biswas, V. V. Prasad, O. Raz, and R. Rajesh, Phys. Rev. E 102, 012906 (2020). [35] A. Biswas, V. V. Prasad, and R. Rajesh, EPL 136, 46001 (2021). [36] E. Momp´o, M. A. L´opez Casta˜no, A. Torrente, F. Vega Reyes, A. Lasanta, Phys. Fluids 33, 062005 (2021). [37] A. Meg´ıas and A. Santos, Front. Phys. 10, 971671 (2022). [38] A. Patr´on, B. S´anchez-Rey, C. A. Plata and A. Prados, EPL 143, 61002 (2023). [39] T. Keller, V. Torggler, S. B. J¨ager, S. Sch¨utz, H. Ritsch, and G. Morigi, New J. Phys. 20, 025004 (2018). [40] A. Santos and A. Prados, Phys. Fluids 32, 072010 (2020). [41] A. Patr´on, B. S´anchez-Rey and A. Prados, Phys. Rev. E 104, 064127 (2021). [42] S. Takada, H. Hayakawa and A. Santos, Phys. Rev. E 103, 032901 (2021). [43] S. Takada, EPJ Web. Conf. 249, 04001 (2021). [44] Z. Lu and O. Raz, Proc. Natl. Acad. Sci. U.S.A. 114, 5083 (2017). [45] I. Klich, O. Raz, O. Hirschberg, and M. Vucelja, Phys. Rev. X 9, 021060 (2019). [46] D. M. Busiello, D. Gupta, amd A. Maritan, New J. Phys. 23, 103012 (2021). [47] J. Lin, K. Li, J. He, J. Ren and J. Wang, Phys. Rev. E 105, 014104 (2022). [48] A. Biswas, R. Rajesh and A. Pal, J. Chem. Phys. 159, 044120 (2023). [49] R. G. Gonz´alez, N. Khalil and V. Garz´o, Phys. Fluids 33, 053301 (2021).. [50] M. Baity-Jesi, E. Calore, A. Cruz, L. A. Fernandez, J. M. Gil-Narvi´on, A. Gordillo-Guerrero, D. I˜niguez, A. Lasanta, A. Maiorano, E. Marinari, V. Martin-Mayor, J. Moreno-Gordo, A. Mu˜noz Sudupe, D. Navarro, G. Parisi, S. Perez-Gaviro, F. Ricci-Tersenghi, J. J. RuizLorenzo, S. F. Schifano, B. Seoane, A. Taranc´on, R. Tripiccione, and D. Yllanes, Proc. Natl. Acad. Sci. U.S.A. 116, 15350 (2019). [51] P. A. Greaney, G. Lani, G. Cicero, and J. C. Grossman, Metall. Mater. Trans. A 42, 3907 (2011). [52] Z.-Y. Yang and J.-X. Hou, Phys. Rev. E 101, 052106 (2020). [53] Z.-Y. Yang and J.-X. Hou, Phys. Rev. E 105, 014119 (2022). [54] I. Gonz´alez-Adalid Pemrt´ın, E. Momp´o, A. Lasanta, V. Mart´ın-Mayor and J. Salas, Phys. Rev. E 104, 044114 (2021). [55] R. Che´etrite, A. Kumar and J. Bechhoefer, Front. Phys. 9, 654271 (2021). [56] R. Holtzman and O. Raz, Communications Physics 5, 280 (2022). [57] A. Meg´ıas, A. Santos, and A. Prados, Phys. Rev. E 105, 054140 (2022). [58] A. Biswas, V. V. Prasad, and R. Rajesh, arXiv:2303.10900 (2023). [59] I. G. Pemartin, E. Momp´o, A. Lasanta, V. MartinMayor and J. Salas, arXiv.2308.04094. [60] A. Nava and M. Fabrizio, Phys. Rev. B. 100, 125102 (2019). [61] F. Carollo, A. Lasanta, and I. Lesanovsky, Phys. Rev. Lett. 127, 060401 (2021). [62] S. K.Manikandan, Phys. Rev. Res. 3, 043108 (2021). [63] F. Ivander, N. Anto-Sztrikacs, and D. Segal, arXiv:2301.06135 (2023). [64] F. Ares, S. Murciano, and P. Calabrese, Nat. Commun. 14, 2036 (2023). [65] A. K. Chatterjee, S. Takada and H. Hayakawa, Phys. Rev. Lett. 131, 080402 (2023). [66] S. Murciano, F. Ares, I. Klich and P. Calabrese, arXiv:2310.07513. [67] C. Rylands, K. Klobas, F. Ares, P. Calabrese, S. Murciano and B. Bertini, arXiv:2310.04419. [68] W. D. Heiss, J. Phys. A: Math. Theor. 45, 444016 (2012). [69] K. Hashimoto, K. Kanki, H. Hayakawa and T. Petrosky, Prog. Theor. Exp. Phys. 2015, 023A02 (2015). [70] J. Doppler, A. Mailybaev, J. B¨ohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, M. Moyseyev, and S. Rotter, Nature 537, 76–79 (2016). [71] Q. Zhong, J. Ren, M. Khajavikhan, D. N. Christodoulides, S. K. Ozdemir, and R. El-Ganainy, ¨ Phys. Rev. Lett. 122, 153902 (2019). [72] A. McDonald and A. A. Clerk, Nat. Commun. 11, 5382 (2020). [73] S. Soleymani, Q. Zhong, M. Mokim, S. Rotter, R. ElGanainy, and S. K. Ozdemir, ¨ Nat. Commun. 13, 599 (2022). [74] K. Ding, C. Fang, and G. Ma, Nat. Rev. Phys. 4, 745 (2022). [75] F. Minganti, A. Miranowicz, R. W. Chhajlany, and F. Nori, Phys. Rev. A 100, 062131 (2019). [76] I. I. Arkhipov, A. Miranowicz, F. Minganti and F. Nori, Phys. Rev. A 101, 013812 (2020). [77] I. I. Arkhipov, F. Minganti, A. Miranowicz and F. Nori, Phys. Rev. A 104, 012205 (2021). [78] M. M¨uller and I. rotter, J. Phys. A: Math. Theor. 41, 244018 (2008). [79] M. Naghiloo, M. Abbasi, Y. N. Joglekar and K. W. Murch, Nature Phys. 15, 1232 (2019). [80] M. Saha, B. K. Agarwalla, M. Kulkarni and A. Purkayastha, Phys. Rev. Lett. 130, 187101 (2023). [81] F. Roccati, A. Purkayastha, G. Massimo Palma and F. Ciccarello, Eur. Phys. J. Spec. Top. 232, 1783 (2023). [82] M. A. Miri and A. Al`u, Science 363, eaar7709 (2019). [83] S. K. Ozdemir, S. Rotter, F. Nori, and L. Yang ¨ Nat. Mater. 18, 783–798 (2019). [84] Yan-Li Zhou, Xiao-Die Yu, Chun-Wang Wu, Xie-Qian Li, Jie Zhang, Weibin Li, and Ping-Xing Chen, Phys. Rev. Res. 5, 043036 (2023). [85] A. Lapolla and A. Godec, Phys. Rev. Lett. 125, 110602 (2020). [86] T. Van Vu and Y. Hasegawa, Phys. Rev. Res. 3, 043160 (2021). [87] S. Kochsiek, F. Carollo and I. Lesanovsky, Phys. Rev. A 106, 012207 (2022). [88] F. Ghimenti and F. van Wijland, Phys. Rev. E 105, 054137 (2022). [89] R. Bao and Z. Hou, arXiv:2212.11170. [90] C. M. Bender, Rep. Prog. Phys. 70, 947 (2007). [91] N. Moiseyev, Non-Hermitian Quantum Mechanics. Cambridge University Press; Cambridge, UK: 2011. [92] R. Nakano, N. Hatano and T. Petrosky, Int. J. Theor. Phys. 50, 1134 (2011). [93] N. Hatano and T. Petrosky, AIP Conf. Proc. 1648 200005 (2015). [94] X. Li, Y. Li, and J. Jin, Phys. Rev. A 107, 032219 (2023). [95] N. Hatano, Mol. Phys. 117, 2121 (2019). [96] S. Kullback and R. A. Leibler, Ann. Math. Statist. 22, 79 (1951). [97] P. Strasberg, Quantum Stochastic Thermodynamics: Foundations and Selected Applications (Oxford University Press, Oxford, 2022). [98] T. Sagawa, Entropy, Divergence and Majorization in Classical and Quantum Thermodynamics (Springer Berlin, 2022). [99] N. Shiraishi, An Introduction to Stochastic Thermodynamics: From Basic to Advanced (Springer, 2023). [100] R. Yoshii and H. Hayakawa, Phys. Rev. Res. 5, 033014 (2023). [101] G. Lindblad, Commun. Math. Phys. 48, 119 (1976). [102] D. Manzano, AIP Advances 10, 025106 (2020) . [103] V. Gorini, A. Kossakowski and E. C. G. Sudarshan, J. Math. Phys. 17, 821 (1976). [104] D. Chruscinski and S. Pascazio, Open Sys. Inf. Dyn. 24, 1740001 (2017). [105] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, Adv. Comput. Math. 5, 329 (1996). [106] A. A. More, Chem. Eng. Sci. 61, 5515 (2006). [107] M. Asadian, H. Saeedi, M. Yadegari and M. Shojaee, J. Cryst. Growth 396, 61 (2014). [108] A. Braun, A. Wokaun and H. G. Hermanns, Appl. Math. Model. 27, 47 (2003). [109] A. Bot, B. Dewi and P. Venema, ACS Omega. 6, 7862 (2021). [110] J. Moran, M. Romeijnders, P. Le Doussal, F. P. Pijpers, U. Weitzel, D. Panja and J-P. Bouchaud, arXiv:2309.15070. [111] Shun-ichi Amari, Information Geometry and Its Applications, Applied Mathematical Sciences. Vol. 194. (Springer Japan, 2016). [112] K. P. Burnham and D. R. Anderson, Model Selection and Multi-Model Inference, (Springer, 2002). [113] A. N. Soklakov, Entropy 22, 860 (2020). [114] L. D. Landau, and E. M. Lifshitz, Statistical Physics. Part 1, (Pergamon Press, Oxford, 449, 1980). [115] G. Gauthier, M. T. Reeves, X. Yu, A. S. Bradley, M. A. Baker, T. A. Bell, H. Rubinsztein-Dunlop, M. J. Davis, and T. W. Neely, Science. 364, 1264 (2019). [116] S. P. Johnstone, A. J. Groszek, P. T. Starkey, C. J. Billinton, T. P. Simula, and K. Helmerson, Science. 365, 1267 (2019). [117] P. Medley, D. M. Weld, H. Miyake, D. E. Pritchard, and W. Ketterle, Phys. Rev. Lett. 106, 195301 (2011). [118] M. M. Ali, W-M. Huang, and W-M. Zhang, Sci. Rep. 10, 13500 (2020). [119] A. Gyenis, P. S. Mundada, A. Di Paolo, T. M. Hazard, X. You, D. I. Schuster, J. Koch, A. Blais and A. A. Houck, PRX Quantum 2, 010339 (2021).

Multiple Quantum Mpemba Effect: Exceptional Points and Oscillations: References

Why the Facebook/Cambridge Analytica Data Scandal is Awesome News

This paper is available on arxiv under CC 4.0 license.
Authors:
(1) Sam Bowyer, Equal contribution, Department of Mathematics and sam.bowyer@bristol.ac.uk;
(2)  Thomas Heap, Equal contribution, Department of Computer Science University of Bristol and thomas.heap@bristol.ac.uk;
(3) Laurence Aitchison, Department of Computer Science University of Bristol and laurence.aitchison@bristol.ac.uk. Table of Links Abstract & Introduction
Related work
Background
Methods
Experiments
"Conclusion, Limitations, and References"
Derivations
Algorithms
Experimental Datasets And Model Conclusion and Limitations We gave a new and far simpler method for computing posterior moments, marginals and samples in massively parallel importance sampling based on differentiating a slightly modified marginal likelihood estimator. The method has limitations, in that while it is considerably more effective than e.g. VI, HMC and global importance sampling, it is more complex. Additionally, at least a naive implementation may be quite costly in terms of memory consumption, limiting how the number of importance samples we can draw for each variable. That said, it should be possible to eliminate almost all of this overhead by careful optimizations to avoid allocating large intermediate tensors, following the strategy in KeOps (Charlier et al. 2021). References Aitchison, L. 2019. Tensor Monte Carlo: particle methods for the GPU era. Advances in Neural Information Processing Systems, 32. Albarghouthi, A.; D’Antoni, L.; Drews, S.; and Nori, A. V. 2017. Fairsquare: probabilistic verification of program fairness. Proceedings of the ACM on Programming Languages, 1(OOPSLA): 1–30. Andrieu, C.; Doucet, A.; and Holenstein, R. 2010. Particle markov chain monte carlo methods. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 72(3): 269–342. Bidyuk, B.; and Dechter, R. 2007. Cutset sampling for Bayesian networks. Journal of Artificial Intelligence Research, 28: 1–48. Bornschein, J.; and Bengio, Y. 2014. Reweighted wakesleep. arXiv preprint arXiv:1406.2751. Bradbury, J.; Frostig, R.; Hawkins, P.; Johnson, M. J.; Leary, C.; Maclaurin, D.; Necula, G.; Paszke, A.; VanderPlas, J.; Wanderman-Milne, S.; and Zhang, Q. 2018. JAX: composable transformations of Python+NumPy programs. Burda, Y.; Grosse, R.; and Salakhutdinov, R. 2015. Importance weighted autoencoders. arXiv preprint arXiv:1509.00519. Charlier, B.; Feydy, J.; Glaunes, J. A.; Collin, F.-D.; and Du- ` rif, G. 2021. Kernel Operations on the GPU, with Autodiff,  without Memory Overflows. Journal of Machine Learning Research. Chatterjee, S.; and Diaconis, P. 2018. The sample size required in importance sampling. The Annals of Applied Probability, 28(2): 1099–1135. Claret, G.; Rajamani, S. K.; Nori, A. V.; Gordon, A. D.; and Borgstrom, J. 2013. Bayesian inference using data flow anal- ¨ ysis. In Proceedings of the 2013 9th joint meeting on foundations of software engineering, 92–102. Corenflos, A.; Chopin, N.; and Sarkk ¨ a, S. 2022. De- ¨ Sequentialized Monte Carlo: a parallel-in time particle smoother. arXiv preprint arXiv:2202.02264. Crucinio, F. R.; and Johansen, A. M. 2023. Properties of Marginal Sequential Monte Carlo Methods. ArXiv:2303.03498 [math, stat]. Dawid, A. P. 1992. Applications of a general propagation algorithm for probabilistic expert systems. Statistics and computing, 2(1): 25–36. Dechter, R.; Broka, F.; Kask, K.; and Ihler, A. 2018. Abstraction Sampling in Graphical Models. AAAI. Del Moral, P.; Doucet, A.; and Jasra, A. 2006. Sequential Monte Carlo Samplers. Journal of the Royal Statistical Society Series B: Statistical Methodology, 68(3): 411–436. DOE. 2023. Bus Breakdown and Delays. url https://data.cityofnewyork.us/Transportation/BusBreakdown-and-Delays/ez4e-fazm Accessed on: 05.05.2023. Doucet, A.; Johansen, A. M.; et al. 2009. A tutorial on particle filtering and smoothing: Fifteen years later. Handbook of nonlinear filtering, 12(656-704): 3. Geffner, T.; and Domke, J. 2022. Variational Inference with Locally Enhanced Bounds for Hierarchical Models. arXiv preprint arXiv:2203.04432. Gehr, T.; Misailovic, S.; and Vechev, M. 2016. PSI: Exact symbolic inference for probabilistic programs. In International Conference on Computer Aided Verification, 62–83. Springer. Geldenhuys, J.; Dwyer, M. B.; and Visser, W. 2012. Probabilistic symbolic execution. In Proceedings of the 2012 International Symposium on Software Testing and Analysis, 166–176. Gogate, V.; and Dechter, R. 2012. Importance samplingbased estimation over and/or search spaces for graphical models. Artificial Intelligence, 184: 38–77. Goodman, N. D.; and Stuhlmuller, A. 2014. The design and implementation of probabilistic programming languages. Gordon, N. J.; Salmond, D. J.; and Smith, A. F. 1993. Novel approach to nonlinear/non-Gaussian Bayesian state estimation. In IEE proceedings F (radar and signal processing), volume 140, 107–113. IET. Harper, F. M.; and Konstan, J. A. 2015. The movielens datasets: History and context. Acm transactions on interactive intelligent systems (tiis), 5(4): 1–19. Heap, T.; and Laurence, A. 2023. Massively parallel reweighted wake-sleep. UAI. Hoffman, M. D.; Gelman, A.; et al. 2014. The No-UTurn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo. J. Mach. Learn. Res., 15(1): 1593–1623. Holtzen, S.; Van den Broeck, G.; and Millstein, T. 2020. Scaling exact inference for discrete probabilistic programs. Proceedings of the ACM on Programming Languages, 4(OOPSLA): 1–31. Jordan, M. I.; Ghahramani, Z.; Jaakkola, T. S.; and Saul, L. K. 1999. An introduction to variational methods for graphical models. Machine learning, 37(2): 183–233. Kingma, D. P.; Welling, M.; et al. 2019. An introduction to variational autoencoders. Foundations and Trends® in Machine Learning, 12(4): 307–392. Kuntz, J.; Crucinio, F. R.; and Johansen, A. M. 2023. The divide-and-conquer sequential Monte Carlo algorithm: theoretical properties and limit theorems. ArXiv:2110.15782 [math, stat]. Lai, J.; Domke, J.; and Sheldon, D. 2022. Variational Marginal Particle Filters. In International Conference on Artificial Intelligence and Statistics, 875–895. PMLR. Le, T. A.; Igl, M.; Rainforth, T.; Jin, T.; and Wood, F. 2017. Auto-encoding sequential monte carlo. arXiv preprint arXiv:1705.10306. Le, T. A.; Kosiorek, A. R.; Siddharth, N.; Teh, Y. W.; and Wood, F. 2020. Revisiting reweighted wake sleep for models with stochastic control flow. In Uncertainty in Artificial Intelligence, 1039–1049. PMLR. Lindsten, F.; Johansen, A. M.; Naesseth, C. A.; Kirkpatrick, B.; Schon, T. B.; Aston, J.; and Bouchard-Cotˆ e, A. 2017. ´ Divide-and-conquer with sequential Monte Carlo. Journal of Computational and Graphical Statistics, 26(2): 445–458. Maddison, C. J.; Lawson, J.; Tucker, G.; Heess, N.; Norouzi, M.; Mnih, A.; Doucet, A.; and Teh, Y. 2017. Filtering variational objectives. Advances in Neural Information Processing Systems, 30. Naesseth, C.; Linderman, S.; Ranganath, R.; and Blei, D. 2018. Variational sequential monte carlo. In International conference on artificial intelligence and statistics, 968–977. PMLR. Narayanan, P.; Carette, J.; Romano, W.; Shan, C.-c.; and Zinkov, R. 2016. Probabilistic inference by program transformation in Hakaru (system description). In International Symposium on Functional and Logic Programming, 62–79. Springer Obermeyer, F.; Bingham, E.; Jankowiak, M.; Pradhan, N.; Chiu, J.; Rush, A.; and Goodman, N. 2019. Tensor variable elimination for plated factor graphs. In International Conference on Machine Learning, 4871–4880. PMLR. Pfeffer, A. 2005. The design and implementation of IBAL: A generalpurpose probabilistic programming language. In Harvard Univesity. Citeseer. Salvatier, J.; Wiecki, T. V.; and Fonnesbeck, C. 2016. Probabilistic programming in Python using PyMC3. PeerJ Computer Science, 2: e55. Publisher: PeerJ Inc. Sankaranarayanan, S.; Chakarov, A.; and Gulwani, S. 2013. Static analysis for probabilistic programs: inferring whole program properties from finitely many paths. In Proceedings of the 34th ACM SIGPLAN conference on Programming language design and implementation, 447–458. Wang, D.; Hoffmann, J.; and Reps, T. 2018. PMAF: an algebraic framework for static analysis of probabilistic programs. ACM SIGPLAN Notices, 53(4): 513–528. Weinberg, S. 1995. The quantum theory of fields, volume 2. Cambridge university press. Zavatone-Veth, J.; Canatar, A.; Ruben, B.; and Pehlevan, C. 2021. Asymptotics of representation learning in finite Bayesian neural networks. Advances in neural information processing systems, 34: 24765–24777. This paper is available on arxiv under CC 4.0 license. Authors: (1) Sam Bowyer, Equal contribution, Department of Mathematics and sam.bowyer@bristol.ac.uk; (2)  Thomas Heap, Equal contribution, Department of Computer Science University of Bristol and thomas.heap@bristol.ac.uk; (3) Laurence Aitchison, Department of Computer Science University of Bristol and laurence.aitchison@bristol.ac.uk. This paper is available on arxiv under CC 4.0 license. Authors: Authors: (1) Sam Bowyer, Equal contribution, Department of Mathematics and sam.bowyer@bristol.ac.uk; (2)  Thomas Heap, Equal contribution, Department of Computer Science University of Bristol and thomas.heap@bristol.ac.uk; (3) Laurence Aitchison, Department of Computer Science University of Bristol and laurence.aitchison@bristol.ac.uk. Table of Links Abstract & Introduction Related work Background Methods Experiments "Conclusion, Limitations, and References" Derivations Algorithms Experimental Datasets And Model Abstract & Introduction Abstract & Introduction Related work Related work Background Background Methods Methods Experiments Experiments "Conclusion, Limitations, and References" "Conclusion, Limitations, and References" Derivations Derivations Algorithms Algorithms Experimental Datasets And Model Experimental Datasets And Model Conclusion and Limitations We gave a new and far simpler method for computing posterior moments, marginals and samples in massively parallel importance sampling based on differentiating a slightly modified marginal likelihood estimator. The method has limitations, in that while it is considerably more effective than e.g. VI, HMC and global importance sampling, it is more complex. Additionally, at least a naive implementation may be quite costly in terms of memory consumption, limiting how the number of importance samples we can draw for each variable. That said, it should be possible to eliminate almost all of this overhead by careful optimizations to avoid allocating large intermediate tensors, following the strategy in KeOps (Charlier et al. 2021). References Aitchison, L. 2019. Tensor Monte Carlo: particle methods for the GPU era. Advances in Neural Information Processing Systems, 32. Albarghouthi, A.; D’Antoni, L.; Drews, S.; and Nori, A. V. 2017. Fairsquare: probabilistic verification of program fairness. Proceedings of the ACM on Programming Languages, 1(OOPSLA): 1–30. Andrieu, C.; Doucet, A.; and Holenstein, R. 2010. Particle markov chain monte carlo methods. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 72(3): 269–342. Bidyuk, B.; and Dechter, R. 2007. Cutset sampling for Bayesian networks. Journal of Artificial Intelligence Research, 28: 1–48. Bornschein, J.; and Bengio, Y. 2014. Reweighted wakesleep. arXiv preprint arXiv:1406.2751. Bradbury, J.; Frostig, R.; Hawkins, P.; Johnson, M. J.; Leary, C.; Maclaurin, D.; Necula, G.; Paszke, A.; VanderPlas, J.; Wanderman-Milne, S.; and Zhang, Q. 2018. JAX: composable transformations of Python+NumPy programs. Burda, Y.; Grosse, R.; and Salakhutdinov, R. 2015. Importance weighted autoencoders. arXiv preprint arXiv:1509.00519. Charlier, B.; Feydy, J.; Glaunes, J. A.; Collin, F.-D.; and Du- ` rif, G. 2021. Kernel Operations on the GPU, with Autodiff,  without Memory Overflows. Journal of Machine Learning Research. Chatterjee, S.; and Diaconis, P. 2018. The sample size required in importance sampling. The Annals of Applied Probability, 28(2): 1099–1135. Claret, G.; Rajamani, S. K.; Nori, A. V.; Gordon, A. D.; and Borgstrom, J. 2013. Bayesian inference using data flow anal- ¨ ysis. In Proceedings of the 2013 9th joint meeting on foundations of software engineering, 92–102. Corenflos, A.; Chopin, N.; and Sarkk ¨ a, S. 2022. De- ¨ Sequentialized Monte Carlo: a parallel-in time particle smoother. arXiv preprint arXiv:2202.02264. Crucinio, F. R.; and Johansen, A. M. 2023. Properties of Marginal Sequential Monte Carlo Methods. ArXiv:2303.03498 [math, stat]. Dawid, A. P. 1992. Applications of a general propagation algorithm for probabilistic expert systems. Statistics and computing, 2(1): 25–36. Dechter, R.; Broka, F.; Kask, K.; and Ihler, A. 2018. Abstraction Sampling in Graphical Models. AAAI. Del Moral, P.; Doucet, A.; and Jasra, A. 2006. Sequential Monte Carlo Samplers. Journal of the Royal Statistical Society Series B: Statistical Methodology, 68(3): 411–436. DOE. 2023. Bus Breakdown and Delays. url https://data.cityofnewyork.us/Transportation/BusBreakdown-and-Delays/ez4e-fazm Accessed on: 05.05.2023. Doucet, A.; Johansen, A. M.; et al. 2009. A tutorial on particle filtering and smoothing: Fifteen years later. Handbook of nonlinear filtering, 12(656-704): 3. Geffner, T.; and Domke, J. 2022. Variational Inference with Locally Enhanced Bounds for Hierarchical Models. arXiv preprint arXiv:2203.04432. Gehr, T.; Misailovic, S.; and Vechev, M. 2016. PSI: Exact symbolic inference for probabilistic programs. In International Conference on Computer Aided Verification, 62–83. Springer. Geldenhuys, J.; Dwyer, M. B.; and Visser, W. 2012. Probabilistic symbolic execution. In Proceedings of the 2012 International Symposium on Software Testing and Analysis, 166–176. Gogate, V.; and Dechter, R. 2012. Importance samplingbased estimation over and/or search spaces for graphical models. Artificial Intelligence, 184: 38–77. Goodman, N. D.; and Stuhlmuller, A. 2014. The design and implementation of probabilistic programming languages. Gordon, N. J.; Salmond, D. J.; and Smith, A. F. 1993. Novel approach to nonlinear/non-Gaussian Bayesian state estimation. In IEE proceedings F (radar and signal processing), volume 140, 107–113. IET. Harper, F. M.; and Konstan, J. A. 2015. The movielens datasets: History and context. Acm transactions on interactive intelligent systems (tiis), 5(4): 1–19. Heap, T.; and Laurence, A. 2023. Massively parallel reweighted wake-sleep. UAI. Hoffman, M. D.; Gelman, A.; et al. 2014. The No-UTurn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo. J. Mach. Learn. Res., 15(1): 1593–1623. Holtzen, S.; Van den Broeck, G.; and Millstein, T. 2020. Scaling exact inference for discrete probabilistic programs. Proceedings of the ACM on Programming Languages, 4(OOPSLA): 1–31. Jordan, M. I.; Ghahramani, Z.; Jaakkola, T. S.; and Saul, L. K. 1999. An introduction to variational methods for graphical models. Machine learning, 37(2): 183–233. Kingma, D. P.; Welling, M.; et al. 2019. An introduction to variational autoencoders. Foundations and Trends® in Machine Learning, 12(4): 307–392. Kuntz, J.; Crucinio, F. R.; and Johansen, A. M. 2023. The divide-and-conquer sequential Monte Carlo algorithm: theoretical properties and limit theorems. ArXiv:2110.15782 [math, stat]. Lai, J.; Domke, J.; and Sheldon, D. 2022. Variational Marginal Particle Filters. In International Conference on Artificial Intelligence and Statistics, 875–895. PMLR. Le, T. A.; Igl, M.; Rainforth, T.; Jin, T.; and Wood, F. 2017. Auto-encoding sequential monte carlo. arXiv preprint arXiv:1705.10306. Le, T. A.; Kosiorek, A. R.; Siddharth, N.; Teh, Y. W.; and Wood, F. 2020. Revisiting reweighted wake sleep for models with stochastic control flow. In Uncertainty in Artificial Intelligence, 1039–1049. PMLR. Lindsten, F.; Johansen, A. M.; Naesseth, C. A.; Kirkpatrick, B.; Schon, T. B.; Aston, J.; and Bouchard-Cotˆ e, A. 2017. ´ Divide-and-conquer with sequential Monte Carlo. Journal of Computational and Graphical Statistics, 26(2): 445–458. Maddison, C. J.; Lawson, J.; Tucker, G.; Heess, N.; Norouzi, M.; Mnih, A.; Doucet, A.; and Teh, Y. 2017. Filtering variational objectives. Advances in Neural Information Processing Systems, 30. Naesseth, C.; Linderman, S.; Ranganath, R.; and Blei, D. 2018. Variational sequential monte carlo. In International conference on artificial intelligence and statistics, 968–977. PMLR. Narayanan, P.; Carette, J.; Romano, W.; Shan, C.-c.; and Zinkov, R. 2016. Probabilistic inference by program transformation in Hakaru (system description). In International Symposium on Functional and Logic Programming, 62–79. Springer Obermeyer, F.; Bingham, E.; Jankowiak, M.; Pradhan, N.; Chiu, J.; Rush, A.; and Goodman, N. 2019. Tensor variable elimination for plated factor graphs. In International Conference on Machine Learning, 4871–4880. PMLR. Pfeffer, A. 2005. The design and implementation of IBAL: A generalpurpose probabilistic programming language. In Harvard Univesity. Citeseer. Salvatier, J.; Wiecki, T. V.; and Fonnesbeck, C. 2016. Probabilistic programming in Python using PyMC3. PeerJ Computer Science, 2: e55. Publisher: PeerJ Inc. Sankaranarayanan, S.; Chakarov, A.; and Gulwani, S. 2013. Static analysis for probabilistic programs: inferring whole program properties from finitely many paths. In Proceedings of the 34th ACM SIGPLAN conference on Programming language design and implementation, 447–458. Wang, D.; Hoffmann, J.; and Reps, T. 2018. PMAF: an algebraic framework for static analysis of probabilistic programs. ACM SIGPLAN Notices, 53(4): 513–528. Weinberg, S. 1995. The quantum theory of fields, volume 2. Cambridge university press. Zavatone-Veth, J.; Canatar, A.; Ruben, B.; and Pehlevan, C. 2021. Asymptotics of representation learning in finite Bayesian neural networks. Advances in neural information processing systems, 34: 24765–24777.

Using Autodiff to Estimate Posterior Moments: Conclusion, Limitations, and References

This paper is under CC 4.0 license. available on arxiv Authors: (1) Mounika Vanamala, Department of Computer Science, University of Wisconsin-Eau Claire, United States; (2) Keith Bryant, Department of Computer Science, University of Wisconsin-Eau Claire, United States; (3) Alex Caravella, Department of Computer Science, University of Wisconsin-Eau Claire, United States. Table of Links Abstract & Introduction Discussions Conclusions, Acknowledgment, and References Conclusion Upon recognizing the significance of cyber security vulnerability controls during the software requirement phase, the CAPEC software vulnerability repository emerged as the most practical repository for this study. The arrangement of attack patterns thus facilitates precise identification and seamless referral back to CAPEC for recommended defense strategies. We define and elaborate on topic modeling, as well as unsupervised and supervised ML methods, showcasing recent research instances and the applicability of these approaches. As our research continues, our efforts will involve the implementation of supervised machine learning. The CAPEC repository provides a prelabeled dataset, a valuable asset for training data set implementation. Supervised ML offers the added benefit of proficiently utilizing metrics to fine-tune the ML process, thus enabling thorough evaluation and process enhancement. A training set for the SRS document must either be crafted or located for supervised ML execution. Given the absence of a comparable research framework employing supervised ML, our future endeavors will assess and compare results stemming from Naïve Bayes and RF ML methodologies. Naïve Bayes showcases statistical prowess across both large and small data sets, making it suitable for the modest data set of SRS documents as well as the larger data set encompassing CAPEC Vulnerabilities. RF's capacity to counteract overfitting aligns well with the intricate data from CAPEC. The algorithm returning the most accurate recommendations for CAPEC attack patterns from an SRS document will be harnessed to deploy an automated tool for result processing and visualization. Acknowledgment Funding Information Author’s Contributions Acquisition of data and analysis and interpretation of data and content written. Keith Bryant and Alex Caravella: Conception and design of the article, intellectual content generation, critically reviewed the article. Keith Bryant, Alex Caravella, and Mounika Vanamala: Contribution to intellectual content ideation and reviewed the article along with the coordination for publication. Mounika Vanamala: Ethics This article is original and contains unpublished material. The corresponding author confirms that all of the other authors have read and approved the manuscript and that no ethical issues are involved. References Al-Sabahi, K., Zuping, Z., & Kang, Y. (2018). Latent semantic analysis approach for document summarization based on word embeddings. arXiv preprint arXiv:1807.02748. https://doi.org/10.3837/tiis.2019.01.015 Alyami, H., Nadeem, M., Alharbi, A., Alosaimi, W., Ansari, M. T. J., Pandey, D., ... & Khan, R. A. (2021). The evaluation of software security through quantum computing techniques: A durability perspective. Applied Sciences, 11(24), 11784. https://doi.org/10.3390/app112411784 Asim, M. N., Ghani, M. U., Ibrahim, M. A., Mahmood, W., Dengel, A., & Ahmed, S. (2021). Benchmarking performance of machine and deep learning-based methodologies for Urdu text document classification. Neural Computing and Applications, 33, 5437-5469. https://doi.org/10.1007/s00521-020-05321-8 Bedi, G. (2018). A guide to Text Classification (NLP) using SVM and Naive Bayes with Python. Medium, Nov. Bellaouar, S., Bellaouar, M. M., & Ghada, I. E. (2021, February). Topic modeling: Comparison of LSA and LDA on scientific publications. In 2021 4th International Conference on Data Storage and Data Engineering (pp. 59-64). https://doi.org/10.1145/3456146.3456156 CISA. (2021). c? | CISA. https://www.cisa.gov/uscert/ncas/tips/ST04-001 CVE. (2022). https://cve.mitre.org Delli, U., & Chang, S. (2018). Automated process monitoring in 3D printing using supervised machine learning. Procedia Manufacturing, 26, 865-870. https://doi.org/10.1016/j.promfg.2018.07.111 Guo, Y., & Li, J. (2021). Distributed Latent Dirichlet Allocation on Streams. ACM Transactions on Knowledge Discovery from Data (TKDD), 16(1), 1-20. https://doi.org/10.1145/3451528 Prasad, S. G., Badrinarayanan, M. K., & Sharmila, V. C. (2022). Efficacy and Security Effectiveness: Key Parameters in Evaluation of Network Security. International Journal of Performability Engineering, 18(4), 282. https://doi.org/10.23940/ijpe.22.04.p6.282288 IBM. (2019). What is machine learning? https://www.ibm.com/topics/machinelearning?lnk=fle Mallet, J., Pryor, L., Dave, R., Seliya, N., Vanamala, M., & Sowells-Boone, E. (2022, March). Hold on and swipe: A touch-movement based continuous authentication schema based on machine learning. In 2022 Asia Conference on Algorithms, Computing and Machine Learning (CACML) (pp. 442-447). IEEE. https://doi.org/10.1109/CACML55074.2022.00081 Kanakogi, K., Washizaki, H., Fukazawa, Y., Ogata, S., Okubo, T., Kato, T., ... & Yoshioka, N. (2022). Comparative Evaluation of NLP-Based Approaches for Linking CAPEC Attack Patterns from CVE Vulnerability Information. Applied Sciences, 12(7), 3400. https://doi.org/10.3390/app12073400 Kim, D., & Im, T. (2022). A Systematic Review of Virtual Reality-Based Education Research Using Latent Dirichlet Allocation: Focus on Topic Modeling Technique. Mobile Information Systems, 2022. https://doi.org/10.1155/2022/1201852 Krzeszewska, U., Poniszewska-Marańda, A., & Ochelska-Mierzejewska, J. (2022). Systematic comparison of vectorization methods in classification context. Applied Sciences, 12(10), 5119. https://doi.org/10.3390/app12105119 León-Paredes, G. A., Barbosa-Santillán, L. I., & SánchezEscobar, J. J. (2017). A heterogeneous system based on latent semantic analysis using GPU and multiCPU. Scientific Programming, 2017. https://doi.org/10.1155/2017/8131390 Livingston, F. (2005). Implementation of Breiman’s random forest machine learning algorithm. ECE591Q Machine Learning Journal Paper, 1-13. Macsai, D. 2012. The most important company you’ve never heard of. 1 Minute Read. Fast Company. https://www.fastcompany.com/3017927/30mitre McAllister, P., Zheng, H., Bond, R., & Moorhead, A. (2018). Combining deep residual neural network features with supervised machine learning algorithms to classify diverse food image datasets. Computers in Biology and Medicine, 95, 217-233. https://doi.org/10.1016/j.compbiomed.2018.02.008 Mounika, V., Yuan, X., & Bandaru, K. (2019, December). Analyzing CVE database using unsupervised topic modelling. In 2019 International Conference on Computational Science and Computational Intelligence (CSCI) (pp. 72-77). IEEE. https://doi.org/10.1109/CSCI49370.2019.00019 MITRE ATT&CK®. (2022). https://attack.mitre.org Mohamed, A. E. (2017). Comparative study of four supervised machine learning techniques for classification. International Journal of Applied, 7(2), 1-15. https://www.ijastnet.com/journal/index/859 NIST. (2022). About NIST. https://www.nist.gov/about-nist Prakash, A., Singh, N. K., & Saha, S. K. (2022). Automatic extraction of similar poetry for study of literary texts: An experiment on Hindi poetry. ETRI Journal, 44(3), 413-425. https://doi.org/10.4218/etrij.2019-0396 Rahman, A. S., Shamrat, F. J. M., Tasnim, Z., Roy, J., & Hossain, S. A. (2019). A comparative study on liver disease prediction using supervised machine learning algorithms. International Journal of Scientific & Technology Research, 8(11), 419-422. http://www.ijstr.org/final-print/nov2019/AComparative-Study-On-Liver-Disease-PredictionUsing-Supervised-Machine-LearningAlgorithms.pdf Rustam, F., A. Reshi, S. Mehmood, S. Ullah, B. On, W. Aslam and G. Choi. 2020. COVID-19 Future Forecasting Using Supervised Machine Learning Models. IEEE Access, pp: 101489-99. https://doi.org/10.1109/ACCESS.2020.2997311 Sanguri, K., Bhuyan, A., & Patra, S. (2020). A semantic similarity adjusted document co-citation analysis: a case of tourism supply chain. Scientometrics, 125(1), 233-269. https://doi.org/10.1007/s11192-020-03608-0 Schrider, D. R., & Kern, A. D. (2018). Supervised machine learning for population genetics: a new paradigm. Trends in Genetics, 34(4), 301-312. https://doi.org/10.1016/j.tig.2017.12.005 Shalev-Shwartz, S., & Ben-David, S. (2014). Understanding machine learning: From theory to algorithms. Cambridge university press. https://www.cs.huji.ac.il/~shais/UnderstandingMach ineLearning/ Sharma, C., Sharma, S., & Sakshi. (2022). Latent DIRICHLET allocation (LDA) based information modelling on BLOCKCHAIN technology: A review of trends and research patterns used in integration. Multimedia Tools and Applications, 81(25), 36805-36831. https://doi.org/10.1007/s11042-022-13500-z Siddiqui, N., Dave, R., Vanamala, M., & Seliya, N. (2022). Machine and deep learning applications to mouse dynamics for continuous user authentication. Machine Learning and Knowledge Extraction, 4(2), 502-518. https://doi.org/10.3390/make4020023 Sweeney, E. M., Vogelstein, J. T., Cuzzocreo, J. L., Calabresi, P. A., Reich, D. S., Crainiceanu, C. M., & Shinohara, R. T. (2014). A comparison of supervised machine learning algorithms and feature vectors for MS lesion segmentation using multimodal structural MRI. PloS One, 9(4), e95753. https://doi.org/10.1371/journal.pone.0095753 Uddin, S., Khan, A., Hossain, M. E., & Moni, M. A. (2019). Comparing different supervised machine learning algorithms for disease prediction. BMC Medical Informatics and Decision Making, 19(1), 1-16. https://doi.org/10.1186/s12911-019-1004-8 Ullah, F., Wang, J., Farhan, M., Jabbar, S., Naseer, M. K., & Asif, M. (2020). LSA based smart assessment methodology for SDN infrastructure in IoT environment. International Journal of Parallel Programming, 48, 162-177. https://doi.org/ 10.1007/s10766-018-0570-1 Ullah, F., Jabbar, S., & Mostarda, L. (2021). An intelligent decision support system for software plagiarism detection in academia. International Journal of Intelligent Systems, 36(6), 2730-2752 https://doi.org/10.1002/int.22399 Vanamala, M., Gilmore, J., Yuan, X., & Roy, K. (2020a, December). Recommending attack patterns for software requirements document. In 2020 International Conference on Computational Science and Computational Intelligence (CSCI) (pp. 1813-1818). IEEE. https://doi.org/10.1109/CSCI51800.2020.00334 Vanamala, M., Yuan, X., & Roy, K. (2020b, August). Topic modeling and classification of Common Vulnerabilities and Exposures database. In 2020 International Conference on Artificial Intelligence, Big Data, Computing and Data Communication Systems (icABCD) (pp. 1-5). IEEE. https://doi.org/10.1109/icABCD49160.2020.9183814 Zhu, L., He, Y., & Zhou, D. (2020). A neural generative model for joint learning topics and topic-specific word embeddings. Transactions of the Association for Computational Linguistics, 8, 471-485. https://doi.org/10.1162/tacl_a_00326

CAMBRIDGE QUANTUM

cambridgequantum.com

-- employees

Since n.d.

COMPANY RANKING

CAMBRIDGE QUANTUM

EVERGREEN INDEX #2103

Representing this company? Click here to claim and customize this page!

Cambridge Quantum WIKI

Apply to get your Tech Company News Page live on HackerNoon.com today!

CAMBRIDGE QUANTUM

cambridgequantum.com

-- employees

Since n.d.

COMPANY RANKING

CAMBRIDGE QUANTUM

EVERGREEN INDEX #2103

RELATED COMPANIESRANK

STORIES ABOUT CAMBRIDGE QUANTUM

Representing this company? Click here to claim and customize this page!

Cambridge Quantum WIKI

Apply to get your Tech Company News Page live on HackerNoon.com today!