The New Quantum Era

In this episode of The New Quantum Era, Kevin Rowney and Sebastian Hassinger are joined by Dr. Ieva Čepaitė to delve into the nuanced world of quantum physics and computation. Dr. Čepaitė discusses her journey into quantum computing and her work on counterdiabatic methods used to optimize the control of many body quantum states. She provides an overview of the landscape of new algorithms available within the field. She points out the importance of understanding the hardware to implement a quantum algorithm effectively. The focus then shifts to a discussion on adiabatic and counterdiabatic systems, providing a detailed understanding of both methods. The conversation concludes with a speculative take on future breakthroughs that could emerge with respect to quantum algorithms.

00:31 Introduction and Overview of the Interview
02:43 Dr. Čepaitė's Journey into Quantum Computing
05:23 Dr. Čepaitė's Diverse Experience in Quantum Computing
09:37 The Challenges and Opportunities in Quantum Computing
11:50 Understanding Adiabatic and Counterdiabatic Systems
15:15 The Potential of Counterdiabatic Techniques in Quantum Computing
25:49 The Future of Quantum Algorithms
32:55 The Role of Quantum Machine Learning
35:48 Closing Remarks and Reflections

In this interview, independent quantum information science researcher and consultant, Dr. Cassandra Grenade, shares their journey from triple majoring in physics, math, and computer science to their current consulting work with their firm, Dual Space Solutions. She discusses the concept behind the Quantum Intermediate Representation project (QIR), a tool which represents quantum programs and allows language designers to work independently of specific quantum processor details. Cassandra explains how QIR can solve the 'N to M' problem, where multiple language designs must interface with multiple quantum hardware architectures, thereby preventing the need for creating numerous unique compilers. Further, she dives into the evolution and future of quantum computing, highlighting the need for an industry-wide shift in understanding a quantum computer as more than just a circuit-based entity.

00:02 Introduction and Guest Background
00:22 Cassandra's Journey into Quantum Computing
01:40 The Birth of Dual Space Solutions
05:35 The Importance of Interdisciplinary Approach in Quantum Computing
08:14 The Challenges and Solutions in Quantum Computing
10:42 The Role of Quantum Intermediate Representation (QIR)
15:56 The Impact of QIR on Quantum Computing
19:01 The Future of Quantum Computing with QIR

Misty Wahl of the Unitary Fund joins us for this episode to talk about quantum error mitigation strategies like zero noise extrapolation (ZNE) and probabilistic error reduction using the Mitiq open source framework. Misty is a lead contributor the the Mitiq project as well as an author on a number of recent papers on the topic. We'll discuss the current state of the art, potential future strategies that leverage machine learning and quantum error correction, and how the Mitiq framework makes it easier to code up and compare mitigation strategies on a wide variety of qubits and SDKs.

You can find a sampling of Misty's reasearch papers and talk on her personal website, mistywahl.com

• Error mitigation in quantum computing with Misty Wall. 0:02
  • Misty Wahl, technical staff at Unitary Fund, discusses Mitiq project for error mitigation in quantum computers.
  • Misty discusses the growth of quantum computing as a field, with a focus on the Unitary Fund and its role in developing error mitigation techniques.
• Non-traditional background in quantum computing. 3:31
  • Misty Wahl shares her non-traditional background in mechanical engineering and project management, transitioning to quantum software development and research through self-study and online courses.
  • Misty joined Mitiq as a full-time technical staff member in March 2022, contributing to quantum error mitigation and software development through their experience with unitary hack.
  • Unitary Hack is a unique event hosted by Unitary Fund, where participants can tag issues in their GitHub repos and community can choose to solve them, providing valuable experience and connections in the quantum computing field.
• Quantum error mitigation techniques and software frameworks. 8:31
  • Misty Wahl describes her experience with the Mitiq framework
  • Misty explains how zero noise extrapolation works
  • Misty Wahl: By intentionally adding noise to quantum computations, researchers can extrapolate to the zero noise limit and estimate the optimal value of an expectation value.
• Quantum error mitigation techniques. 21:57
  • Misty believes that error mitigation will be crucial in the transition to fault-tolerant quantum computers, and will be used to enhance results at every step.
  • Misty presents a technique combining quantum error mitigation and quantum error correction to scale the distance of the surface code and improve error rate.
• Quantum computing, open source, and research funding. 28:56
  • Unitary Fund is building an open-source quantum community through community calls on Discord, with the goal of fostering collaboration and advancing quantum computing.
  • Unitary Fund is a 501(c)(3) nonprofit that funds research and development projects in AI, blockchain, and more through government grants and corporate sponsorships.

In this episode, Kevin and Sebastian are joined by Alex Keesling, CEO of QuEra Computing, for a discussion about his work with neutral atom arrays for simulation and computation. Alex describes his very early introduction to quantum information science as a high school student in Mexico, which kicked off a defining fascination with the field. At MIT as an undergraduate he started working with photonic systems, and as a PdD student with Misha Lukin at Harvard he played an instrumental role in the "atom array" project that eventually was spun out as QuEra. Today, QuEra's Aquila device has 256 atoms in its array that can be used as for analog Hamiltonian simulations, and is accessible on the cloud via AWS' Braket service. Alex explains in detail how these devices work, what physics breakthroughs they rely on for their operation, and where they may be going in the future with work underway on digital gates for universal computation. Additionally Alex takes us through some of the incredible scientific results these devices have already made possible, and discusses what the future of both scientific and commercial applications might hold.

The QuEra team published a deep dive into their Aquila device and its capabilities in a paper called Aquila: QuEra's 256-qubit neutral-atom quantum computer. 

In this episode of The New Quantum Era, hosts Sebastian Hassinger and Kevin Rowney interview Daniel Stick, a researcher at Sandia National Lab. They discuss the fascinating world of ion traps, a novel approach to quantum computing architecture. Stick explains the concept of suspending atoms inside a radio frequency Paul trap and utilizing laser pulses to manipulate their qubit states. The conversation also delves into the advantages and limitations of ion traps compared to other architectures. Stick shares exciting advancements in their technology, including the enchilada trap, developed as part of the Quantum Systems Accelerator project. Tune in to learn more about the cutting-edge research happening in the field of quantum computing.

[00:07:14] Large scale ion trap. 
[00:10:29] Entangling gates. 
[00:14:14] Major innovations in magneto optical systems. 
[00:17:30] The Name "Enchilada" 
[00:21:16] Combining chains for collective gates. 
[00:27:02] Sympathetic cooling and decoherence. 
[00:30:16] Unique CMOS application. 
[00:33:08] CMOS compatible photonics. 
[00:38:04] More breakthroughs on accuracy. 
[00:41:39] Scaling quantum computing systems. 
[00:45:00] Private industry and technology scaling. 
[00:51:36] Ion trap technology progress. 
[00:54:39] Spreading the word and building community.

• 00:01:15 - "So these architectures have, I think, powerful advantages versus other architectures."
• 00:18:30 - "So that was the name."
• 00:23:34 - "That's correct. That's that is one of the selling points for trapped ion quantum computing is that there is no threshold temperature at which you make the qubit go from behaving really well to behaving, you know, above which things would operate really poorly."
• 00:35:37 - "That is the grand vision that you've got this chip sitting inside of a chamber, and a bunch of digital signals go in and out of it."
• 00:38:40 - "What's a few exponents between friends anyway?"
• 00:41:39 - "That is one of the things that we have to think about is our gates are just, I don't know, 100 times to a thousand times slower than superconducting quantum computing systems or solid state quantum computing systems and ways to get around that have to leverage other kind of other attempts that are not limited by the physical speeds that are possible with an ion trap."
• 00:48:43 - "Do you have a paperclip, Kevin? That's all you need."

Title: Operating at the Quantum Limit with Dr. Dana Anderson

“In 25 to 30 years, quantum is going to be in the kitchen, sitting next to the toaster.” — Dr. Dana Anderson

Description: Welcome to another episode of The New Quantum Era Podcast hosted by Kevin Rowney and Sebastian Hassinger. Today, they are joined by Dr. Dana Anderson to talk about quantum computation, simulation, and sensing technologies using ultracold neutral atoms. Dr. Anderson is Chief Strategy Officer of Infleqtion, which was founded in 2007 as ColdQuanta and recently changed its name after acquiring Super.tech. Dr. Anderson is an applied physicist trained in quantum optics with extensive experience in optical neural networks, signal processing, precision measurement, and what he calls the field of “atomtronics.”

Key Takeaways:

[3:34] Dr. Anderson shares how he found his passion in physics and his entry point to quantum information science in general.

[5:13] How do lasers make atoms cold?

[7:13] Does Dr. Anderson think that what was learned from building atomic clocks and quantum devices has accelerated the development and maturation of the technologies behind the neutral atom arrays?

[10:44] Dr. Anderson talks about the optical lattice.

[12:41] Dr. Anderson addresses the early dawn of the transistor and the parallels with what he calls our age of atomtronics.

[14:00] Does Dr. Anderson think components on the optical side continue to shrink?

[15:17] Dr. Anderson explains how he uses machine learning to train an interferometer.

[17:44] Would machine learning assist in qubit control?

[25:05] What kind of new sensing technologies will emerge into the market?

[27:31] Dr. Anderson shares NASA developments regarding climate change.

[29:31] There will be a home-use application for quantum (and it will be boring, according to Dr. Anderson).

[31:48] Dr. Anderson discusses the benefits of meeting quantum and machine learning.

[36:06] Dr. Anderson helps us understand how the Infleqtion platform and quantum computation could emerge as a set of practical outcomes.

[45:02] Sebastian and Dr. Anderson discuss Infleqtion’s acquisition of Super.tech and what they have been working on.

[47:18] What does Dr. Anderson see on the horizon for the next 12 to 24 months for neutral atoms?

Mentioned in this episode:

Visit The New Quantum Era Podcast

The Nobel Prize in physics for Bose Einstein Condensates
Learn more about Infleqtion
NASA Cold Atom Lab 

Tweetables and Quotes:

“Every atom is a qubit, and every atom is just like every other atom, and it is as perfect as it could be.“ — Dr. Dana Anderson

“Roughly speaking, the way to think about everything Infleqtion can be boiled down to atomtronics.” — Dr. Dana Anderson

“If you are not operating at a quantum limit, you are not competitive .” — Dr. Dana Anderson

If anyone needs no introduction on a podcast about quantum computing, it's John Preskill. His paper "Quantum Computing in the NISQ era and beyond," published in 2018, is the source of the acronym "NISQ," for Noisy, Intermediate Scale Quantum" computers -- basically everything we are going to build until we get to effective error correction. It's been cited almost 6000 times since, and remains essential reading to this day.

John is a particle physicist and professor at Caltech whose central interests are actually cosmology, quantum matter, and quantum gravity -- he sees quantum computing as a powerful means to gain more understanding of the fundamental behavior of our universe. We discuss the information paradox of black holes, quantum error correction, some history of the field, and the work he's doing with his PhD student Robert (Hsin-Yuan) Huang using machine learning to estimate various properties of quantum systems. 

• How did you become interested in quantum information? 5:13
• The discovery of Shor’s algorithm. 10:11
• Quantum error correction. 15:51
• Black holes and it from qubit. 21:19
• Is there a parallel between error correcting codes and holographic projection of three dimensions? 27:27
• The difference between theory and experiment in quantum matter. 38:56
• Scientific applications of quantum computing. 55:58

Notable links:

• The Physics of Quantum Information, adapted from John's talk at the Solvay Conference on the Physics of Information
• Quantum Computing 40 Years Later, an update to John's NISQ paper on the occasion of the 40th anniversary of the conference at Endicott, the Physics of Computation.
• Lecture notes for John's class on quantum computing at Caltech, PH229
• Predicting many properties of a quantum system from very few measurements, one of the papers Robert Huang has published with John, appearing in Nature Physics

Tweetables and Quotes:

“The idea that you can solve problems efficiently that you'd never be able to solve because it's a quantum world and not a world governed by classical physics, I thought that was one of the coolest ideas I'd ever encountered.” — John Preskill

“There's something different about quantum information than ordinary information. You can't look at it without disturbing it.” — John Preskill

“Ideas which were being developed without fundamental physics, necessarily in mind, like quantum error correction, have turned out to be very relevant in other areas of physics.” — John Preskill

“Thinking about quantum error correction in the context of gravitation led us to construct new types of codes which weren't previously known. “ — John Preskill

“With quantum computers and quantum simulators, we can start to investigate new types of matter, new phases, which are far from equilibrium.“ — John Preskill.

 Welcome to another episode of The New Quantum Era Podcast hosted by Kevin Rowney and Sebastian Hassinger. Today, they are joined by another distinguished researcher, Dr. Harry Buhrman. Dr. Buhrman is a professor at the University of Amsterdam, he's a director at the CWI, and he's the director at Qusoft as well. He's got a long and illustrious career in quantum information. Today, Dr. Buhrman takes us through some of his earlier work and some of his areas of interest, and he also discloses details of his recent paper which was going to be called Ultra Fast Quantum Circuits for Quantum State Preparation, but was posted to the arXiv as State preparation by shallow circuits using feed forward, which provides fascinating results with respect to the core architecture divided into four layers and time complexity around that framework.

Key Takeaways:

[4:45] Sebastian introduces Dr. Harry Buhrman.

[5:31] How did Dr. Buhrman become interested in Quantum Computing?

[9:31] Dr. Buhrman remembers the first time he heard about the complexity class known as fast quantum polynomial time, or BQP.

[11:35]  Dr. Buhrman and Richard Cleve started working on communication complexity.

[14:14] Dr. Buhrman discusses the opportunity that arose after Shor’s algorithm.

[14:53] Dr. Buhrman has also written biology papers explaining how he became involved in this field.

[18:05] Is quantum computation and quantum algorithms the main focus now regarding Dr. Buhrman’s areas of study?

[20:06] Software and hardware are codependent, so codesigning is needed.

[20:58]. What are the big unsolved problems in the areas of time complexity and hierarchy for quantum? 

[24:50] Does Dr. Buhrman think it's possible that there could be a future where some of the classical time complexity problems could be powerfully informed by quantum information science and Quantum Time complexity discovery?

[27:32] Does Dr. Buhrman think that, over time, the distinction between classical information theory and quantum information theory will erode?

[28:50] Dr. Burhman talks about his Team's most recent paper.

[33:55]  Dr. Buhrman’s group is using tmid-circuit measurement and classical fan out to extend the amount of computation time 

[35:04] How does this approach differ from VQE or QAOA?

[38:35] About Dr. Buhrman’s current paper, is he thinking through algorithms that may be able to be implemented in at least toy problems sort of scale to try this theory out and implementation?

{39:22] Sebastian talks about  QubiC, an open-source Lawrence Berkeley National Lab project.

[41:14]  Dr. Buhrman recognizes he is very much amazed by the fact that when he started in this field in the mid-late 90s, it was considered very esoteric and beautiful but probably wouldn't lead to anything practical.

[43:49] Dr. Buhrman assures that there is a chance that those intractable problems for classical computing also remain intractable for quantum computers.

[44:24] What's the next big frontier for Dr. Buhrman and his team?

[47:03] Dr. Buhrman explains Quantum Position Verification used for implementing secure communication protocols.

[50:56] Sebastian comments on the hilarious and interesting titles for papers Dr. Buhrman comes up with.

[53:10] Kevin and Sebastian share the highlights of an incredible conversation with Dr. Buhrman.

Mentioned in this episode:

Visit The New Quantum Era Podcast

Quantum entanglement and communication complexity

The first peptides: the evolutionary transition between prebiotic amino acids and early proteins

A Qubit, a Coin, and an Advice String Walk Into a Relational Problem

Six hypotheses in search of a theorem

Tweetables and Quotes:

“ Biological processes are quantum mechanical, and sometimes you need the quantum mechanical description to understand them, and indeed, quantum computers could be of great help in simulating them and understanding them better than we currently do.“ — Dr. Harry Buhrman

“There's a huge gap between what we can do and what we can prove is true.“ — Dr. Harry Buhrman

“Our problems have become bigger but also more interesting, I would say.“ — Dr. Harry Buhrman

“We're not the first ones to see that having mid-computation measurements plus classical feed forwards actually is very useful and can help you solve problems or generate states that if you don't have this  are impossible  to make.” — Dr. Harry Buhrman

“Big companies are very interested in QC not only for building quantum computers but also figuring out whether it is useful from a software point of view. ” — Dr. Harry Buhrman

Welcome to another episode of The New Quantum Era Podcast hosted by Kevin Rowney and Sebastian Hassinger. Today, they are joined by an outstanding European researcher: Professor Leo Kouwenhoven.

Leo is a professor in Applied Physics specialized in the field of Quantum NanoScience at TU Delft. Leo got his Ph.D. in Mesoscopic Physics at Delft. He was a postdoc researcher at the University of California at Berkeley and a visiting professor at Harvard. Highlights in Leo’s career include the discovery of conductance quantization in quantum point contacts, Coulomb blockade in quantum dots, artificial atoms, the Kondo effect in quantum dots, Spin qubits, induced superconductivity in nanowires and nanotubes, spin-orbit qubits in nanowires and nanotubes and Majoranas in nanowires. Leo and his group found evidence of Majoranas detailed in a paper from 2012. He lead the Microsoft hardware R&D effort, working on topological qubits using Majorana zero modes from 2016 to 2022. His current focus at Delft is on topological effects in solid-state devices, such as the emergence of Majoranas and topological qubits.

Key Takeaways:

[2:53] Kevin and Sebastian share their appreciation about how quantum computing was represented in the episode Joan is Awful of the TV show Black Mirror. 

[6:04] Leo shares how he got interested in the field of quantum computing.

[9:40] Leo discusses how much he knew about the work done in theoretical quantum computing in the mid to late 90s.

[14:37] The advantage of superconducting qubits is that you have a large number of electrons in the circuit you are manipulating.

[15:34] Measurability can be easier but “it always comes with a price”.

[17:05] Leo admits the coherence was insufficient, and he shares how they tried to improve it.

[19:15] What is the feature of silicon that makes it valuable for Quantum Computing?

[22:12] Leo shares the benefits of a hybrid system (combining super connectivity and semi-connectors).

[23:10] Leo discusses how he became interested in Majoranas.

[27:30] Leo addresses the main research agenda destination regarding Majoranas.

[28:22] Was the Majoranas fundamental particle found?

[33:21] The potential for theory and application is so huge. What's Leo’s sense about the prospects for these avenues of inquiry research?

[36:25] Leo explains the non-abelian property that Majoranas zero modes have.

[40:18] Leo addresses the two groups of gate operations needed for universal computing.

[41:22] Leo gives his opinion regarding the timeframe for the appearance of commercially viable outcomes in this domain. 

[47:16] Sebastian reflects on the maturation of the neutral atom systems, considering them as the first realization of Feynman's vision from 1981 regarding the fact that in order to simulate a natural system, there is a need for a quantum computer to do it.

[48:08] Can we build machines that can help us simulate the dynamics of quantum systems that might help us understand more what the challenges are in Majorana Qubit? 

[51:01] Does Leo think there's any value in Majorana braiding simulations to try to understand the dynamics of the system or overcome the challenges?

[53:50] There is room for optimism in Quantum Computing.

[56:24] Leo talks about the dream of topological Majoranas qubit.  

[58:16] Kevin and Sebastian share the highlights of an insightful conversation with Leo Kouwenhoven.

Mentioned in this episode:

Visit The New Quantum Era Podcast

Black Mirror: Joan is Awful

Learn more about Leo Kouwenhoven
Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices

Tweetables and Quotes:

“The advantage of the superconducting qubits is that you have a large number of electrons in the circuit you are manipulating, which can make measurability easier, but it always comes with a price.”— Leo Kouwenhoven

“I read that making qubits was too much engineering when it should be something more fundamental… so now we think qubits are fundamental?!” — Leo Kouwenhoven

“Problems are there to be solved; they only exist to be solved. People in classical electronics also solved all their problems, so why can’t we? ” — Leo Kouwenhoven

Description: Welcome to another episode of The New Quantum Era Podcast hosted by Kevin Rowney and Sebastian Hassinger. Today, they are joined by Scott Aaronson, who is a leading authority in the space of Quantum Computing, a fascinating person with a long list of relevant achievements. Scott is also the author of an outstanding blog called Shtetl-Optimize and a book named Quantum Computing Since Democritus.

Scott helped design Google Quantum Supremacy, but his work exceeds it; he is involved in Complexity Theory and Computer Science and is just extremely good at connecting, explaining, and digging deeper into concepts.

Key Takeaways:

[3:38] How did Scott get into quantum computing?

[11:35] Scott talks about the moment when the question arose: Does nature work this way?

[14:28] Scott shares when he realized he wanted to dig deeper into Quantum Computing.

[15:56] Scott remembers when he proved the limitation of quantum algorithms for a variation of Grover's search problem.

[18:43] Scott realized that his competitive advantage was the ability to explain how things work.

[20:01] Scott explains the collision problem.

[21:33] Scott defines the birthday paradox.

[23:24] Scott discusses the dividing line between serious and non-serious quantum computing research.

[24:11]  What's Scott’s relative level of faith and optimism that the areas of topological quantum computing and measurement-based quantum computation are going to produce?

[28:33] Scott talks about what he thinks will be the source of the first practical quantum speed-up. 

[31:55] Scott didn’t imagine that being a complexity theorist would become exponential.

[36:14] Is Scott optimistic about quantum walks? 

[40:11] Has Scott returned to his machine learning and AI roots but is now trying to explain the concepts? 

[42:03] Scott was asked: ‘What is it going to take to get you to stop wasting your life on quantum computing?’

[44:50] Scott talks about the future need to prevent  AI misuse. and his role in Open AI

[47:41] Scott emphasizes the need for an external source that can point out your errors.

[50:13] Scott shares his thoughts about the possible risks and misuses of GPT.

[51:40] Scott made GPT to take a Quantum Computing exam; what did surprise him about the answers? It did much better on conceptual questions than on calculation questions

[55:55] What kind of validation will we be able to give GPT?

[56:22] Scott explains how RLHF (Reinforced Learning from Human Feedback) works.

[59:28] Does Scott feel that there's room for optimism that educators can have a decent tool to hunt down this kind of plagiarism?

[1:02:08] Is there anything that Scott is excited about seeing implemented on 1000 gate-based qubits with a decent amount of error mitigation? 

[1:04:05] Scott shares his interest in designing better quantum supremacy experiments.

[1:07:43] Could these quantum supremacy experiments (based on random circuit sampling) already deliver a scalable advantage? 

[1:10:58] Kevin and Sebastian share the highlights of a fun and enlightening conversation with Scott Aaronson.

Mentioned in this episode:

Visit The New Quantum Era Podcast

Check Shtetl-Optimize

Quantum Computing Since Democritus, Scott Aaronson

Learn more about the Adiabatic Algorithm result by Hastings and the Quantum Walk Algorithm result by Childs et Al.

Tweetables and Quotes:

“The dividing line between serious and nonserious quantum computing research is, are you asking the question of, ‘Can you actually be the best that a classical computer could do at the same desk? “ — Scott Aaronson

“My first big result in quantum computing that got me into the field was to prove that Prasad Hoyer tap algorithm for the collision problem was optimal.”  — Scott Aaronson

“ Quantum Walks are  a way of achieving Grover type speed ups at a wider range of problems than you would have expected.” — Scott Aaronson

“AI safety is now a subject where you can get feedback.”  — Scott Aaronson

“We don't have any theorems that would explain the recent successes of deep learning, the best way we can explain why is that none of the theorems rule it out.” — Scott Aaronson

Welcome to another episode of The New Quantum Era Podcast hosted by Kevin Rowney and Sebastian Hassinger.

In this episode, we are joined by Dorit Aharonov, a professor at the Hebrew University of Jerusalem and one of the pioneers of quantum computing. She's also the Chief Science Officer at QEDMA, a quantum startup based in Israel. Dorit is one of the major movers and shakers of quantum error correction and co-author of the important Threshold Theorem for quantum error correction. Kevin, Sebastian, and Dorit talk about her recent work on the theoretical foundations of random circuit sampling.

Key Takeaways:

[4:22] Dorit shares her path into quantum information and computing.

[8:27]  Dorit explains the threshold theorem in an easy-to-understand manner.

[16:35] The velocity of error correction versus the generation of errors in the computation could depend on physical implementation, or the algorithm. Maybe even both.

[18:53] A more powerful assertion Dorit makes is that there's a deeper connection between the phases of matter and the transition between solid and liquid and these quantum error correction thresholds.

[19:51] A lot of the foundations of classical error correction were laid down in the mid-40s in Von Neumann's work when the IAS system was being built. Dorit still sees the echoes of that.

[22:35] We might be witnessing a growing momentum around the powerful expression of new quantum error correction technologies.

[25:28] Dorit talks about the difference between error mitigation and error correction.

[26:55] Dorit explains the idea of the reset gate.

[30:22] It might be safe to say that challenges are primarily engineering in nature and that we have enough science to enable that engineering to get to fault tolerance.

[31:50] Dorit discusses a possible timeline for this engineering to get to fault tolerance.

[34:07] Is Dorit an NISQ optimist or a pessimist when it comes to real-world applications?

[39:21] Dorit addresses the difference between practical and asymptotic quantum advantage.

[41:30] Dorit shares what the paper on random circuit sampling shows.

[45:25] Dorit explains why the machine learning algorithms that were dequantized are  treacherous.

[49:56] Dorit shows optimism regarding the possibility of seeing evidence of a quantum event.

[52:25] Dorit admits to finding constructive interference between working in the industry and working on theoretical questions.

[53:50] Is there something Dorit is excited about in the next year or two that will be another step forward?

[56:50] Dorit talks about concrete examples of experiments and sensors that might be arriving thanks to quantum computing advancements.

[1:00:35] Sebastian and Kevin share the highlights of a fantastic conversation with Dorit.

Mentioned in this episode:

Visit The New Quantum Era

The New Quantum Era Podcast

Limitations of Noisy Reversible Computation Dorit Aharonov, Michael Ben-Or, Russell Impagliazzo, Norm Nisan

The Complexity of NISQ, Sitan Chen, Jordan Cotler, Hsin-Yuan, and  Jerry Li
A polynomial-time classical algorithm for noisy random circuit sampling Dorit Aharonov, Xun Gao, Zueph Landau, Yunchao Liu, Umesh Vazirani 

QEDMA

Tweetables and Quotes:

“Nobody actually believed that it was possible to correct errors that occur on quantum states because of the lack of reversibility. ” —  Dorit Aharonov

“it's a physics phenomenon… below a certain threshold, we can think of this as if the system is capable of some completely different behavior, like ice and water. It's just like a phase transition -- below that, there would be macroscopic entanglement and … ability to control large scale quantum correlations. And above it, this would not be possible.”  — Dorit Aharonov

Welcome to another episode of The New Quantum Era Podcast hosted by Kevin Rowney and Sebastian Hassinger. Today, they are joined by James Whitfield, who's a professor at Dartmouth College and is a colleague of Sebastian’s at Amazon Web Services’ quantum team. James has a quantum chemistry background, and, as a result, he brings that sensibility to his work in quantum information science.

In today’s episode, they cover three main topics:

• They talk about the specific areas of quantum chemistry where progress in quantum computation can be seen towards cracking key problems.
• They address the intuitive nature of perceiving entanglement within quantum states and how those manifest in quantum algorithms (excellent material for people trying to get on top of that challenging concept).
• James shares his perspectives on enhancing pedagogy in Quantum Information Science, both in the K -12 range and at the graduate level. 

Key Takeaways:

[4:06] James talks about his background.

[6:37] What's the simplest way to explain what quantum chemistry is?

[8:18] James shares framing remarks on the merit of quantum computing in these early phases regarding its applicability to physical chemistry. 

[10:30] James talks about the concept of time evolution.

[11:13] James explains the differences between the dynamical nature and the optimization nature of a problem.

[13:06] James speaks of what happens inside of quantum time evolution.

[14:54] Geometry optimization is only one problem that people discuss.

[16:47] James talks about the ‘clamped nuclei’ approximation.

[17:33] James describes the two ways of thinking about the Schrodinger equation.

[19:59] What types of things would we be able to do if we could model time intervals? 

[24:09] Does James think that, in terms of time evolutions,  fairly large numbers of fault-tolerant qubits are needed to do useful calculations? Or is there a class of problems that NISQ or even Analog Devices like QuEra could be helpful with?

[27:13] What is entanglement entropy? And what does that mean for computation?

[30:48] Why do people believe in the extra power of quantum computing?

[32:37] James defines coherence and decoherence.

[34:25] James explains why measuring the growth rate of entanglement entropy over time is one way to capture the richness of the other quantum state.

[36:42] James talks about the application of quantum chemistry.

[42:55] James believes that, eventually, these will all converge.

[43:54] James shares one of his projects about how we use quantum computers to benchmark what people do today.

[45:37] The hard part is not the implementation; James explains why.

[47:53] James uses the analogy of the robotics challenge.

[48:41] James talks about the event called: Quantum Computing Quantum Chemistry Benchmark. 2023.

[49:25] Is there an optimum starting point for quantum education? 

[52:45] James works with no negative probabilities.

[55:05] James talks about quantum mechanics and atomic physics.

[56:25] Quantum and AI often get grouped into the same category in terms of technology.

[57:46] James shares what he enjoys the most about his work.

[59:30] Does James think that eventually, software will eat all of these disciplines of science related to quantum information, and we will end up with scientists writing code, and that code will solve problems in chemistry, physics, or other scientific areas through writing software?

[1:02:40] Kevin and Sebastian share the highlights of a fantastic conversation with James Whitfield.

Mentioned in this episode:

Visit The New Quantum Era Podcast
Computational Complexity in Electronic Structure James Whitfield, Peter J. Love, Alan Aspuru-Guzik
Limitations of Linear Cross-Entropy as a Measure for Quantum Advantage Xun Gao, Marcin Kalinowski, Chi-Ning Chou, Mikhail D. Lukin, Boaz Barak, Soonwon Choi
Understanding the Schrodinger equation as a kinematic statement: A probability-first approach to quantum James Daniel Whitfield

2023 Quantum Chemistry on Quantum Computers Benchmarking Contest

Tweetables and Quotes:

“To actually get what the strength of that spring should be, you need to know what the electrons are doing, and that's where electronic structure comes in, and this is where a lot of the effort inside of quantum computing has gone in.”.   —  James Whitfield

“ In terms of their justification for believing in the extra power of quantum computing, the soul of the claim for many people is largely founded on the capacity of these systems to witness entanglement and have a richer notion of state, which is harder to express classically.” — Kevin Rowney

“Quantum and AI often get grouped into the same category in terms of technology.”  — Sebastian Hassinger.

“There are still fantastic scientists who take entire journeys inside their head, building mathematical structures, they don't bother to code it up, and then they give it to someone else who codes it up.”   —  James Whitfield.

We continue our stimulating conversation with Joe Fitzsimons, CEO and founder of Horizon Quantum Computing. After last episode's exploration of Joe's reasoned case for an optimistic future for quantum computing, we dig into Horizon's development of compiling tools that Joe hopes will unlock broad performance advantages from future quantum devices.

Computer History Museum
McCullough-Pitts paper on artificial neurons
A guide to the HHL algorithm from the excellent qiskit open source textbook

Kevin and Sebastian are joined by Joe Fitzsimons, founder and CEO of Horizon Quantum Computing, a startup based in Singapore. Joe recently posted a thread on Twitter responding to some of the reactions to a recent Time cover story about quantum computing. We were really struck by his level-headed optimism and so we wanted to dig in deeper. This is part one of our conversation with Joe, where he explains the reasoning behind his optimism for the future of the technology.

Mentioned in the episode
Global Risk Institute 2022 Quantum Threat Timeline Report 
The Center for Quantum Technologies in Singapore
Wikipedia page on 2 nanometer process for microprocessor fabrication

Kevin and Sebastian are joined by Grant Salton, a quantum researcher at AWS, who helps us understand a recent paper from Google and Caltech whose authors describe a simulation of a wormhole on Google's Sycamore quantum computer. The paper stirred some controversy and push back on the misunderstanding of the claims being made, and Grant walks us through a sub-domain of quantum information science called "it from qubit," which seeks to bridge elements of astrophysics with concepts from quantum information.

Mentioned in the episode:
The Nature paper from Google and Caltech describing the wormhole experiment and findings.
Some context from Caltech blog.
John Wheeler's paper: "Information, Physics, Quantum: The Search for Links" which coined "it from bit."
A BBC article describing the "quantum hair" solution to Hawking's black hole information paradox.
The Edge of All We Know, a terrific documentary that traces the efforts to solve the information paradox in parallel with the effort to capture an image of a black hole. 

Key Takeaways:

[3:38] Nathalie shares how she found her way into the field of quantum technology.

[6:25] Nathalie talks about the key moment in the landscape towards being a believer in Quantum Technology.

[8:29] Nathalie talks about certain things that made her change her mind.

[12:20] Nathalie speaks about her particular entry into the science field.

[18:09] How far up the stack does Nathalie’s interest lie, and how does that inform what she has been doing down at the materials?

[22:54] Nathalie shares the story about NSF.

[25:48] What is wrong with Niobium?

[27:12]  Nathalie explains the difficulty of surface physics and surface chemistry in this domain.

[32:30] Is there a way to describe conceptually how a vacancy in a diamond can be used as a two-level system or for a cubit, or as a sensing device?

[37:03] Why is it called a color center? 

[37:59] Nathalie talks about the genesis of her paper which includes material science foundations for the quantum information process.

[42:35] Can Nathalie make any speculations based on what she learned from the review paper?

[46:54] Is it true that manipulating diamonds is really slow?

[48:28] Sebastian talks about the way they met Nathalie.

[49:29] Are there things that either educators or industry participants in this stage of quantum computing and quantum information technologies can do to help make this area work better than the other fields have in the past? 

[55:58] Sebastian and Kevin share the highlights of an amazing conversation with Nathalie DeLeon.

Mentioned in this episode:

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Co-Design Center for Quantum Advantage 

Tweetables and Quotes:

“If you could do a quantum version of erasure conversion, you can actually get extremely high thresholds.“ — Nathalie DeLeon

“The fact that,  in some sense, fault tolerance is a phase, a transition is a quantum phase transition, right? You have a fundamentally different system before and after you turn on your error correction. .“ — Nathalie DeLeon

Welcome to another episode of The New Quantum Era Podcast hosted by Kevin Rowney and Sebastian Hassinger. Today, they are joined by Steve Girvin, professor of Physics at Yale who has a central role in the Yale Quantum Institute, which has been ground zero for the recent development in superconducting qubits. 

The topics we had initially planned needed some adjustment, because on the day of the interview, the Nobel Prize in Physics was awarded to three scientists for their work experimentally verifying the theory behind entanglement, the source of much of quantum computing's power. Alain Aspect, John F. Clauser, and Anton Zeilinger were recognized for their experiments in an area that has broad implications for secure information transfer and quantum computing. 

Sebastian, Kevin, and Steve have an interesting talk about some of the history of the superconducting qubits and the transmon in particular, which is a basis for most of the modern superconducting qubits on the market. They also cover the topic of diversity, quality, and inclusion.

Key Takeaways:

[3:43] Steve introduces himself.

[5:23] Steve shares his primary domains of research.

[9:50]  Was there a sort of self-awareness in the Yale group that Steve and his team were taking radically? Were they considering a different approach that could solve some of the challenges of the other models that existed at the time?

[14:38] Steve talks about how relatively quickly the hardware can be fabricated to be able to crank out, iterations, variations, and experiments. 

[17:27] Is there room for optimism about the new dimensions of research related to MER material science?  

[19:25] Steve shares his thoughts on the news about the 2022 Nobel Prize in Physics.

[22:18] Steve talks about how some of the epistemological questions that these paradoxes present, feel really mind-bending to many people on the outside of physics.

[25:38] Steve addresses how hard it is to predict the future.

[27:21] Does Steve consider himself an optimist about the progress of quantum computing?

[30:10] How can we get reliable performance out of an inherently, very unreliable system?

[33:22] Steve helps us fill in the narrative, in the history of where GKP codes are situated and their significance to contemporary developments.

[41:14] Steve talks about the basic steps of the algorithm to do the error correction.

[44:01] The history of computer science is very, uh, white, male, and, uh, dominated in nature, Steve shares his thoughts about diversity, equity, and inclusion.

[48:34] What we can do to change the composition of the field when the underlying foundations of the way science is done in the lab have a such rigid history of hierarchy, power structures, and power dynamics that are so easily abused?

[55:02] Sebastian and Kevin share their thoughts on an amazing conversation with Steve Girvin,  

Mentioned in this episode:

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Turing's Cathedral: The Origins of the Digital Universe, George Dyson

Documentary: Picture a Scientist

Tweetables and Quotes:

“A very productive part of my childhood was having nothing to do, but to dream.“ — Steve Girvin 

“The simpler you keep things, the easier it's to do things “ — Steve Girvin 

“Einstein really made massive contributions to the development of the quantum theory. “ — Steve Girvin 

“The way we test whether our quantum computer is a quantum computer is checking first thing in the morning to calibrate it, if it's doing the thing that Einstein said was impossible then, it's working.“ — Steve Girvin 

“Looking ahead, it's very, very hard to predict where this is going, but along the way, there's such fantastic. basic science and quantum.” — Steve Girvin

“When you're doing a hiring search, it's not about adding constraints, like interviewing more women…It's about removing constraints. You should look wider. There's a theorem that if you release constraints, the optimum cannot get worse, it can only get better. ” — Steve Girvin

Key Takeaways:

[3:23] James introduces himself.

[4:20] James talks about his engagement in game development using the public IBM Cloud quantum systems.

[5:40] James explains why he said he expected the field of quantum computing to be more accessible by starting with hobbyists.

[7:02] James talks about the theory behind quantum computing.

[8:23] James speaks of how to engage people in quantum computing by proving Einstein was wrong in how he saw quantum mechanics.

[12:39] What are some of the things that James has seen that were sort of super inventive ways to use quantum computing in a game context?

[14:20] James talks about the quantum emoji generator.

[15:26] James shares his opinion in regard to Quantum Chess.

[16:48] James talks about a new game called Quantum Odyssey

[18:08] James shares an experience working with kids when he was at the University of Basel.

[19:55] James talks about his passion for quantum error correction.

[20:41] James tells the difference between quantum error correction and quantum error mitigation.

[24:18] Sebastian talks about mitigation strategies.

[27:00] Could it be that lots of the statistical tradecraft with respect to analyzing data and attempting to interpret its meaning in the presence of acknowledged errors and the signal is perhaps a foundational part of QAM? 

[28:01] What are the major and most interesting themes to James these days? 

[29:36] James explains the threshold theorem.

[34:33] What is the current math result in terms of the threshold of error occurrence that you need to get to get over the hump?

[35:16] James talks about the experimental results where people have built minimal examples of quantum error-correcting codes

[36:01] James talks about a recent experiment made at IBM quantum.

[36:40] What does surface code mean?

[39:20] Are there any other types of errors that quantum error correction has to struggle with? Or are the bit flip and phase error the two main aspects?

[41:55] James talks about the recent research on silicon spin qubits.

[45:39] Sebastian and Kevin share the highlights of an amazing conversation with James.

Mentioned in this episode:

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Stephen Hawking faces Paul Rudd in epic chess match (feat. Keanu Reeves)

Tweetables and Quotes:

“It's better if we start off by building a little bit of intuition, and then bringing in the maths, it's important to bring in the maths but I think it's better when the maths is describing an intuition that people already have and that's the starting point.” — James Wootton

“There have been experimental results already where people have built minimal examples of quantum error correcting codes and showing that they have a beneficial effect. So that's what happens when the noise is low enough. “ — James Wootton

Title: Molecular and Subatomic Physics with David Mazziotti

Description: Welcome to another episode of The New Quantum Era Podcast hosted by Kevin Rowney and Sebastian Hassinger. Today, they are joined by David Mazziotti, a physicist, and research team leader at the University of Chicago. He generously showed up with some deeply fascinating material for your consideration. Professor Mazziotti is a highly accomplished scholar, researcher, and mentor. This interview with David is a ringside seat on one of the most interesting recent research on molecular and subatomic physics that is now being explored by scholars using quantum computers. 

Today, David talks about how he got interested in quantum computing, his current findings and experiments, and his optimistic perspective about the possibility of breakthroughs in the near future of quantum.

Key Takeaways:

[6:13] David talks about his background.

[11:32] David’s first professor role was teaching quantum chemistry.

[12:30] David speaks about the first time he used quantum computing hardware to perform experimentation, like simulation of quantum chemistry.

[14:42] David Talks about his first foray into quantum computing.

[16:45] What measurements is David doing inside the quantum computer to register that data on the polytopes?

[18:58] Where did the inspiration come from for using limited hardware with limited capabilities (from a gate and noise perspective) in a really creative way to do really sophisticated simulations? 

[24:19] What are the major engineering or commercial applications?

[28:43] David talks about his collaboration on a couple of papers on a generalizable system for free a simulation of open quantum systems.

[31:24] Is there something that can be done on a standard quantum computer to simulate open systems? Is new hardware needed?

[33:40] Is it possible for David to speculate if there will be brand new algorithmic breakthroughs for clever classical optimization problems? 

[35:10] David shares the publication of  a new paper on communications physics.

[37:15] Can we make progress with noisy quantum computers?

[40:39] David speaks about how he and his team ended up getting a spectrumscoptic noise “fingerprint” of each of their IBM Quantum computers on which they were doing an experiment. What does derive from the spectrum of the QC?

[42:28] Is David  programming the pulses or is he using gates?

[43:42] Is the fingerprint like a qubit? 

[45:22] David believes that a more holistic perspective on the noise could be the way to control noise better.

[48:30] David’s work has been on superconducting hardware, is it applicable to  trapped ions or neutral atoms or Rydberg atom systems that are coming out in the next year? And hopefully to photonic systems down the road? 

[49:46] Is David’s work on superconducting hardware applicable to quantum sensing devices?

[52:41] David shares his excitement about the evolution of quantum computing in the next couple of years.

[56:19] For listeners who want to explore  some of the code and are qiskit literate? Is any of the stuff that David has  mentioned available open source style? 

[59:05] David speaks of his work on reduced density matrix theory. 

[1:00:26] If David could wish for any new hardware in the next year, what would he want? 

[1:04:53] Sebastian  and Kevin share their insights from a mind blowing conversation with David Mazziotti.

Mentioned in this episode:

Visit The New Quantum Era Podcast

Learn more about David Mazziotti’s work at his group’s website and check out their github repo

Tweetables and Quotes:

“So a polytope is basically a convex object with a bunch of flat sides and on one side, there's this polytope that's forbidden, on the other side, one that's allowed, and then there's this hyperplane in the middle called the Borland-Dennis Inequality, and you just don't want the points to go through.”  — David Mazziotti

“In superconductivity, electrons form Cooper pairs, and these Cooper pairs of electrons all end up in a global quantum state and that allows you to send electricity into the superconductor, and actually have a current then come out from a macroscopic distance away, but not have any loss due to friction because you're really sending an electron into a global quantum state that's entangled with the electron that's coming out on the other side at the same time.”  — David Mazziotti

“It's only after 2000, that people were able to realize excitation condensates by pumping them with light with radiation. And then in the last few years, since 2017, they've been able to prepare them in the laboratory, even without pumping them with radiation, either using strong magnetic fields or, actually, in some cases, not using any magnetic fields at all. But using Creative Chemistry.” — David Mazziotti

“The quantum computer  gives one an ability  to look at some things that before were really more just a theoretical dream” — David Mazziotti

“Can we make progress with noisy quantum computers? I think that's one of the central questions, because ultimately, quantum computers are always going to be somewhat noisy to some extent.” — David Mazziotti

 Welcome to another episode of The New Quantum Era Podcast hosted by Kevin Rowney and Sebastian Hassinger. Today, they are joined by Cesar A. Rodriguez  Rosario, Chief Scientific Officer at Strangeworks, who is discussing the parallels between quantum computing and the development of the classical computing world at the stage of vacuum tubes and the invention of the transistor.

Cesar Rodriguez is a great example of somebody who is knowledgeable about the space of Quantum Computing and sees its possibility but he's got a decent level of guarded optimism and even skepticism on some of these results, which sometimes run fits and starts and sometimes even go backward.

Key Takeaways:

[4:33] Cesar shares what brought him into Quantum Computing.

[5:35] Cesar talks about his academic background

[11:39] Coming from computer engineering and having an unconventional journey through quantum physics, does Cesar consider he has a different perspective on the field today? 

[15:28] Given the current stage of technology, what does Cesar think of the role of foreign theorists? 

[17:37] How does Cesar view the reliability and the breakthrough potential of the currently existing crop of algorithms given the current limits?

[18:57] Cesar explains what QUantum Advantage is.

[21:08] From the landscape of the current algorithms out there, does  Cesar feel like there's an imminent breakthrough in these scare algorithms? 

[23:05] Will there going to be more "dequantized" algorithms?

[24:35] Cesar shares what he calls Quantum Value.

[25:21] Looking at the theory landscape, what are the most exciting things to Cesar?

[29:20] Does everything still fit into the general buckets of VQE and QAOA? Are there other categories that are emerging that are distinct enough from those two approaches that they have their own acronym yet?

[30:52] What does quantization mean?

[33:49] Cesar explains why quantum computers are fundamentally better at some problems than classical computers.

[37:33] Cesar defines the molecular geometry problem

[39:50] Cesar speaks of the beginning of Quantum Computing.

[42:53] Cesar talks about a recent major breakthrough.

[45:48] Cesar talks about the complexity of photonics.

[48:28] Cesar shares the challenge of speed.

[52:10] Kevin and Sebastian share the highlights of an interesting conversation with Cesar A. Rodriguez Rosario.

Resources:
Visit The New Quantum Era Podcast

Google's 2019 quantum supremacy experiment
A classical attack on Google's supremacy claim
An overview of Quantum supremacy
The variational quantum eigensolver algorithm paper from Alan Asperu-Guzik's group at Harvard
Eddie Farhi and Jeffrey Goldstone's Quantum Approximate Optimization Algorithm paper
Nature paper on error correction on spin qubits in diamond
The Chip, by T. R. Reid is a terrific book for understanding the early history of classical computing.

Tweetables and Quotes:

“You can use some qubits and their quality is really, really good. You can connect them very, very efficiently, and you can connect as many as you want, in a way that scales, we have to do all those things… and nobody has cracked the code for all these bullet points.” — Cesar A. Rodriguez Rosario

“Ideally, what's going to happen is that once we have the scalable error corrected qubits and all that, then you don't have to be a theorist anymore, and then I'm going to be a full-time quantum engineer and that will be healthy, I want that to happen since that would mean that the industry succeeded.”  — Cesar A. Rodriguez Rosario

“It's okay, that things are not useful, yet, there's nothing wrong with that, because we're still working towards that.” — Cesar A. Rodriguez Rosario

Key Takeaways:

[8:25] Nick Bronn does a quick introduction about himself.

[9:23] At what point in Nick’s academic career did he find he was attracted to quantum computing rather than the condensed matter physical started to get drawn into the field?

[13:27] When Nick joined IBM, did they have a functioning superconducting qubit? Was there a transmon that was operational at that point? Or was it still building the first one in IBM?

[17:23] How a transmon qubit does its thing?

[20:27] Nick explains the DiVincenzo criteria.

[25:25] Nick explains how you can build whatever wavefunction you want with transmon qubits.

[28:40] Nick mentioned transitioning from experimental to more, such as the theory and the software. What was the motivator for Nick to get more involved in how to program these things?

[33:43] How would Nick recommend somebody who has not done a few decades in the lab doing the kind of necessary work to acquire his intuition on factors and what kind of budget they should have for certain resources to know to avoid one idiom of code versus another? 

[36:27] Is there a way to encourage people to include a Jupyter Notebook with their code in the papers they post to the arxiv?

[41:25] Nick shares about his work in trying to actually create Majorana braiding on the superconducting qubits.

[46:10] Nick talks about other techniques such as variational algorithms.

[48:14] What are we going to see in the short to medium term, what will the big breakthroughs be? 

[51:01] Nick is trying to simulate Majoranas state using the qubits. Would there be any learnings there or applications that would help in terms of error mitigation or error correction? 

[53:33] Nick shares his thoughts on Majoranas and the very strong theoretical justification for their existence. 

[56:16] Nick encourages physicists to learn to code, and developers to learn physics.

[58:01] Sebastian and Kevin share the highlights of an amazing conversation with Nick Bronn.

Links
Nick's video on error correction
DiVincenzo's criteria
Qiskit site, an incredible resource for learning!
The paper Nick mentioned by Bryce Fuller and Antonio Mezzocapo, Second-quantized fermionic operators with polylogarithmic qubit and gate complexity
The paper where Nick collaborated with David Pekker on simulating Majorana braiding on IBM's superconducting qubits. 

Tweetables and Quotes:

“We're supposed to think about quantum computers as being a digital type of thing, you have these fundamental universal gates set, and that is not necessarily a continuous thing. But if you understand how the physics of these microwave operations work, then sometimes you can frame certain problems in a more efficient way, and reduce the overall amount of error that you incur.” — Nick Bronn

“We do have a large community of quantum computing users. And, and it's kind of, it's moving so fast that it's not even, it's not very easy to kind of convey what the best way to do everything is, l there's no standard operating procedure, no kind of best practices.” — Nick Bronn

“Physicists are not good coders, but just know enough to be dangerous.” — Nick Bronn

“What is incredibly interesting about the condensed matter of physics is that they allow you to understand the properties of materials, even crazy materials, like superconductors with relatively simple models.” — Nick Bronn