S2E11 - Quantum Cryptography with Dr Mina Doosti

How can we use quantum mechanics to generate unbreakable encryption? Take a listen to Season 2, Episode 11 of insideQuantum to find out!

This week, Dr Mina Doosti, a Chancellor’s Fellow at the University of Edinburgh, tells us all about her work in quantum cryptography, and what it will take to make the communications systems of the future quantum secure.

Dr Doosti obtained her undergraduate degree from the Sharif University of Technology, followed by a PhD at the University of Edinburgh, and she is now a Chancellor’s Fellow at the University of Edinburgh in the School of Informatics.

Transcript

🟢 Ellen Derbyshire (00:06): Hello there and welcome to insideQuantum, the podcast telling the human stories behind the latest developments in quantum technologies. I’m Dr. Ellen Derbyshire , and I’ll be your host for this episode.

(00:18): In previous episodes, we’ve talked about quantum communication, machine learning, and benchmarking quantum systems, but we haven’t yet delved deeply into the area of security of quantum information and the necessary research area of quantum cryptography. It turns out that quantum devices, although challenging to control, have the potential to be much more secure than classical devices. And understanding the fundamental reasons for this are important for securely transferring information. Today’s guest works in the field of quantum cryptography and her research has encompassed topics such as machine learning and quantum foundations. It’s a pleasure to be joined today by Dr. Mina Doosti, a Chancellor’s fellow at the University of Edinburgh. Hi Mina, and thank you for joining us today.

🟣 Mina Doosti (01:06): Hi, Ellen. Thank you so much. Thanks for having me. It’s a pleasure.

🟢 Ellen Derbyshire (01:11): It’s a real pleasure to have you here. So before we get into the details of security of quantum information, let’s first talk about your journey to this point. Maybe you could tell us a bit about how you got to where you are now and give us a quick summary of your career up until now.

🟣 Mina Doosti (01:29): Okay, thank you. Should I do the long version or short version?

🟢 Ellen Derbyshire (01:34): As broad or narrow as you want.

🟣 Mina Doosti (01:37): Okay. So I guess the long version is that I had a love for mathematics and physics from very young age, and I think this interest is channeled into the love for physics mostly in high school. I got interested in big words like string theory and particle physics and things like that. Long story short, when I was going to pick a major for my undergrad, I was very sure that I’m going to do physics. And actually one of the reasons was that I wanted to understand this mysterious theory called quantum mechanics, which talks about the physics of subatomic world and things like that. So yeah, so that was the plan from high school. So when I went to undergrad, I went to Sharif University of Technology in Iran, which is kind of like the MIT of Iran. So it’s the best technical university in the country.

(02:49): And I studied physics and I think I was so much waiting for taking the quantum mechanics course to finally understand what’s it about. And when I finally had it, I felt like this is not enough, so I need to learn more about it. And I think, so this was my journey more in the undergrad. But the thing is that I don’t think my plan was to work on quantum computing from the beginning. It was mostly that I wanted to do quantum field theory and particle physics and high energy physics. But then I more tried to learn about what people do in this field on the research side. And further and further I felt like maybe this is not what I want to do, or I found the problem that people are working on the research side are not as exciting to me, although the big picture of the field is very exciting, but the day-to-day work and the things you have to work on and the kind of math you have to solve is not what I exactly want.

(03:58): So I tried to look for other areas of physics that I might find exciting. So I tried cosmology, biophysics, lots of different things, and I was kind of becoming desperate because I couldn’t find, none of them would really click to me as something I would really like. And what I thought that maybe I’d give it a go with quantum computing. So we had a good quantum information and computing group in the university, and I thought maybe I do my undergrad project in that area. And well, there was a professor with whom I did my undergrad project Laleh Memarzadeh, and I really liked her as well. So I was like, why not? Let’s give it a go. And while I started my undergrad project that summer, there was a conference going on, an international conference on quantum information and quantum computing in Iran. And she suggested to me that maybe you should come to this conference and see what people are doing.

(05:07): And because it’s a big conference, a lot of international researchers are coming from all over the world. And I was like, yeah, that sounds good. So I went to the conference and that’s I think where everything clicked. I became so happy. I was listening to the talks, not understanding most of it, but one thing that it really resonated with me was that how diverse is the field? And I felt like there’s a lot of innovations there. So the kind of questions people are working on.

🟢 Ellen Derbyshire (05:46): Do you mean diverse scientifically?

🟣 Mina Doosti (05:49): Yeah, diverse scientifically and the kind of tools and ideas they use in the research. It seems very interesting to me. And then I was like, okay, this is what I want to do finally. And it had lots of quantum as well, which I loved. So it was perfect. So then I did my master also in the same university, but with Vahid Karimipour, who is the head of this quantum information and computing group in Iran.

(06:28): He’s also kind of the father of quantum information in Iran is one of the first person who have worked on this area. So I did my master’s there and towards the end of it, I wanted to go abroad for my PhD. So I decided for that, that I didn’t want to continue doing my PhD there. I wanted to go abroad and get the experience of working in an international environment and having the chance to collaborate with more researchers. So that was the plan. I applied for a couple of places, including Professor Elham Kashefi, who she became my PhD supervisor. And I actually met Elham in Iran before I applied to her.

🟢 Ellen Derbyshire (07:21): Oh I didn’t know that.

🟣 Mina Doosti (07:21): Yeah, maybe a year before that I applied to her or something. She came for a conference, she gave a talk and she was talking about how we can verify and check these quantum computers when we have them and things like that, which to me sounded really cool at the time. And I was like, oh, this stuff are really interesting. So I applied and well, she accepted me and I remember the first interview we had together. I also knew that she works on quantum cryptography a lot. I told her that, look, I’m really also interested in doing quantum cryptography. And she asked me, oh, okay, good. How much cryptography do you know? And I said, nothing.

🟢 Ellen Derbyshire (08:17): Well, it’s good to be honest.

🟣 Mina Doosti (08:19): Yes, but I said, I’m really, really interested to learn. And that was another thing that I really kind of had a passion for ever since I was a kid, like codes and riddles and things like that. And I always liked to know what’s it about, more in depth scientifically. So yeah, so she was really one of the people who I really wished that she would accept me as a PhD student. And she did. Then I did my PhD with her at the University of Edinburgh, which was a really amazing journey, I met lots of people including yourself.

(09:04): And so when I finished my PhD, I got two different fellowship, one from Perimeter Institute and one from QuICS in Maryland. And actually, so I was really close to go to perimeter. It was my dream institute again from back the day when I was very young. But for some reason, after lots of thinking and thinking and thinking, I decided to go to Maryland. So I accepted the Maryland offer, but I actually didn’t get there. I waited for the visa for a very long time, more than a year for a visa that never came. And well, I couldn’t actually join in person, but while I was waiting, I joined as a postdoc at the University of Edinburgh. So I worked as a postdoc there till kind of like now. I recently applied for a Chancellor’s fellow position in the University of Edinburgh, and I became a Chancellor Fellow officially from a couple of days ago.

🟢 Ellen Derbyshire (10:29): Yes, congratulations!

🟣 Mina Doosti (10:31): Thank you. So yeah, I think that was the long version.

🟢 Ellen Derbyshire (10:37): There’s a lot to delve into there. So I have a few follow-ups. So one, you said that you were interested in quantum mechanics from before your undergrad, which is interesting because a lot of people don’t hear too much about it. Is there a reason you knew kind of more or…you knew about this area of physics?

🟣 Mina Doosti (11:07): So I think it was because as I said, I was reading lots of popular science books. I was trying to also read some non-popular science books, but it was very hard, so I couldn’t really understand them very well. But I was trying. But I think the reason I got into quantum mechanics was that I was reading about string theory and particle physics, and then I realized that in order to understand those first you need to understand quantum mechanics. So that became my passion for it. I was like, okay, so this is something I really need to understand. And as I said, even when I went to undergrad when I had the quantum mechanic course, I was like, no, that’s not it. I have to really understand it. So I actually learned quantum mechanics from Feynman lectures.

🟢 Ellen Derbyshire (12:04): Oh, a great way to learn.

🟣 Mina Doosti (12:07): Yes.

🟢 Ellen Derbyshire (12:07): Yeah. I think I had a very similar journey with particle physics and all the pop science books. Definitely. So also maybe for the listeners, they might not know so much about QuICS, but as you said, it is in Maryland in the US, and it’s so horrible that you had to wait such a long time and still didn’t get a visa. Is this something that you come across a lot and that people come across a lot within the field?

🟣 Mina Doosti (12:40): Yes. I think one thing might be that, well, because of the field, and the other thing is particularly I come across a lot, I think because, well, I’m from Iran, I have to apply for Visa for almost everywhere. And it takes a long time, usually. Well, it never took so long, but in this particular case, but I mean, I think it’s a big limitation for researchers. It puts a big hold on you being able to travel for conferences and visits and things like that.

🟢 Ellen Derbyshire (13:24): Definitely.

🟣 Mina Doosti (13:26): But yeah, what can you do?

🟢 Ellen Derbyshire (13:28): Well. Yeah, no, but definitely it’s a huge barrier, especially from researchers who are coming from Iran and as you said, with Sharif University. I mean Iran is so, there’s so many great students and researchers coming from there too. Yeah, that’s true. So yeah, it’s a real shame, something that I wish we could change in this podcast conversation, but of course we can’t. So also, you mentioned how your interests have always been learning and code breaking and quantum mechanics as you mentioned. But I also wonder what you think you might be doing if you weren’t a quantum cryptographer, because I think that you could be so many different things.

🟣 Mina Doosti (14:22): Well, thanks. I think the answer to this question is not that hard for me, but it might be very different from what you expect or what the audience expects. So if I wasn’t doing what I’m doing now, at the moment, I was probably a musician, and I think I almost went on that road in high school. I was really close to go to art high school instead of the normal high school. But then I decided that maybe this is something that I’m better at. To be fair, it was also a bit of fear of not being, maybe like good enough as a violin player while in science it was more comfortable. But I really love that and I still love that, and music is still a big part of my life. So I think, yeah, I would probably be a musician in an alternative life.

🟢 Ellen Derbyshire (15:34): Yes. I was hoping you would say that. For the audience, Mina is a very good musician. She writes and plays beautiful music.

🟣 Mina Doosti (15:43): Thank you.

🟢 Ellen Derbyshire (15:44): And I think, I mean it’s sometimes said also that if you are good at music, then you might also be good at maths. So they do go hand in hand, I would say. Okay. So we’ve talked a lot about your journey until this point, and I would love to hear more about your work now. So if I were to summarize your research work in a single phrase, I might very broadly choose quantum cryptography or quantum cryptanalysis. And perhaps you can break down for us what these two words mean and the difference between them and talk a bit more about what you’re working on specifically.

🟣 Mina Doosti (16:27): Okay, that’s a great question. That’s a great way to put it. Yeah, let’s break it - so quantum cryptography, I’d say is the science and art of making systems and schemes that have some cryptographic functionality. And in order to do that, you use quantum systems or so it can be only quantum systems or it can be hybrid combination of classical system or quantum system. And the objective is to design a secure system. So security always doesn’t mean encryption, which is when you have a message and you hide it with a secret key and then no one can open it or read it, and then somebody else decrypt it, meaning that reads the message. So that’s maybe cryptography for most of people when they think about it, but it’s a lot more than that. So there’s a lot of other things you can do that they need to be secure.

(17:41): For instance, if I don’t see you on the zoom, and I want to make sure that I’m talking to you, there are cryptographic methods that can ensure that, and that’s called entity authentication, for instance. So there’s a lot of functionality, but the goal is to make these secure schemes in a way that we can trust them and we can achieve what we want to do. And quantum cryptography is using the laws of physics, in particular quantum mechanics and quantum systems to make that happen. So that’s quantum cryptography the way I define it.

🟢 Ellen Derbyshire (18:19): I think that’s a very good summary.

🟣 Mina Doosti (18:20): Thank you. So quantum crypto analysis on the other hand is analyzing the security of these quantum or quantum classical systems. But as you can imagine for analyzing security, we need to understand how we can break the systems. So as much as quantum cryptography is about designing system or making systems secure, quantum cryptanalysis would be about breaking systems.

(18:51): So you would have to think of yourself as a hacker and try to use quantum devices and classical algorithms, everything you have to break the systems because when you break them, you would understand how to make them secure again, and that’s how it goes, I think. So in that area, I think that’s maybe where my research focuses on more and more recently is that. So one thing that it becomes related to breaking systems is how you would learn from a quantum system. Because, imagine I have a cryptographic scheme and somehow I can embed it into a quantum system. You can assume that it’s a large system including lots of qubits and what you want. So now that your crypto system is embedded in such a system, in order to break it, you would have to learn this process. So you would have to try to interact with it through some input and output and then try to extract that secret parameter or try to be able to predict how it would behave. And that’s how cryptanalysis becomes relevant to learning and things like machine learning and learning theory, which is the more recent focus of my recent work. So I’m trying to somehow formally relate these two fields together and use learning algorithms as a tool for cryptanalysis.

🟢 Ellen Derbyshire (20:48): I see. Thank you. No, that’s really cleared up a lot for me as well, especially about cryptanalysis. And it sounds like you need a lot of creativity and curiosity to also try to break these systems, and I can completely see where learning theory comes into that. So what would you say that the fundamental goals of your field of research are? So you’ve mentioned that we want to have secure cryptographic functions or protocols, and is it the hope that by understanding the fundamental nature of these quantum systems, i e, by quantum learning, we can develop stronger security tools?

🟣 Mina Doosti (21:44): Yes, that’s correct. I think one of the main goals of the field of quantum cryptography and quantum cryptanalysis is that you build secure system with the help of quantum devices. But I would argue that this is more than just about security, it’s about reliability in the long term. So if I want to break this down, I have to first explain that a lot of modern cryptography is based on some assumptions. So these assumptions come from mathematics and come from the fact that some problems we believe that they’re hard to solve.

(22:34): And there’s usually some sort of asymmetry between that. These are problems that are hard to solve, but they’re easy to check somehow. So a lot of modern cryptography is based on that. But some of these problems, it turned out that if you have a quantum computer, they’re not hard to solve anymore. And that has changed a lot. The whole field of cryptography as we know, and quantum cryptography becomes really important. So now there’s other sets of assumption that we believe quantum computers cannot break, but still the idea of how I can make a crypto systems that is secure and I don’t need to rely on such assumptions is also a big part of quantum cryptography. And that comes with the fact that in quantum cryptography, a lot of times you can have provable security, so you can actually prove relying on some minimal assumption and some assumptions that come from the physics itself, that your system is secure and it will remain secure, which is I think an important and interesting idea.

(23:53): So yes, I mean designing such systems is a big question of the field, but also because we want to actually use the systems in practice and make it accessible, it is very important to design these classical systems or hybrid quantum classical systems that are efficient and are practical to use because otherwise they will only stay in the lab. So that’s from the practical side of it, but to me personally, I think there’s one more goal to the field, and that is that crypto brings a lot of nice and interesting math with it, which we can actually use to understand quantum information and quantum mechanics better. So we can use crypto and all the crypto toolkits as a good way to attack other problems in other areas, again, like learning theory or things like that. So I think another goal of the field to me is something like that.

🟢 Ellen Derbyshire (25:10): Right. So essentially, so it’s not just security, it’s reliability and efficiency in the long term, but as you mentioned, it’s also beautiful maths and developing new techniques and tools that we can apply to different situations. And so what kind of theoretical tools do you use in your field? Maybe you could discuss a few of them on a high level.

🟣 Mina Doosti (25:44): So I use a lot of tools from quantum information theory, things like entropy, the concept of uncertainty and something we call uncertainty relation or uncertainty, entropic relation. So I mean to explain it very, very roughly, it’s the idea that we have uncertainty in quantum mechanic that we cannot observe some observables at the same time with arbitrary precision. And this idea can be formalized in the information theory way and it will give you some inequalities which you can use for your proofs and lots of things like that, which is actually really beautiful. So yeah, there’s that and other tools also in quantum information, but broadly quantum information. I use a lot of tools, techniques, proof techniques and tricks and everything from classical cryptography. And so a lot of also concepts in cryptography, they actually come from classical notion and then we try to redefine them in the quantum world and see how they mean what they mean now, when we have quantum parties playing these games or quantum devices that act differently, but it’s often very important to know the classical ones.

🟢 Ellen Derbyshire (27:30): Right! And so these were things that you didn’t know about before you started your PhD, presumably - the cryptography side?

🟣 Mina Doosti (27:38): Yes. No, I didn’t. No, I didn’t. I learned along the way these things and I found them very exciting, very interesting. As I said, to me, crypto is a lot of nice maths and that’s what I like about it. Maybe that’s what I even more care about it than what these functionalities actually achieve at the end, but…

🟢 Ellen Derbyshire (28:06): It’s more about the journey.

🟣 Mina Doosti (28:08): Yes, it’s more about the journey. Exactly.

🟢 Ellen Derbyshire (28:12): So that kind of leads onto the next question I had because your work really lies between cryptography and quantum computing or quantum information theory, or more broadly, computer science and physics, but very broadly. So have you found that these communities have a lot in common? Has it been challenging to combine them? And I mean, did you feel a bit overwhelmed at the start or just excited when you were learning?

🟣 Mina Doosti (28:43): Absolutely, yes. I mean, I feel these communities do have similarities, lots of similarities, but they’re also very different. I think they’re very different in their mindsets, I would say even. So in computer science, what I find out is that it’s a lot about formalizing, simplifying, clarifying the problem, and then attack it step by step almost like a machine solving the problem, but it gives you a very structured, nice, beautiful way of thinking about problems and solving them and writing proofs, which I didn’t know and I learned, and I’m really grateful that I learned how to do that. On the other hand, physics is a lot about, at least to me, is a lot about intuition. It’s a lot about the big picture, and it’s a lot about you close your eyes and you try to imagine what’s happening even in a small research problems. So even when I’m thinking about unitary’s and channels as a physicist, I close my eyes and try to understand how it would behave and this difference in the mindset, I think it spreads out throughout the field and throughout all the researchers who are working on the field. And it makes them very different, but also very, very interesting and very interesting and challenging at the same time to interact with each other. And I think in that respect, I was lucky that I got to be both. So I learned how to be a computer scientist, and I was born and bred a physicist.

🟢 Ellen Derbyshire (30:38): Yes. I love that. Yeah. I think you touched on something really nice there, the differences between them and how it spreads. And it seems to me, at least in within quantum computing, you kind of have more of an overlap, maybe the intuition and the imagination of the physicists. They’re trying to also become a bit more concrete as the computer scientists and vice versa, whereas in the wider fields, they might be a little more separate. I don’t know.

🟣 Mina Doosti (31:18): Yeah, I think I agree, and I think that’s what I like about the field, and I think it would be very important if we keep doing this.

🟢 Ellen Derbyshire (31:28): Yes. Definitely.

🟣 Mina Doosti (31:29): And we keep talking to each other because for instance, maybe some of the proof that I would write as a physicist, if I read them now, I would be like, no, that’s not a proof, that’s not formal.

🟢 Ellen Derbyshire (31:44): So true. Yeah.

🟣 Mina Doosti (31:47): It really changes you when you do it properly. You would realize that. Yes. So no, I think it’s very important that they talk to each other and it’s really beautiful that in this field we have a space for both the species to exist to interact. Yeah, I think I’m kind of hopeful, and I think we got, even in this field over the years, we got better at working with each other and interacting with each other, both physicists and computer scientists.

🟢 Ellen Derbyshire (32:27): Yeah, I think I would agree with that as well. Definitely. And hopefully it can continue that way.

🟣 Mina Doosti (32:35): Yes.

🟢 Ellen Derbyshire (32:37): Okay. So we ask this question to every guest. Physics has historically been a field dominated by white cisgender men, and there’s still a long way to go before we reach a level playing field, so to speak. In your experience, do you think things are changing at all for the better? And perhaps this also touches a bit on the barriers for all different types of people in physics in general?

🟣 Mina Doosti (33:11): I see now that’s a great question. So I think I agree with the first part of it that physics is at least very visibly male dominated. I’d say computer science is even worse in a sense that I think physics is slightly better. You see more diversity in physics than maybe in computer science, but that’s maybe my view. I’m not sure a hundred percent if I’m right. Well, so first of all, personally to me, I think this might maybe sound weird, but it never seemed like a barrier. So it was like seeing less women in the field never stopped me to want that. And if anything, maybe it made me more serious about that I want a place in that space and I want to be a part of it, but I understand that this doesn’t maybe apply to everyone. So maybe young people, when they see less people like themselves being in the high ranked positions in physics or computer science, they might be discouraged, but I think maybe this is the first thing that we should try to learn, and maybe we should try to teach to our children or younger people that we would obviously try to change that, but the main change should come internally from us, and this should not stop us, so we should go for it.

(35:11): No matter if there’s people like us or not, people like us, we should go for it if that’s something we want. And maybe if this comes from within, then it can change things for better in the larger scale. But I mean, that’s my view. Maybe I got this from the fact that I’m coming from a country that gave me a very thick skin to injustice towards women or things like that. But yeah, so I think maybe that’s the way to attack this problem. And I think that second part of the questions that “Do you think it’s got better?”. Yeah, I think it got much better. So I think maybe the main problem is not that you see fewer women, is when you see that in some cases women are not taken seriously. Maybe that’s what is most discouraging and problematic. And I think in that respect, it’s getting much better. And you see women in high ranks and positions in physics and in computer science and who are respected and who do great work, and a lot of people realize that they’re doing great work. And I think this is great. It’s great to see. It’s very encouraging, and it’s great to see that this is changing. Maybe years ago, this wasn’t the case, even though you had really a strong record and you were very distinguished in your field, even then you wouldn’t have been taken as seriously. But I don’t think that’s the case now, and I think it’s a good change.

🟢 Ellen Derbyshire (37:09): Yeah, no, I do see what you mean.

(37:15): I think I can completely understand your perspective, and I think that you are right that perhaps the amount of women in the field hasn’t changed that much, but the way that the people’s research is viewed has changed. I do think that when you are the minority in any situation, you can sometimes question any reaction that people have to you and view it in terms of that. So it can sometimes play with your mind a bit, and maybe that’s where that thick skin you talked about comes in very handy because you’ve had to face so many different things that you’re very determined just to get through your goal and say, look, I know this. I know this about myself, and I’m going to do it regardless.

🟣 Mina Doosti (38:10): Yeah, I think so. Yeah, I think it’s really important that people face this problem internally, and I think maybe ideally when we would overcome this problem is when we define ourselves as scientists, irrespective of gender, but really respective of it, not trying to maybe empower one minority over the other one, or I think that’s not the way it should go. The way it should go is that I’m a scientist who cares what gender do I have, and maybe that’s the best practice to go towards, but I understand that maybe in order to get there, we need to have

🟢 Ellen Derbyshire (39:04): A bit more representation.

🟣 Mina Doosti (39:05): Yeah, yeah. Have a bit more representation.

🟢 Ellen Derbyshire (39:08): It becomes easier to do that. But I think you’re right. I think that’s definitely the end goal where we can just talk about science without even having to have this conversation. That would be incredible. Okay, so now for the super deep question, talking about internal things, if you could go back in time and give yourself just one piece of advice, what would it be? It can be a bit more than one.

🟣 Mina Doosti (39:38): Okay. That’s a good cheat. One piece of advice. So the first piece of advice I would give myself is that don’t take any advice.

🟢 Ellen Derbyshire (39:51): I love that.

🟣 Mina Doosti (39:53): Yeah, I think I’ve came a long way with just that advice of not taking any advice, but maybe if I am able to cheat a little bit, I would say one more thing, and that’s don’t lose your passion, because I feel like when you’re passionate about something, it doesn’t matter even how good at it you are, you’ll get good at it, and it’s not even about being good at something. I think if you have passion, you’re doing something that’s probably meaningful, whereas if you lose your passion, you’ll just become a brick in the wall, even in science. So I think the way to go is just, I mean, at least to me, I would say to me, I’ll probably say to other people, don’t lose your passion.

🟢 Ellen Derbyshire (40:48): I think that’s a really good piece of advice. If you lose it, then try to find it again.

🟣 Mina Doosti (40:56): I agree. Yes, exactly.

🟢 Ellen Derbyshire (40:59): Okay, so if our audience wants to learn more, where can they find you on the internet? We can put links in our website.

🟣 Mina Doosti (41:07): So I am on LinkedIn, but other than that, I’m not a very social media person publicly at least, but I recently put up a website. It’s minadoosti.github.io. Yeah, you can find more about my research and some of the stuff there.

🟢 Ellen Derbyshire (41:28): Perfect. Thank you very much to Dr. Mina Doosti for her time today.

🟣 Mina Doosti (41:32): Thanks a lot for having me.

🟢 Ellen Derbyshire (41:34): Thanks also to the Unitary Fund for supporting this podcast. If you’ve enjoyed today’s episode, please consider liking, sharing and subscribing wherever you like to listen to your podcasts. It really helps get our guest stories out to as wide an audience as possible. I hope you’ll join us again for our next episode. And until then, this has been insideQuantum. I’ve been Dr. Ellen Derbyshire , and thank you very much for listening. Goodbye!