Complexity-Theoretic Foundations of Quantum Supremacy Experiments
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Limits on Efficient Computation in the Physical World
Limits on Efficient Computation in the Physical World by Scott Joel Aaronson Bachelor of Science (Cornell University) 2000 A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Computer Science in the GRADUATE DIVISION of the UNIVERSITY of CALIFORNIA, BERKELEY Committee in charge: Professor Umesh Vazirani, Chair Professor Luca Trevisan Professor K. Birgitta Whaley Fall 2004 The dissertation of Scott Joel Aaronson is approved: Chair Date Date Date University of California, Berkeley Fall 2004 Limits on Efficient Computation in the Physical World Copyright 2004 by Scott Joel Aaronson 1 Abstract Limits on Efficient Computation in the Physical World by Scott Joel Aaronson Doctor of Philosophy in Computer Science University of California, Berkeley Professor Umesh Vazirani, Chair More than a speculative technology, quantum computing seems to challenge our most basic intuitions about how the physical world should behave. In this thesis I show that, while some intuitions from classical computer science must be jettisoned in the light of modern physics, many others emerge nearly unscathed; and I use powerful tools from computational complexity theory to help determine which are which. In the first part of the thesis, I attack the common belief that quantum computing resembles classical exponential parallelism, by showing that quantum computers would face serious limitations on a wider range of problems than was previously known. In partic- ular, any quantum algorithm that solves the collision problem—that of deciding whether a sequence of n integers is one-to-one or two-to-one—must query the sequence Ω n1/5 times. -
Marching Towards Quantum Supremacy
Princeton Center for Theoretical Science The Princeton Center for Theoretical Science is dedicated to exploring the frontiers of theory in the natural sciences. Its purpose is to promote interaction among theorists and seed new directions in research, especially in areas cutting across traditional disciplinary boundaries. The Center is home to a corps of Center Postdoctoral Fellows, chosen from nominations made by senior theoretical scientists around the world. A group of senior Faculty Fellows, chosen from science and engineering departments across the campus, are responsible for guiding the Center. Center activities include focused topical programs chosen from proposals by Princeton faculty across the natural sciences. The Center is located on the fourth floor of Jadwin Hall, in the heart of the campus “science neighborhood”. The Center hopes to become the focus for innovation and cross-fertilization in theoretical natural science at Princeton. Faculty Fellows Igor Klebanov, Director Ned Wingreen, Associate Director Marching Towards Quantum Andrei Bernevig Jeremy Goodman Duncan Haldane Supremacy Andrew Houck Mariangela Lisanti Thanos Panagiotopoulos Frans Pretorius November 13-15, 2019 Center Postdoctoral Fellows Ricard Alert-Zenon 2018-2021 PCTS Seminar Room Nathan Benjamin 2018-2021 Andrew Chael 2019-2022 Jadwin Hall, Fourth Floor, Room 407 Amos Chan 2019-2022 Fani Dosopoulou 2018-2021 Biao Lian 2017-2020 Program Organizers Vladimir Narovlansky 2019-2022 Sergey Frolov Sabrina Pasterski 2019-2022 Abhinav Prem 2018-2021 Michael Gullans -
Closed Timelike Curves Make Quantum and Classical Computing Equivalent
Closed Timelike Curves Make Quantum and Classical Computing Equivalent Scott Aaronson∗ John Watrous† MIT University of Waterloo Abstract While closed timelike curves (CTCs) are not known to exist, studying their consequences has led to nontrivial insights in general relativity, quantum information, and other areas. In this paper we show that if CTCs existed, then quantum computers would be no more powerful than classical computers: both would have the (extremely large) power of the complexity class PSPACE, consisting of all problems solvable by a conventional computer using a polynomial amount of memory. This solves an open problem proposed by one of us in 2005, and gives an essentially complete understanding of computational complexity in the presence of CTCs. Following the work of Deutsch, we treat a CTC as simply a region of spacetime where a “causal consistency” condition is imposed, meaning that Nature has to produce a (probabilistic or quantum) fixed-point of some evolution operator. Our conclusion is then a consequence of the following theorem: given any quantum circuit (not necessarily unitary), a fixed-point of the circuit can be (implicitly) computed in polynomial space. This theorem might have independent applications in quantum information. 1 Introduction The possibility of closed timelike curves (CTCs) within general relativity and quantum gravity theories has been studied for almost a century [11, 15, 13]. A different line of research has sought to understand the implications of CTCs, supposing they existed, for quantum mechanics, computation, and information [9, 8, 5]. In this paper we contribute to the latter topic, by giving the first complete characterization of the computational power of CTCs. -
Computational Pseudorandomness, the Wormhole Growth Paradox, and Constraints on the Ads/CFT Duality
Computational Pseudorandomness, the Wormhole Growth Paradox, and Constraints on the AdS/CFT Duality Adam Bouland Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, 617 Soda Hall, Berkeley, CA 94720, U.S.A. [email protected] Bill Fefferman Department of Computer Science, University of Chicago, 5730 S Ellis Ave, Chicago, IL 60637, U.S.A. [email protected] Umesh Vazirani Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, 671 Soda Hall, Berkeley, CA 94720, U.S.A. [email protected] Abstract The AdS/CFT correspondence is central to efforts to reconcile gravity and quantum mechanics, a fundamental goal of physics. It posits a duality between a gravitational theory in Anti de Sitter (AdS) space and a quantum mechanical conformal field theory (CFT), embodied in a map known as the AdS/CFT dictionary mapping states to states and operators to operators. This dictionary map is not well understood and has only been computed on special, structured instances. In this work we introduce cryptographic ideas to the study of AdS/CFT, and provide evidence that either the dictionary must be exponentially hard to compute, or else the quantum Extended Church-Turing thesis must be false in quantum gravity. Our argument has its origins in a fundamental paradox in the AdS/CFT correspondence known as the wormhole growth paradox. The paradox is that the CFT is believed to be “scrambling” – i.e. the expectation value of local operators equilibrates in polynomial time – whereas the gravity theory is not, because the interiors of certain black holes known as “wormholes” do not equilibrate and instead their volume grows at a linear rate for at least an exponential amount of time. -
NP-Complete Problems and Physical Reality
NP-complete Problems and Physical Reality Scott Aaronson∗ Abstract Can NP-complete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adia- batic algorithms, quantum-mechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, Malament-Hogarth spacetimes, quantum gravity, closed timelike curves, and “anthropic computing.” The section on soap bubbles even includes some “experimental” re- sults. While I do not believe that any of the proposals will let us solve NP-complete problems efficiently, I argue that by studying them, we can learn something not only about computation but also about physics. 1 Introduction “Let a computer smear—with the right kind of quantum randomness—and you create, in effect, a ‘parallel’ machine with an astronomical number of processors . All you have to do is be sure that when you collapse the system, you choose the version that happened to find the needle in the mathematical haystack.” —From Quarantine [31], a 1992 science-fiction novel by Greg Egan If I had to debate the science writer John Horgan’s claim that basic science is coming to an end [48], my argument would lean heavily on one fact: it has been only a decade since we learned that quantum computers could factor integers in polynomial time. In my (unbiased) opinion, the showdown that quantum computing has forced—between our deepest intuitions about computers on the one hand, and our best-confirmed theory of the physical world on the other—constitutes one of the most exciting scientific dramas of our time. -
ANGLAIS Durée : 2 Heures
CONCOURS D’ADMISSION 2021 FILIERE UNIVERSITAIRE INTERNATIONALE FORMATION FRANCOPHONE FUI-FF_ Session 2_Printemps Épreuve n°4 ANGLAIS Durée : 2 heures L’utilisation de dictionnaires et traducteurs électroniques n’est pas autorisée pour cette épreuve Physicists Need to Be More Careful with How They Name Things The popular term “quantum supremacy,” which refers to quantum computers outperforming classical ones, is uncomfortably reminiscent of “white supremacy” Adapted from Scientific American, By Ian Durham, Daniel Garisto, Karoline Wiesner on February 20, 2021 In 2012, quantum physicist John Preskill wrote, “We hope to hasten the day when well controlled quantum systems can perform tasks surpassing what can be done in the classical world.” Less than a decade later, two quantum computing systems have met that mark: Google’s Sycamore, and the University of Science and Technology of China’s Jiǔzhāng. Both solved 5 narrowly designed problems that are, so far as we know, impossible for classical computers to solve quickly. How quickly? How “impossible” ? To solve a problem that took Jiǔzhāng 200 seconds, even the fastest supercomputers are estimated to take at least two billion years. Describing what then may have seemed a far-off goal, Preskill gave it a name: “quantum supremacy.” In a blog post at the time, he explained “I’m not completely happy with this term, 10 and would be glad if readers could suggest something better.” We’re not happy with it either, and we believe that the physics community should be more careful with its language, for both social and scientific reasons. Even in the abstruse realms of matter and energy, language matters because physics is done by people. -
Quantum Lower Bound for Recursive Fourier Sampling
Quantum Lower Bound for Recursive Fourier Sampling Scott Aaronson Institute for Advanced Study, Princeton [email protected] Abstract One of the earliest quantum algorithms was discovered by Bernstein and Vazirani, for a problem called Recursive Fourier Sampling. This paper shows that the Bernstein-Vazirani algorithm is not far from optimal. The moral is that the need to “uncompute” garbage can impose a fundamental limit on efficient quantum computation. The proof introduces a new parameter of Boolean functions called the “nonparity coefficient,” which might be of independent interest. Like a classical algorithm, a quantum algorithm can solve problems recursively by calling itself as a sub- routine. When this is done, though, the algorithm typically needs to call itself twice for each subproblem to be solved. The second call’s purpose is to uncompute ‘garbage’ left over by the first call, and thereby enable interference between different branches of the computation. Of course, a factor of 2 increase in running time hardly seems like a big deal, when set against the speedups promised by quantum computing. The problem is that these factors of 2 multiply, with each level of recursion producing an additional factor. Thus, one might wonder whether the uncomputing step is really necessary, or whether a cleverly designed algorithm might avoid it. This paper gives the first nontrivial example in which recursive uncomputation is provably necessary. The example concerns a long-neglected problem called Recursive Fourier Sampling (henceforth RFS), which was introduced by Bernstein and Vazirani [5] in 1993 to prove the first oracle separation between BPP and BQP. Many surveys on quantum computing pass directly from the Deutsch-Jozsa algorithm [8] to the dramatic results of Simon [14] and Shor [13], without even mentioning RFS. -
Quantum Error Correction1
QUANtum error correction1 Quantum Error Correction James Sayre M.D. Department of Physics, University of Washington Author Note James Sayre Department of Physics University of Washington QUANtum error correction2 Abstract Quantum Computing holds enormous promise. Theoretically these computers should be able to scale exponentially in speed and storage. This spectacular capability comes at a price. Qubits are the basic storage unit. The information they store degrades over time due to decoherence and quantum noise. This degradation must be controlled if not eliminated. An entire field of inquiry is devoted to this is the function of quantum error correction. This paper starts with a discussion of the theoretical underpinnings of quantum error correction. Various methods are then described in detail. Keywords: Quantum Computing, Error Correction, Threshold Theorem, Shor’s Algorithm, Steane Code, Shor’s Code. QUANtum error correction3 Quantum Error Correction Quantum Computing holds enormous promise. A useful quantum computer will have the capability to perform searches, factorizations and quantum mechanical simulations far beyond the capability of classical computers [1] [2]. For instance, such a computer running Shor’s algorithm will be able to factorize a large number in polynomial time [3]. This is a tremendous speed advantage over a classical computer that requires exponential time to complete the same task. In theory a computation that requires millions of years on a classical computer could take hours on a quantum computer. The basic building block of a quantum computer is a qubit. Analogous to a classical bit, the qubit combines with other qubits to store numbers. Unlike classical computers, that storage grows exponentially with the number of qubits. -
The Power of Unentanglement
The Power of Unentanglement Scott Aaronson∗ Salman Beigi Andrew Drucker Bill Fefferman MIT MIT MIT University of Chicago Peter Shor MIT Abstract The class QMA (k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Does QMA (k) = QMA (2) for k 2? Can QMA (k) protocols be amplified to exponentially small error? ≥ In this paper, we make progress on all of the above questions. We give a protocol by which a verifier can be convinced that a 3Sat formula of size m is • satisfiable, with constant soundness, given O (√m) unentangled quantum witnesses with O (log m) qubits each. Our protocol relies on the existence of very short PCPs. We show that assuming a weak version of thee Additivity Conjecture from quantum infor- • mation theory, any QMA (2) protocol can be amplified to exponentially small error, and QMA (k)= QMA (2) for all k 2. ≥ We prove the nonexistence of “perfect disentanglers” for simulating multiple Merlins with • one. 1 Introduction Quantum entanglement is often described as a complicated, hard-to-understand resource. But ironically, many questions in quantum computing are easiest to answer assuming unlimited entan- glement, and become much more difficult if entanglement is not allowed! One way to understand this is that Hilbert space—the space of all quantum states—has extremely useful linear-algebraic properties, and when we restrict to the set of separable states we lose many of those properties. -
A Blueprint for Demonstrating Quantum Supremacy with Superconducting Qubits
Title: A blueprint for demonstrating quantum supremacy with superconducting qubits Abstract: Long coherence times and high fidelity control recently achieved in scalable superconducting circuits paved the way for the growing number of experimental studies of many-qubit quantum coherent phenomena in these devices. Albeit full implementation of quantum error correction and fault tolerant quantum computation remains a challenge the near term pre-error correction devices could allow new fundamental experiments despite inevitable accumulation of errors. One such open question foundational for quantum computing is achieving the so called quantum supremacy, an experimental demonstration of a computational task that takes polynomial time on the quantum computer whereas the best classical algorithm would require exponential time and/or resources. It is possible to formulate such a task for a quantum computer consisting of less than a 100 qubits. The computational task we consider is to provide approximate samples from a non-trivial quantum distribution. This is a generalization for the case of superconducting circuits of ideas behind boson sampling protocol for quantum optics introduced by Arkhipov and Aaronson. In this presentation we discuss a proof-of-principle demonstration of such a sampling task on a 9-qubit chain of superconducting gmon qubits developed by Google. We discuss theoretical analysis of the driven evolution of the device resulting in output approximating samples from a uniform distribution in the Hilbert space, a quantum chaotic state. We analyze quantum chaotic characteristics of the output of the circuit and the time required to generate a sufficiently complex quantum distribution. We demonstrate that the classical simulation of the sampling output requires exponential resources by connecting the task of calculating the output amplitudes to the sign problem of the Quantum Monte Carlo method. -
Google Researchers Have Reportedly Achieved Quantum Supremacy
Google Researchers Have Reportedly Achieved Quantum Supremacy HydromedusanIs Rayner chattering and diffidentwhen Laird Munroe intersperse tarmacs: slantly? which PetrFidel collets is unexploited insinuatingly enough? while cislunar Bruno interfold well-timed or trichinises atop. By the optimum solution to computing, nothing in helping and some to their supremacy have researchers reportedly solved All developments are approaching to google researchers have reportedly achieved quantum supremacy for? The google reportedly started thinking along with companies which quantities can. But google research, achieving quantum supremacy. From Google led by John Martinis have demonstrated quantum supremacy for. The device named Jiuzhang reportedly conducted a calculation in 200. Google says an advanced computer has achieved quantum supremacy. Quantum computer and threw its 20-qubit system available online at the must of 2017. I'd advice to put why quantum computers don't have an mat in solving. Did Google achieve quantum supremacy? Achieving quantum supremacy means we've was able to shout one. Who has achieved quantum supremacy? John Martinis Reportedly Leaves Google Quantum Effort. Google team has reportedly reached quantum supremacy with the 53-qubit. Galaxy a big step closer to google researchers do the other side of control signals constantly moving data is. Breakthrough In Quantum Computing Google Researchers. Erik Lucero Research Scientist and Lead Production Quantum Hardware. Try and researchers. Google Reportedly Attains 'Quantum Supremacy' Slashdot. But google achieved quantum supremacy achievement if someone were looking for a momentous achievement is going online. Francisco Luis Bentez on Twitter Google researchers have. The announcement came true a tub that was reportedly published on the NASA website. -
Jim Ormond 212-626-0505 [email protected]
Contact: Jim Ormond 212-626-0505 [email protected] ACM PRIZE AWARDED TO PIONEER IN QUANTUM COMPUTING UT Austin’s Scott Aaronson Recognized for Far-Reaching Theoretical, Technical and Educational Contributions New York, NY, April 14, 2021 – ACM, the Association for Computing Machinery, today announced that Scott Aaronson has been named the recipient of the 2020 ACM Prize in Computing for groundbreaking contributions to quantum computing. Aaronson is the David J. Bruton Jr. Centennial Professor of Computer Science at the University of Texas at Austin. The goal of quantum computing is to harness the laws of quantum physics to build devices that can solve problems that classical computers either cannot solve, or not solve in any reasonable amount of time. Aaronson showed how results from computational complexity theory can provide new insights into the laws of quantum physics, and brought clarity to what quantum computers will, and will not, be able to do. Aaronson helped develop the concept of quantum supremacy, which denotes the milestone that is achieved when a quantum device can solve a problem that no classical computer can solve in a reasonable amount of time. Aaronson established many of the theoretical foundations of quantum supremacy experiments. Such experiments allow scientists to give convincing evidence that quantum computers provide exponential speedups without having to first build a full fault-tolerant quantum computer. The ACM Prize in Computing recognizes early-to-mid-career computer scientists whose research contributions have fundamental impact and broad implications. The award carries a prize of $250,000, from an endowment provided by Infosys Ltd.