CS395T Topics in Quantum and Classical Complexity Theory
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Front Matter of the Book Contains a Detailed Table of Contents, Which We Encourage You to Browse
Cambridge University Press 978-1-107-00217-3 - Quantum Computation and Quantum Information: 10th Anniversary Edition Michael A. Nielsen & Isaac L. Chuang Frontmatter More information Quantum Computation and Quantum Information 10th Anniversary Edition One of the most cited books in physics of all time, Quantum Computation and Quantum Information remains the best textbook in this exciting field of science. This 10th Anniversary Edition includes a new Introduction and Afterword from the authors setting the work in context. This comprehensive textbook describes such remarkable effects as fast quantum algorithms, quantum teleportation, quantum cryptography, and quantum error-correction. Quantum mechanics and computer science are introduced, before moving on to describe what a quantum computer is, how it can be used to solve problems faster than “classical” computers, and its real-world implementation. It concludes with an in-depth treatment of quantum information. Containing a wealth of figures and exercises, this well-known textbook is ideal for courses on the subject, and will interest beginning graduate students and researchers in physics, computer science, mathematics, and electrical engineering. MICHAEL NIELSEN was educated at the University of Queensland, and as a Fulbright Scholar at the University of New Mexico. He worked at Los Alamos National Laboratory, as the Richard Chace Tolman Fellow at Caltech, was Foundation Professor of Quantum Information Science and a Federation Fellow at the University of Queensland, and a Senior Faculty Member at the Perimeter Institute for Theoretical Physics. He left Perimeter Institute to write a book about open science and now lives in Toronto. ISAAC CHUANG is a Professor at the Massachusetts Institute of Technology, jointly appointed in Electrical Engineering & Computer Science, and in Physics. -
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. -
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. -
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. -
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. -
Complexity-Theoretic Foundations of Quantum Supremacy Experiments
Complexity-Theoretic Foundations of Quantum Supremacy Experiments Scott Aaronson∗ Lijie Cheny Abstract In the near future, there will likely be special-purpose quantum computers with 40-50 high-quality qubits. This paper lays general theoretical foundations for how to use such devices to demonstrate “quantum supremacy”: that is, a clear quantum speedup for some task, motivated by the goal of over- turning the Extended Church-Turing Thesis as confidently as possible. First, we study the hardness of sampling the output distribution of a random quantum circuit, a- long the lines of a recent proposal by the Quantum AI group at Google. We show that there’s a nat- ural average-case hardness assumption, which has nothing to do with sampling, yet implies that no polynomial-time classical algorithm can pass a statistical test that the quantum sampling procedure’s outputs do pass. Compared to previous work—for example, on BosonSampling and IQP—the central advantage is that we can now talk directly about the observed outputs, rather than about the distribution being sampled. Second, in an attempt to refute our hardness assumption, we give a new algorithm, inspired by Sav- itch’s Theorem, for simulating a general quantum circuit with n qubits and depth d in polynomial space and dO(n) time. We then discuss why this and other known algorithms fail to refute our assumption. Third, resolving an open problem of Aaronson and Arkhipov, we show that any strong quantum supremacy theorem—of the form “if approximate quantum sampling is classically easy, then the poly- nomial hierarchy collapses”—must be non-relativizing. -
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. -
QMA/Qpoly ⊆ PSPACE/Poly : De-Merlinizing Quantum Protocols
QMA/qpoly PSPACE/poly: De-Merlinizing Quantum Protocols ⊆ Scott Aaronson∗ University of Waterloo Abstract But what if Alice is a quantum advisor, who can send Bob a quantum state ψf ? Even in that case, Ambainis This paper introduces a new technique for removing ex- et al. [4] showed that Alice| i has to send Ω (2n/n) qubits for istential quantifiers over quantum states. Using this tech- Bob to succeed with probability at least 2/3 on every x. nique, we show that there is no way to pack an exponential Subsequently Nayak [11] improved this to Ω (2n), mean- number of bits into a polynomial-size quantum state, in such ing that there is no quantum improvement over the classical a way that the value of any one of those bits can later be bound. Since 2n qubits is too many for Alice to communi- proven with the help of a polynomial-size quantum witness. cate during her weekly meetings with Bob, it seems Bob is We also show that any problem in QMA with polynomial- out of luck. size quantum advice, is also in PSPACE with polynomial- So in desperation, Bob turns for help to Merlin, the star size classical advice. This builds on our earlier result that student in his department. Merlin knows f as well as x, BQP/qpoly PP/poly, and offers an intriguing counter- and can thus evaluate f (x). The trouble is that Merlin point to the recent⊆ discovery of Raz that QIP/qpoly = ALL. would prefer to take credit for evaluating f (x) himself, so Finally, we show that QCMA/qpoly PP/poly and that he might deliberately mislead Bob. -
Impossibility of Succinct Quantum Proofs for Collision-Freeness
Impossibility of Succinct Quantum Proofs for Collision-Freeness Scott Aaronson∗ Abstract We show that any quantum algorithm to decide whether a function f :[n] [n] is a permutation or far from a permutation must make Ω n1/3/w queries to f, even if the→ algorithm is given a w-qubit quantum witness in support of f being a permutation. This implies that there exists an oracle A such that SZKA QMAA, answering an eight-year-old open question of the author. Indeed, we show that relative6⊂ to some oracle, SZK is not in the counting class A0PP defined by Vyalyi. The proof is a fairly simple extension of the quantum lower bound for the collision problem. 1 Introduction The collision problem is to decide whether a black-box function f : [n] [n] is one-to-one (i.e., → a permutation) or two-to-one function, promised that one of these is the case. Together with its close variants, the collision problem is one of the central problems studied in quantum computing theory; it abstractly models numerous other problems such as graph isomorphism and the breaking of cryptographic hash functions. In this paper, we will mostly deal with a slight generalization of the collision problem that we call the Permutation Testing Problem, or PTP. This is a property testing problem, in which we are promised that f : [n] [n] is either a permutation or far from any permutation, and are asked → to decide which is the case. In 1997, Brassard, Høyer, and Tapp [8] gave a quantum algorithm for the collision problem that makes O n1/3 queries to f, an improvement over the Θ (√n) randomized query complexity that follows from the birthday paradox. -
Quantum Computing for All?
books & arts Quantum computing for all? Quantum Computing Nevertheless, the sophistication of the recent high-profile work on intermediate- for Everyone mathematics is at the level advertised, and scale quantum computation. This is partly is readily digestible by mathematically out of necessity, as contemporary research competent high-school students. requires more sophisticated mathematics The explanatory innovation used by to understand, but is also in line with By Chris Bernhardt Bernhardt is to dispense with imaginary Bernhardt’s focus on well-established ideas. numbers. For a physical theory almost In the final chapter, Bernhardt provides defined by the introduction of imaginary an overview of the potential impact numbers, this is an audacious move. of quantum computation across many MIT PRESS: 2019. However, even by slicing off the imaginary disciplines, including cryptography, big 216PP. £20.00 part of the qubits (bits of quantum data, chemistry, optimization problems and information), we can still understand the theoretical computer science. Again, this ax Planck was awarded the Nobel aspects of quantum physics that enable is kept brief as the author sensibly abstains Prize in Physics 100 years ago for quantum computing: superposition and from making predictions about future Mhis discovery of energy quanta. entanglement. Bernhardt introduces these technological advances. At the time, notions of digital computing concepts in detail using simple linear algebra. Quantum Computing for Everyone is pure were somewhat premature, and it would Both superposition and entanglement exposition. While it seems that Bernhardt take several decades before the foundations rely on the concept of state. In physics, has no central thesis to put forward, of modern computing systems were broadly speaking, the state is a complete the implicit argument is this: to really established.