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Science | DOI:10.1145/3290407 Don Monroe Quantum Leap A new proof supports a 25-year-old claim N of the unique power of quantum computing.

OPES FOR QUANTUM comput- ing have long been buoyed by the existence of algorithms that would solve some par- ticularly challenging prob- Hlems with exponentially fewer opera- tions than any known algorithm for conventional computers. Many experts believe, but have been unable to prove, that these problems will resist even the cleverest non-quantum algorithms. Recently, researchers have shown the strongest evidence yet that even if con- ventional computers were made much more powerful, they probably still could not efficiently solve some problems that a quantum computer could. That such problems exist is a long- standing conjecture about the greater capability of quantum computers. “It was really the first big conjecture in quantum complexity theory,” said computer scientist Umesh Vazirani of the University of California, Berke- ley, who proposed the conjecture with then-student Ethan Bernstein in the 1993 paper (updated in 1997) that es- tablished the field. That work, now further validated, al model, then or now, “that violates Quantum Resources challenged the cherished thesis that the extended Church-Turing thesis,” Conventional “classical” computers any general computer can simulate any Vazirani said. “It overturned this basic store information as bits than can be other efficiently, since quantum com- fabric of computer science, and said: in one of two states, denoted 0 and 1. puters will sometimes be out of reach ‘here’s a new kid on the block, and it’s In contrast, a quantum degree of free- of conventional emulation. Quantum completely different and able to do to- dom, such as the spin of an electron or

computation is the only computation- tally different things.’” the polarization of a photon, can exist ON GRAPHIC FROM UNIVERSITY OF STRATHCLYDE BASED ASSOCIATES, ANDRIJ BORYS BY IMAGE

10 COMMUNICATIONS OF THE ACM | JANUARY 2019 | VOL. 62 | NO. 1 news concurrently in a weighted mixture of These mathematical arguments can two states. A set of, say, 50 of these “qu- determine if an answer can be assured ACM bits” thus can represent all 250 (~1015) given access to specific resources, such combinations of the individual states. as computational time or the depth of Member Manipulations of this assembly can be a corresponding circuit. An algorithm viewed as simultaneously performing a that is guaranteed to finish in “poly- quadrillion calculations. nomial time,” meaning the runtime News Performing a vast number of com- increases no faster than some fixed PRESERVING HISTORY putations does not do much good, power of the size of the input, can be IN A DIGITAL LIBRARY however, unless a unique answer can regarded as efficient. In contrast, many Edward A. Fox, be extracted. Interrogating the qubits problems, notably those that require a professor of computer forces them into some specific combi- exhaustively searching many combina- science at nation of 0s and 1s, with probabilities torial possibilities, are only known to Virginia that depend on their post-calculation yield to methods whose execution time Polytechnic Institute and weights. Critically, however, different grows exponentially or worse with the State University (Virginia Tech), initial configurations make quantum size of the input. recalls joining ACM more than contributions to the weight that are Complexity theory divides problems 50 years ago. complex numbers, which can cancel into “complexity classes,” depending Fox first became a member of ACM in 1967, while each other out as well as reinforce each on the resources they need. Some of an undergraduate at the other. The challenge is devising an al- the best-known problems belong to the Massachusetts Institute gorithm for which this cancellation class P, consisting of problems whose of Technology (MIT). During occurs for all configurations except solution can be found in polynomial his first year as a member, he launched MIT’s ACM the desired solution, so the eventual time. A much larger class is NP, which Student Chapter. measurement reveals this answer. includes problems for which a pro- In 2017, Fox was named Soon after Bernstein and Vazira- posed solution can be verified as cor- an ACM Fellow, cited for his contributions to information ni’s work, mathematician Peter Shor, rect in polynomial time. NP includes retrieval and digital libraries, working at AT&T Research in New such classic challenges as the traveling the latter a field he helped to Jersey as it spun off from Bell Labs, salesman problem and the graph ver- launch. “A lot of people don’t presented an algorithm that achieved tex coloring problem, which research- know what a digital library is,” Fox says, “So a way to think this goal for determining the fac- ers have been unable to show belong to of it is as an information tors of a large integer. The security P. Many experts strongly suspect that system tailored to a community of public key cryptography schemes polynomial-time algorithms for many of people.” Fox has served in numerous depends on this factorization being problems in NP have not been found positions and capacities within impractically time consuming, so the because they do not exist, in which case ACM over the years. He is potential for a rapid quantum solu- P≠NP. This important question, re- currently co-chair (with Michael tion attracted a lot of attention. garded by many as the most important Nelson) of the ACM Publications Board’s Digital Library and Inspired by this and other concrete open question in theoretical computer Technology Committee, which examples, researchers have been striv- science, remains unresolved, and a works closely with the technical ing to assemble ever-larger systems of $1-million prize from the Clay Math- and publishing staff to review physical qubits in the lab that can pre- ematics Institute awaits its answer. services offered by ACM in the context of competing and serve the delicate complex amplitudes complementary primary and of the qubits long enough to perform secondary online resources. a calculation, and to correct the in- He first became interested To compare in computer science in the mid- evitable errors. In recent years, several 1960s when, as a junior in high competing implementations have got- techniques that have school, he attended a computer ten big enough (dozens of qubits) to yet to be devised course during a study program achieve “quantum supremacy,” mean- at Columbia University. He and machines that went on to earn his bachelor’s ing solving selected problems faster degree in electrical engineering than a conventional computer. have yet to be built, from MIT, and both his master’s and Ph.D. degrees in computer Classifying Complexity computer scientists science from Cornell University. In the future, Fox hopes the Assessing comparative execution rely on computational technologies of information times is complicated by the fact that retrieval, digital libraries, and algorithms continually improve, complexity theory. archiving will be even better integrated, as a means of sometimes dramatically. To compare helping to preserve our history techniques that have yet to be devised and achievements for the future. and machines that have yet to be built, —John Delaney computer scientists rely not on cod- ing but on formal methods known as computational complexity theory.

JANUARY 2019 | VOL. 62 | NO. 1 | COMMUNICATIONS OF THE ACM 11 news

Bernstein and Vazirani defined “we show that there is one problem that a new complexity class called BQP BQP will solve better than PH.” In addi- (Bounded Quantum Polynomial), which “The basic ability tion to choosing the right oracle, he and has access to quantum resources. BQP to do Fourier Tal had to choose a problem that reveals is closely analogous to the conventional quantum computation’s strength—and class BPP (Bounded Probabilistic Poly- transformation, classical computation’s weakness—but nomial), which has access to a perfect that’s the heart they only needed one example. random-number generator and must They adapted an earlier suggestion not give a wrong answer too often. Cur- of the power by Scott Aaronson (then at the Mas- rently, some problems having only of quantum, sachusetts Institute of Technology) stochastic solutions are known, but it in which the computer must deter- is hoped that deterministic, “de-ran- at least most mine if one sequence of bits is (ap- domized” algorithms will eventually be of the algorithms proximately) the Fourier transform of found for them. another. Computing such frequency we know.” spectra is a natural task for quan- Consulting the Oracle tum computations, and Shor’s algo- The relationship of the quantum class rithm exploits precisely this strength BQP to various conventional classes, to identify periodicities that expose however, continues to be studied, long separations. “They are a way for us to prime factors of the target. “The basic after Bernstein and Vazirani suggested understand what kinds of problems ability to do Fourier transformation,” it includes problems beyond the scope are hard to prove and what kinds of re- Fortnow said, “that’s the heart of the of conventional techniques. “We have sults might be possible, but they’re not power of quantum, at least most of our conjectures and we can feel strongly a definite proof technique,” he said. the algorithms we know.” about them, but every so often they are “We didn’t prove a separation between “The hard part is to give the lower wrong,” Vazirani said. “A proof is really these two classes,” Raz agreed. “I can’t bound for the polynomial hierarchy,” something to be celebrated.” imagine that [such a separation] will Raz said. To show that no such algo- The new proof of separation does be proved in our lifetime.” rithm, even with access to the oracle, not apply to the pure versions of BQP “Already there were oracle separa- could solve it efficiently, he and Tal and the other complexity classes ad- tions of BQP and NP, BQP and P, and tweaked Aaronson’s suggestion so they dressed by the Vazirani-Bernstein con- other classes,” Raz said. He and Tal could apply recent discoveries about jecture. Similar to the long-standing now extend the argument to a super- pseudorandom sequences. unproven relationship of P and NP, charged class called the polynomial hi- These and the earlier results illus- “We almost never are able to actually erarchy, or PH. “This is what is stronger trate what quantum computers will be separate these important classes of in our result,” he said. PH can be viewed able to do, once they get very large and complexity theory,” said computer sci- as an infinite ladder of classes, start- perform like the idealized models, Vazi- entist Ran Raz of Princeton University ing with P and NP, in which successive rani said. What is less clear is how to ef- in New Jersey and the Weizmann Insti- rungs can build on the earlier ones by fectively use the less-capable machines tute in Israel. “We don’t know how.” using logical constructions. Later class- that are now being developed. “What Instead, Raz and his former stu- es invoke the earlier ones rather like a will we be able to do with those?” he dent Avishay Tal (now at Stanford subroutine, for example by defining asked. “That’s one of the things that we University) performed what is called problems using them in a phrase such are working hard to try to figure out.” an oracle separation. Like its name- as “for every,” or “there exists.” “Almost sake from ancient Greece (or The Ma- all the problems that we encounter in Further Reading trix movies), an oracle provides an- everyday life are somewhere in the poly- swers to profound questions without nomial hierarchy,” Raz said. Bernstein, E. and Vazirani, E. explaining how it got them. Roughly If all NP problems had polynomial- Quantum Complexity Theory, SIAM J. Comput. 26, 1411 (1997). speaking, Raz and Tal compared the time solutions, though, it turns out that capabilities of quantum and classi- the entire polynomial hierarchy would Shor, P.W. Polynomial-Time Algorithms for Prime cal algorithms that were given access collapse into one class, PH=NP=P. The Factorization and Discrete Logarithms on to an oracle that answers a specific new result, though, shows that oracle- a Quantum Computer, SIAM Journal of question. Provided with this oracle, assisted BQP would still be separate. Computing 26, pp. 1484–1509 (1997). they showed the quantum system “The way I view the Raz-Tal oracle is Raz, R. and Tal, A. could efficiently solve a carefully they’re saying that even if P happened Oracle Separation of BQP and PH, Electronic chosen problem more efficiently to equal to NP—that’s an unlikely Colloquium on Computational Complexity, than the classical system could using case,” Fortnow said, “it’s still possible Report No. 107 (2018). the same oracle. that quantum can do more than classi- Don Monroe is a science and technology writer based in Lance Fortnow, a computer scientist cal machines can.” Boston, MA, USA. at the Georgia Institute of Technology, said hundreds of proofs in complex- What Is It Good For? ity theory have relied upon such oracle “If we choose the right oracle,” Raz said, © 2019 ACM 0001-0782/19/1 $15.00

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