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Linear-Time Algorithm for Quantum 2SAT
THEORY OF COMPUTING, Volume 14 (1), 2018, pp. 1–27 www.theoryofcomputing.org Linear-Time Algorithm for Quantum 2SAT Itai Arad∗ Miklos Santha∗† Aarthi Sundaram∗‡ Shengyu Zhang∗§ Received May 16, 2016; Revised August 22, 2017; Published March 9, 2018 Abstract: A well-known result about satisfiability theory is that the 2-SAT problem can be solved in linear time, despite the NP-hardness of the 3-SAT problem. In the quantum 2-SAT problem, we are given a family of 2-qubit projectors Pi j on a system of n qubits, and the task is to decide whether the Hamiltonian H = ∑Pi j has a 0-eigenvalue, or all eigenvalues are greater than 1=na for some a = O(1). The problem is not only a natural extension of the classical 2-SAT problem to the quantum case, but is also equivalent to the problem of finding a ground state of 2-local frustration-free Hamiltonians of spin 1=2, a well-studied model believed to capture certain key properties in modern condensed matter physics. Bravyi has shown that the quantum 2-SAT problem has a deterministic algorithm of complexity O(n4) in the algebraic model of computation where every arithmetic operation on complex numbers can be performed in unit time, and n is the number of variables. In this paper we give a deterministic algorithm in the algebraic model with running time O(n + m), where m is the number of local projectors, therefore achieving the best possible complexity in that model. We also show that if in the input every number has a constant size representation then the bit complexity of our algorithm is O((n + m)M(n)), where M(n) denotes the complexity of multiplying two n-bit integers. -
Amir Pnueli a Gentle Giant: Lord of the ??S and the ??S
Amir Pnueli A Gentle Giant: Lord of the ??s and the ??s Formal Aspects of Computing Applicable Formal Methods ISSN 0934-5043 Volume 22 Number 6 Form Asp Comp (2010) 22:663-665 DOI 10.1007/ s00165-010-0165-0 1 23 Your article is protected by copyright and all rights are held exclusively by British Computer Society. This e-offprint is for personal use only and shall not be self-archived in electronic repositories. If you wish to self-archive your work, please use the accepted author’s version for posting to your own website or your institution’s repository. You may further deposit the accepted author’s version on a funder’s repository at a funder’s request, provided it is not made publicly available until 12 months after publication. 1 23 Author's personal copy DOI 10.1007/s00165-010-0165-0 BCS © 2010 Formal Aspects Formal Aspects of Computing (2010) 22: 663–665 of Computing Amir Pnueli A Gentle Giant: Lord of the ϕ’s and the ψ’s David Harel Most of this piece focuses on Amir’s personality, since FACS readers are unlikely to need too much introduction to his scientific contributions. Amir completed his PhD in applied mathematics, the tile of his thesis being Calculation of Tides in the Ocean, under Chaim Pekeris, who was the founder of our department at the Weizmann Institute. He then switched to computer sci- ence during a brief stint at Stanford and at IBM Yorktown Heights. He was then at the Weizmann Institute until 1973, with two other young, brilliant computer scientists: Shimon Even and Zohar Manna. -
About the Authors of LNCS Volume 6650
About the Authors of LNCS Volume 6650 2011 Oded Goldreich ([email protected]) is a Meyer W. Weisgal Professor at the Fac- ulty of Mathematics and Computer Science of the Weizmann Institute of Science, Israel. Oded was born in 1957, and completed his graduate studies in 1983 under the supervision of Shimon Even. He was a post-doctoral fellow at MIT (1983-86), a faculty member at the Technion (1986-94), a visiting scientist at MIT (1995-98), and a Radcliffe fellow at Harvard (2003/04). Since 1995, he is a member of the Computer Science and Applied Mathematics Department of the Weizmann Institute. He is the author of the books “Modern Cryptography, Probabilistic Proofs and Pseudorandomness” (Springer, 1998), “Computational Complexity: A Conceptual Perspective” (Cambridge University Press, 2008), and the two-volume work “Foundations of Cryptography” (Cambridge University Press, 2001 and 2004). Lidor Avigad ([email protected]) works in HP software. He completed his master thesis at the Weizmann Institute of Science in 2009. Mihir Bellare ([email protected]) is a Professor at the Department of Computer Science and Engineering of the University of California San Diego, USA. Mihir got a BS from Caltech in 1986 and a Ph.D from MIT in 1991 under the supervision of Silvio Micali. He was a research staff member at IBM Research in New York from 1991 to 1995. He is a co-designer of the HMAC authentication algorithm (used to secure credit-card numbers in Internet shopping) and the OAEP encryption scheme (used for RSA-based encryption). He has received the 2009 ACM Paris Kanellakis Theory and Practice Award, the 2003 RSA Conference Award in Mathematics, and a 1996 David and Lucille Packard Fellowship in Science and Engineering.