New Developments in Quantum Algorithms

In this talk, we describe two recent developments in quantum algorithms. The first new development is a quantum algorithm for evaluating a Boolean formula consisting of AND and OR gates of size N in time O( √ N). This provides quantum speedups for any problem that can be expressed via Boolean formulas. This result can be also extended to span problems, a generalization of Boolean formulas. This provides an optimal quantum algorithm for any Boolean function in the black-box query model. The second new development is a quantum algorithm for solving systems of linear equations. In contrast with traditional algorithms that run in time O(N) where N is the size of the system, the quantum algorithm runs in time O(log N). It outputs a quantum state describing the solution of the system. 1 History of quantum algorithms 1.1 First quantum algorithms Quantum computing (and, more broadly, quantum information science) is a new area at the boundary of computer science and physics. It studies how to apply quantum mechanics to solve problems in computer science and information processing. The area of quantum computing was shaped by the discoveries of two major quantum algorithms in mid-1990s. The first of the these two discoveries was Shor’s polynomial time quantum algorithm for factoring and discrete logarithms. Factoring and discrete logarithm are very hard number theoretic problems. The difficulty of these problems has been used to design cryptosystems (such as RSA and Diffie-Helman key exchange) for secure data transmission over an insecure network (such as Internet). The security of data transmission is based on the assumption that it is hard to factor (or find discrete logarithm of) large numbers. Until recently, this assumption was not in doubt. Mathematicians had tried to devise an efficient way of factoring large numbers for centuries, with no success. In 1994, Shor [56] discovered a fast algorithm for factoring large numbers on a quantum mechanical computer. This shook up the foundations of cryptography. ⋆ Supported by FP7 Marie Curie Grant PIRG02-GA-2007-224886 and ESF project 1DP/1.1.1.2.0/09/APIA/VIAA/044. If a quantum mechanical computer is built, today’s methods for secure data transmission over the Internet will become insecure. Another, equally strikingly discovery was made in 1996, by Lov Grover [34]. He invented a quantum algorithm for speeding up exhaustive search problems. Grover’s algorithm solves a generic exhaustive search problem with N possible solutions in time O( √ N). This provides a quadratic speedup for a range of search problems, from problems that are in P classically to NP-complete problems. Since then, each of the two algorithms has been analyzed in great detail. Shor’s algorithm has been generalized to solve a class of algebraic problems that can be abstracted to Abelian hidden subgroup problem [39]. Besides factoring and discrete logarithm, the instances of Abelian HSP include cryptanalysis of hidden linear equations [18], solving Pell’s equation, principal ideal problem [35] and others. Grover’s algorithm has been generalized to the framework of amplitude amplification [21] and extended to solve problems like approximate counting [23,47] and collision-finding [22]. 1.2 Quantum walks and adiabatic algorithms Later, two new methods for designing quantum algorithms emerged: quantum walks [4,41,9,54,60] and adiabatic algorithms [31]. Quantum walks are quantum generalizations of classical random walks. They have been used to obtain quantum speedups for a number of problems. The typical setting is as follows. Assume that we have a classical Markov chain, on a state-space in which some states are special (marked). The Markov chain starts in a uniformly random state and stops if it reaches a marked state. If the classical Markov chain reaches a marked state in expected time T , then there is a quantum algorithm which can find it in time O( √ T ), assuming some conditions on the Markov chain [58,44,42]. This approach gives quantum speedups for a number of problems: element distinctness [5], search on a grid [13,59], finding triangles in graphs [45], testing matrix multiplication [19] and others. Another application of quantum walks is to the ”glued trees” problem [26]. In this problem, we have a graph G with two particular vertices u, v, designed as the entrance and the exit. The problem is to find the vertex v, if we start at the vertex u. There is a special exponential size graph called ”glued trees” on which any classical algorithm needs exponential time to find v but a quantum algorithm can find v in polynomial time [26]. Adiabatic computation is a physics-based paradigm for quantum algorithms. In this paradigm, we design two quantum systems: – Hsol whose lowest-energy state |ψsol〉 encodes a solution to a computational problem (for example, a satisfying assignment for SAT). – Hstart whose lowest-energy state |ψstart〉 is such that we can easily prepare |ψstart〉. We then prepare |ψstart〉 and slowly transform the forces acting on the quantum system fromHstart toHsol. Adiabatic theorem of quantum mechanics guarantees that, if the transformation is slow enough, |ψstart〉 is transformed into a state close to |ψsol〉 [31]. The key question here is: what is ”slowly enough”? Do we need a polynomial time or an exponential time to transform Hstart to Hsol (thus solving SAT by a quantum algorithm)? This is a subject of an ongoing debate [31,29,2]. Adiabatic computation has been used by D-Wave Systems [30] which claims to have built a 128-bit adiabatic quantum computer. However, the claims of D-Wave have been questioned by many prominent scientists (see e.g. [1]). 1.3 Most recent algorithms Two most recent discoveries in this field are the quantum algorithms for formula evaluation [32] and solving systems of linear equations [36]. Both of those algorithms use the methods from the previous algorithms but do it in a novel, unexpected way. Formula evaluation uses quantum walks but in a form that is quite different from the previous approach (which we described above). Quantum algorithm for formula evaluation uses eigenvalue estimation [46] which is the key technical subroutine of Shor’s factoring algorithm [56] and the related quantum algorithms. But, again, eigenvalue estimation is used in a very unexpected way. These two algorithms are the main focus of this survey. We describe them in detail in sections 2 and 3. 2 Formula evaluation