Quantum Algorithms and Complexity for Continuous Problems
Definition of the Subject
Most continuous mathematical formulations arising in science and engineering can only be solved numerically and therefore approximately. We shall always assume that we are dealing with a numerical approximation to the solution.
Are quantum computers more powerful than classical computers for important scientific problems? How much more powerful? This would answer the question posed by Nielsen and Chuang (2000, p. 47).
Many important scientific and engineering problems have continuous formulations. These problems occur in fields such as physics, chemistry, engineering, and finance. The continuous formulations include path integration, partial differential equations (in particular, the Schrödinger equation), and continuous optimization.
To answer the first question, we must know the classical computational complexity (for brevity, complexity) of the...
KeywordsBoolean Function Quantum Computer Path Integration Quantum Algorithm Query Complexity
We are grateful to Erich Novak, University of Jena, and Henryk Woźniakowski, Columbia University and University of Warsaw, for their very helpful comments. We thank Jason Petras, Columbia University, for checking the complexity estimates appearing in the tables.
- Abrams DS, Williams CP (1999) Fast quantum algorithms for numerical integrals and stochastic processes. http://arXiv.org/quant-ph/9908083
- Babuska I, Osborn J (1991) Eigenvalue problems. In: Ciarlet PG, Lions JL (eds) Handbook of numerical analysis, vol II. North-Holland, Amsterdam, pp 641–787Google Scholar
- Bakhvalov NS (1977) Numerical methods. Mir Publishers, MoscowGoogle Scholar
- Beals R, Buhrman H, Cleve R, Mosca M, de Wolf R (2001) Quantum lower bounds by polynomials. J ACM 48(4):778–797, http://arXiv.org/quant-ph/9802049
- Bessen AJ (2007) On the complexity of classical and quantum algorithms for numerical problems in quantum mechanics. PhD thesis. Department of Computer Science, Columbia UniversityGoogle Scholar
- Dawson CM, Eisert J, Osborne TJ (2007) Unifying variational methods for simulating quantum many-body systems. http://arxiv.org/abs/0705.3456v1
- Heinrich S (2003a) From Monte Carlo to quantum computation. In: Entacher K, Schmid WC, Uhl A (eds) Proceedings of the 3rd IMACS Seminar on Monte Carlo Methods MCM2001, Salzburg. Special Issue of Math Comput Simul 62:219–230Google Scholar
- Heinrich S, Milla B (2007) The randomized complexity of initial value problems. Talk presented at First Joint International Meeting between the American Mathematical Society and the Polish Mathematical Society, WarsawGoogle Scholar
- Heinrich S, Novak E (2002) Optimal summation by deterministic, randomized and quantum algorithms. In: Fang KT, Hickernell FJ, Niederreiter H (eds) Monte Carlo and Quasi-Monte Carlo methods 2000. Springer, BerlinGoogle Scholar
- Knuth DE (1997) The art of computer programming, vol 2, 3rd edn, Seminumerical algorithms. Addison-Wesley Professional, CambridgeGoogle Scholar
- Kwas M (2005) Quantum algorithms and complexity for certain continuous and related discrete problems. PhD thesis. Department of Computer Science, Columbia UniversityGoogle Scholar
- Manin Y (1980) Computable and uncomputable. Sovetskoye Radio, Moscow (in Russian)Google Scholar
- Manin YI (1999) Classical computing, quantum computing, and Shor’s factoring algorithm. http://arXiv.org/quant-ph/9903008
- Nayak A, Wu F (1999) The quantum query complexity of approximating the median and related statistics. In: Proc STOC 1999, Association for Computing Machinery, New York, pp 384–393. http://arXiv.org/quant-ph/9804066
- Papageorgiou A, Traub JF (2005) Qubit complexity of continuous problems. http://arXiv.org/quant-ph/0512082
- Somma R, Ortiz G, Knill E, Gubernatis (2003) Quantum simulations of physics problems. In: Pirich AR, Brant HE (eds) Quantum information and computation. Proc SPIE 2003, vol 5105. The International Society for Optical Engineering, Bellingham, pp 96–103. http://arXiv.org/quant-ph/0304063Google Scholar
- Titschmarsh EC (1958) Eigenfunction expansions associated with second-order differential equations, part B. Oxford University Press, OxfordGoogle Scholar
- Traub JF (1999) A continuous model of computation. Phys Today May:39–43Google Scholar
- Wisner S (1996) Simulations of many-body quantum systems by a quantum computer. http://arXiv.org/quant-ph/96
- Yepez J (2002) An efficient and accurate quantum algorithm for the Dirac equation. http://arXiv.org/quant-ph/0210093