# Continuous quantum computation

Two major motivations for studying continuous quantum computation are:

• Many scientific problems have continuous mathematical formulations. Examples of such formulations are
• In their standard monograph Nielsen and Chuang state "Of particular interest is a decisive answer to the problem whether quantum computers are more powerful than classical computers." To answer this question one must know the classical and quantum computational complexities

We discuss the second motivation. By computational complexity (complexity for brevity) is meant the minimal computational resources needed to solve a problem. Two of the most important resources for quantum computing are qubits and queries. Classical complexity has been extensively studied in information-based complexity. The classical complexity of many continuous problems is known. Therefore, when the quantum complexity of these problems is obtained, the question as to whether quantum computers are more powerful than classical can be answered. Furthermore, it can be established how much more powerful. In contrast, the complexity of discrete problems is typically unknown; one has to settle for the complexity hierarchy. For example, the classical complexity of integer factorization is unknown.

## An example: path integration

Path integration has numerous applications including quantum mechanics, quantum chemistry, statistical mechanics, and computational finance. We want to compute an approximation to within error at most $\scriptstyle\varepsilon$ with probability, say, at least 3/4. Then the following was shown by Traub and Woźniakowski:

• A quantum computer enjoys exponential speedup over the classical worst case and quadratic speedup over the classical randomized case.
• The query complexity is of order $\scriptstyle\varepsilon^{-1}$.
• The qubit complexity is of order $\scriptstyle\varepsilon^{-2}\log\varepsilon^{-1}$.

Thus the qubit complexity of path integration is a second degree polynomial in $\scriptstyle\varepsilon^{-1}$. That seems pretty good but we probably won't have enough qubits for a long time to do new science especially with error correction. Since this is a complexity result we can't do better by inventing a clever new algorithm. But perhaps we can do better by slightly modifying the queries.

In the standard model of quantum computation the probabilistic nature of quantum computation enters only through measurement; the queries are deterministic. In analogy with classical Monte Carlo Woźniakowski introduced the idea of a quantum setting with randomized queries. He showed that in this setting the qubit complexity is of order $\scriptstyle \log\varepsilon^{-1}$, thus achieving an exponential improvement over the qubit complexity in the standard quantum computing setting.

## Applications

Besides path integration there have been numerous recent papers studying algorithms and quantum speedups for continuous problems. These include matrix eigenvalues, phase estimation, the Sturm-Liouville eigenvalue problem, Feynman-Kac path integration, initial value problems, function approximation and high-dimensional integration. See the papers listed below and the references therein.

• Bessen, A. J. (2005), A lower bound for phase estimation, Physical Review A, 71(4), 042313. Also http://arXiv.org/quant-ph/0412008.
• Heinrich, S. (2002), Quantum Summation with an Application to Integration, J. Complexity, 18(1), 1–50. Also http://arXiv.org/quant-ph/0105116.
• Heinrich, S. (2003), Quantum integration in Sobolev spaces, J. Complexity, 19, 19–42.
• Heinrich, S. (2004), Quantum Approximation I. Embeddings of Finite Dimensional $L_p$ Spaces, J. Complexity, 20, 5–26. Also http://arXiv.org/quant-ph/0305030.
• Heinrich, S. (2004), Quantum Approximation II. Sobolev Embeddings, J. Complexity, 20, 27–45. Also http://arXiv.org/quant-ph/0305031.
• Jaksch, P. and Papageorgiou, A. (2003), Eigenvector approximation leading to exponential speedup of quantum eigenvalue calculation, Phys. Rev. Lett., 91, 257902. Also http://arXiv.org/quant-ph/0308016.
• Kacewicz, B. Z. (2005), Randomized and quantum solution of initial value problems, J. Complexity, 21, 740-756.
• Kwas, M., Complexity of multivariate Feynman-Kac Path Integration in Randomized and Quantum settings, 2004. Also http://arXiv.org/quant-ph/0410134.
• Novak, E. (2001), Quantum complexity of integration, J. Complexity, 17, 2–16. Also http://arXiv.org/quant-ph/0008124.
• Novak, E., Sloan, I. H., and Wozniakowski, H., Tractability of Approximation for Weighted orobov Spaces on Classical and Quantum Computers, J. Foundations of Computational Mathematics, 4, 121-156, 2004. Also http://arXiv.org/quant-ph/0206023
• Papageorgiou, A. and Wo´zniakowski, H. (2005), Classical and Quantum Complexity of the Sturm-Liouville Eigenvalue Problem, Quantum Information Processing, 4(2), 87–127. Also http://arXiv.org/quant-ph/0502054.
• Papageorgiou, A. and Wo´zniakowski, H. (2007), The Sturm-Liouville Eigenvalue Problem and NP-Complete Problems in the Quantum Setting with Queries, Quantum Information Processing, 6(2), 101-120. Also http://arXiv.org/quant-ph/0504194.
• Traub, J. F. and Wo´zniakowski, H. (2002), Path integration on a quantum computer, Quantum Information Processing, 1(5), 365–388, 2002. Also http://arXiv.org/quant-ph/0109113.
• Woźniakowski, H. (2006), The Quantum Setting with Randomized Queries for Continuous Problems, Quantum Information Processing, 5(2), 83–130. Also http://arXiv.org/quant-ph/0601196.