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Unformatted text preview: arXiv:0810.3843v2 [quantph] 24 Apr 2009 Approximating Fractional Time Quantum Evolution L. Sheridan Institute for Quantum Computing, University of Waterloo, Waterloo, ON, N2L 3G1, Canada D. Maslov Institute for Quantum Computing, University of Waterloo, Waterloo, ON, N2L 3G1, Canada and National Science Foundation, 4201 Wilson Blvd, Arlington, VA, 22230, USA M. Mosca Institute for Quantum Computing, University of Waterloo, Waterloo, ON, N2L 3G1, Canada Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON, N2L 2Y5, Canada St. Jeromes University, Waterloo, ON N2L 3G3, Canada and Department of Combinatorics & Optimization, University of Waterloo, ON N2L 3G1 (Dated: October 20, 2008) An algorithm is presented for approximating arbitrary powers of a black box unitary operation, U t , where t is a real number, and U is a black box implementing an unknown unitary. The complexity of this algorithm is calculated in terms of the number of calls to the black box, the errors in the approximation, and a certain gap parameter. For general U and large t , one should apply U a total of t times followed by our procedure for approximating the fractional power U t t . An example is also given where for large integers t this method is more efficient than direct application of t copies of U . Further applications and related algorithms are also discussed. I. INTRODUCTION Suppose an nqubit unitary U is implemented by evolving (or simulating the evolution of) a timeindependent Hamiltonian H for a period of time = 1, that is, U = e iH . Then for any t R + , one can implement U t by simply evolving the Hamiltonian for a period of time t . For example, if t = 1 2 , then a square root of U , e i 1 2 H , could be implemented in this way, and in such model of computation the cost would be half of the cost of implementing U . In our work, we question what can be done if U is realized in some other way, such as a nontrivial sequence of timedependent Hamiltonians, or a quantum circuit. In other words, consider the situation when U is given in the form of a black box. The goal is to implement real valued powers t of this unitary operation by making use of the multiple copies of the black box implementing U . The complexity of such a procedure is measured in terms of the total number of calls to the unitary. It is possible to find the t th power of an unknown unitary by first performing a sufficiently precise complete quantum process tomography of a 2 n 2 n dimensional unitary U , which uses O (4 n ) calls to the unitary with various input states [1, 2] and measurements to achieve U with constant precision. The exponential scaling with n is necessary....
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