Bounds for approximation in total variation distance by quantum circuits

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It was recently shown that for reasonable notions of approximation of states and functions by quantum circuits, almost all states and,functions are exponentially hard to approximate. The bounds obtained are asymptotically tight except for the one based on total variation distance (TVD). TVD is the most relevant metric for the performance of a quantum circuit. In this paper we obtain asymptotically tight bounds for TVD. We show that in a natural sense, almost all states are hard to approximate to within a TVD of 2/e -- {epsilon} even for exponentially small {epsilon}. The quantity 2/e -- {epsilon} is asymptotically the ... continued below

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22 p.

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Knill, E. September 1, 1995.

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It was recently shown that for reasonable notions of approximation of states and functions by quantum circuits, almost all states and,functions are exponentially hard to approximate. The bounds obtained are asymptotically tight except for the one based on total variation distance (TVD). TVD is the most relevant metric for the performance of a quantum circuit. In this paper we obtain asymptotically tight bounds for TVD. We show that in a natural sense, almost all states are hard to approximate to within a TVD of 2/e -- {epsilon} even for exponentially small {epsilon}. The quantity 2/e -- {epsilon} is asymptotically the average distance to the uniform distribution. Almost all states with probability amplitudes concentrated in a small fraction of the space are hard to approximate to within a TVD of 2 -- {epsilon}. These results imply that non-uniform quantum circuit complexity is non-trivial in any reasonable model. They also reinforce the notion that the relative information distance between states (which is based on the difficulty of transforming one state to another) fully reflects the dimensionality of the space of qubits, not the number of qubits.

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22 p.

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OSTI as DE95016961

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  • SOAD `96, Atlanta, GA (United States), 28-30 Jan 1996

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  • Other: DE95016961
  • Report No.: LA-UR--95-2724
  • Report No.: CONF-960129--1
  • Grant Number: W-7405-ENG-36
  • Office of Scientific & Technical Information Report Number: 125168
  • Archival Resource Key: ark:/67531/metadc624029

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  • September 1, 1995

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  • June 16, 2015, 7:43 a.m.

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  • Feb. 25, 2016, 1:51 p.m.

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Knill, E. Bounds for approximation in total variation distance by quantum circuits, article, September 1, 1995; New Mexico. (digital.library.unt.edu/ark:/67531/metadc624029/: accessed July 15, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.