chapter4 quantum circuits
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4.7 Simulation of quantum systems
计算的实际应用之一是模拟物理系统。
4.7.1 simulation in action
模拟的核心是微分方程的解。
用经典计算机模拟量子系统是可能的,但通常低效。(我们瞄准……这一类问题,提出了一种有效的模拟方式)、
模拟量子系统的关键挑战是: 必须要求解的微分方程的指数数目。
如 for n qubits, 2n equations.
特点: exponential complexity growth of quantum systems.
结论:量子计算机能有效地模拟量子系统。
4.7.2 the quantum simulation algorithms
(*exp(x)展开项*) f01[n_] := (0.1^n)/(n!) f3[n_] := (3^n)/(n!) f5[n_] := (5^n)/(n!) DiscretePlot[{f1[n], f3[n], f5[n]}, {n, 0, 6} \!\(\* GraphicsBox[{{ {RGBColor[0.368417, 0.506779, 0.709798], PointSize[ 0.019444444444444445`], AbsoluteThickness[1.6], { {RGBColor[0.368417, 0.506779, 0.709798], PointSize[ 0.019444444444444445`], AbsoluteThickness[1.6], Opacity[0.2], LineBox[{}, VertexColors->None]}, {RGBColor[0.368417, 0.506779, 0.709798], PointSize[ 0.019444444444444445`], AbsoluteThickness[1.6], Opacity[0.2], LineBox[{{{0., 1.}, {0., 0}}, {{1., 1.}, {1., 0}}, {{2., 0.5}, {2., 0}}, {{3., 0.16666666666666666`}, {3., 0}}, {{ 4., 0.041666666666666664`}, {4., 0}}, {{5., 0.008333333333333333}, {5., 0}}, {{6., 0.001388888888888889}, {6., 0}}}, VertexColors->None]}}}, {RGBColor[0.880722, 0.611041, 0.142051], PointSize[ 0.019444444444444445`], AbsoluteThickness[1.6], { {RGBColor[0.880722, 0.611041, 0.142051], PointSize[ 0.019444444444444445`], AbsoluteThickness[1.6], Opacity[0.2], LineBox[{}, VertexColors->None]}, {RGBColor[0.880722, 0.611041, 0.142051], PointSize[ 0.019444444444444445`], AbsoluteThickness[1.6], Opacity[0.2], LineBox[{{{0., 1.}, {0., 0}}, {{1., 3.}, {1., 0}}, {{2., 4.5}, {2., 0}}, {{3., 4.5}, {3., 0}}, {{4., 3.375}, { 4., 0}}, {{5., 2.025}, {5., 0}}, {{6., 1.0125}, {6., 0}}}, VertexColors->None]}}}, {RGBColor[0.560181, 0.691569, 0.194885], PointSize[ 0.019444444444444445`], AbsoluteThickness[1.6], { {RGBColor[0.560181, 0.691569, 0.194885], PointSize[ 0.019444444444444445`], AbsoluteThickness[1.6], Opacity[0.2], LineBox[{}, VertexColors->None]}, {RGBColor[0.560181, 0.691569, 0.194885], PointSize[ 0.019444444444444445`], AbsoluteThickness[1.6], Opacity[0.2], LineBox[{{{0., 1.}, {0., 0}}, {{1., 5.}, {1., 0}}, {{2., 12.5}, {2., 0}}, {{3., 20.833333333333332`}, {3., 0}}, {{ 4., 26.041666666666668`}, {4., 0}}, {{5., 26.041666666666668`}, {5., 0}}, {{6., 21.70138888888889}, { 6., 0}}}, VertexColors->None]}}}}, { {RGBColor[0.368417, 0.506779, 0.709798], PointSize[ 0.019444444444444445`], AbsoluteThickness[1.6], {}, PointBox[{{0., 1.}, {1., 1.}, {2., 0.5}, {3., 0.16666666666666666`}, {4., 0.041666666666666664`}, {5., 0.008333333333333333}, {6., 0.001388888888888889}}], {}}, {RGBColor[0.880722, 0.611041, 0.142051], PointSize[ 0.019444444444444445`], AbsoluteThickness[1.6], {}, PointBox[{{0., 1.}, {1., 3.}, {2., 4.5}, {3., 4.5}, {4., 3.375}, {5., 2.025}, {6., 1.0125}}], {}}, {RGBColor[0.560181, 0.691569, 0.194885], PointSize[ 0.019444444444444445`], AbsoluteThickness[1.6], {}, PointBox[{{0., 1.}, {1., 5.}, {2., 12.5}, {3., 20.833333333333332`}, {4., 26.041666666666668`}, {5., 26.041666666666668`}, {6., 21.70138888888889}}], {}}}}, AspectRatio->0.6180339887498948, Axes->True, AxesOrigin->{0, 0}, FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}}, GridLinesStyle->Directive[ GrayLevel[0.5, 0.4]], Method->{"MessagesHead" -> DiscretePlot, "AxisPadding" -> 5.87, "DefaultBoundaryStyle" -> Automatic, "DefaultPlotStyle" -> { Directive[ RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6]], Directive[ RGBColor[0.880722, 0.611041, 0.142051], AbsoluteThickness[1.6]], Directive[ RGBColor[0.560181, 0.691569, 0.194885], AbsoluteThickness[1.6]], Directive[ RGBColor[0.922526, 0.385626, 0.209179], AbsoluteThickness[1.6]], Directive[ RGBColor[0.528488, 0.470624, 0.701351], AbsoluteThickness[1.6]], Directive[ RGBColor[0.772079, 0.431554, 0.102387], AbsoluteThickness[1.6]], Directive[ RGBColor[0.363898, 0.618501, 0.782349], AbsoluteThickness[1.6]], Directive[ RGBColor[1, 0.75, 0], AbsoluteThickness[1.6]], Directive[ RGBColor[0.647624, 0.37816, 0.614037], AbsoluteThickness[1.6]], Directive[ RGBColor[0.571589, 0.586483, 0.], AbsoluteThickness[1.6]], Directive[ RGBColor[0.915, 0.3325, 0.2125], AbsoluteThickness[1.6]], Directive[ RGBColor[0.40082222609352647`, 0.5220066643438841, 0.85], AbsoluteThickness[1.6]], Directive[ RGBColor[0.9728288904374106, 0.621644452187053, 0.07336199581899142], AbsoluteThickness[1.6]], Directive[ RGBColor[0.736782672705901, 0.358, 0.5030266573755369], AbsoluteThickness[1.6]], Directive[ RGBColor[0.28026441037696703`, 0.715, 0.4292089322474965], AbsoluteThickness[1.6]]}, "DomainPadding" -> 5.87, "RangePadding" -> 14.675}, PlotRange->{{0, 6}, {0, 26.041666666666668`}}, PlotRangePadding->{{ Scaled[0.02], Scaled[0.02]}, { Scaled[0.02], Scaled[0.02]}}, Ticks->{Automatic, Automatic}]\)
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