Quantum simulation of real-space dynamics

Quantum simulation is a prominent application of quantum computers. While there is extensive previous work on simulating finite-dimensional systems, less is known about quantum algorithms for real-space dynamics. We conduct a systematic study of such algorithms. In particular, we show that the dynamics of a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math>-dimensional Schrödinger equation with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&amp;#x03B7;</mml:mi></mml:math> particles can be simulated with gate complexity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mover><mml:mi>O</mml:mi><mml:mo stretchy="false">&amp;#x007E;</mml:mo></mml:mover></mml:mrow><mml:mrow class="MJX-TeXAtom-OPEN"><mml:mo maxsize="1.2em" minsize="1.2em">(</mml:mo></mml:mrow><mml:mi>&amp;#x03B7;</mml:mi><mml:mi>d</mml:mi><mml:mi>F</mml:mi><mml:mtext>poly</mml:mtext><mml:mo stretchy="false">(</mml:mo><mml:mi>log</mml:mi><mml:mo>&amp;#x2061;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>g</mml:mi><mml:mo>&amp;#x2032;</mml:mo></mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mi>&amp;#x03F5;</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mrow class="MJX-TeXAtom-CLOSE"><mml:mo maxsize="1.2em" minsize="1.2em">)</mml:mo></mml:mrow></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&amp;#x03F5;</mml:mi></mml:math> is the discretization error, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>g</mml:mi><mml:mo>&amp;#x2032;</mml:mo></mml:msup></mml:math> controls the higher-order derivatives of the wave function, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>F</mml:mi></mml:math> measures the time-integrated strength of the potential. Compared to the best previous results, this exponentially improves the dependence on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&amp;#x03F5;</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>g</mml:mi><mml:mo>&amp;#x2032;</mml:mo></mml:msup></mml:math> from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>poly</mml:mtext><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>g</mml:mi><mml:mo>&amp;#x2032;</mml:mo></mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mi>&amp;#x03F5;</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>poly</mml:mtext><mml:mo stretchy="false">(</mml:mo><mml:mi>log</mml:mi><mml:mo>&amp;#x2061;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>g</mml:mi><mml:mo>&amp;#x2032;</mml:mo></mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mi>&amp;#x03F5;</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math> and polynomially improves the dependence on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>T</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math>, while maintaining best known performance with respect to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&amp;#x03B7;</mml:mi></mml:math>. For the case of Coulomb interactions, we give an algorithm using <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>&amp;#x03B7;</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo>+</mml:mo><mml:mi>&amp;#x03B7;</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>T</mml:mi><mml:mtext>poly</mml:mtext><mml:mo stretchy="false">(</mml:mo><mml:mi>log</mml:mi><mml:mo>&amp;#x2061;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>&amp;#x03B7;</mml:mi><mml:mi>d</mml:mi><mml:mi>T</mml:mi><mml:msup><mml:mi>g</mml:mi><mml:mo>&amp;#x2032;</mml:mo></mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">&amp;#x0394;</mml:mi><mml:mi>&amp;#x03F5;</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mi mathvariant="normal">&amp;#x0394;</mml:mi></mml:math> one- and two-qubit gates, and another using <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>&amp;#x03B7;</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mn>4</mml:mn><mml:mi>d</mml:mi><mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>d</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>T</mml:mi><mml:mtext>poly</mml:mtext><mml:mo stretchy="false">(</mml:mo><mml:mi>log</mml:mi><mml:mo>&amp;#x2061;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>&amp;#x03B7;</mml:mi><mml:mi>d</mml:mi><mml:mi>T</mml:mi><mml:msup><mml:mi>g</mml:mi><mml:mo>&amp;#x2032;</mml:mo></mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">&amp;#x0394;</mml:mi><mml:mi>&amp;#x03F5;</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mi mathvariant="normal">&amp;#x0394;</mml:mi></mml:math> one- and two-qubit gates and QRAM operations, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>T</mml:mi></mml:math> is the evolution time and the parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">&amp;#x0394;</mml:mi></mml:math> regulates the unbounded Coulomb interaction. We give applications to several computational problems, including faster real-space simulation of quantum chemistry, rigorous analysis of discretization error for simulation of a uniform electron gas, and a quadratic improvement to a quantum algorithm for escaping saddle points in nonconvex optimization.