Quantum simulation in the semi-classical regime
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Abstract
<jats:p>Solving the time-dependent Schrödinger equation is an important application area for quantum algorithms. We consider Schrödinger's equation in the semi-classical regime. Here the solutions exhibit strong multiple-scale behavior due to a small parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi class="MJX-variant">ℏ</mml:mi></mml:math>, in the sense that the dynamics of the quantum states and the induced observables can occur on different spatial and temporal scales. Such a Schrödinger equation finds many applications, including in Born-Oppenheimer molecular dynamics and Ehrenfest dynamics. This paper considers quantum analogues of pseudo-spectral (PS) methods on classical computers. Estimates on the gate counts in terms of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi class="MJX-variant">ℏ</mml:mi></mml:math> and the precision <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ε</mml:mi></mml:math> are obtained. It is found that the number of required qubits, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi></mml:math>, scales only logarithmically with respect to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi class="MJX-variant">ℏ</mml:mi></mml:math>. When the solution has bounded derivatives up to order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ℓ</mml:mi></mml:math>, the symmetric Trotting method has gate complexity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo maxsize="1.623em" minsize="1.623em">(</mml:mo></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">(</mml:mo><mml:mi>ε</mml:mi><mml:mi class="MJX-variant">ℏ</mml:mi><mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">p</mml:mi><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">l</mml:mi><mml:mi mathvariant="normal">y</mml:mi><mml:mi mathvariant="normal">l</mml:mi><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">g</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>ε</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>−</mml:mo><mml:mfrac><mml:mn>3</mml:mn><mml:mrow><mml:mn>2</mml:mn><mml:mi>ℓ</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:msup><mml:mi class="MJX-variant">ℏ</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mn>2</mml:mn><mml:mi>ℓ</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo maxsize="1.623em" minsize="1.623em">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math> provided that the diagonal unitary operators in the pseudo-spectral methods can be implemented with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">p</mml:mi><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">l</mml:mi><mml:mi mathvariant="normal">y</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> operations. When physical observables are the desired outcomes, however, the step size in the time integration can be chosen independently of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi class="MJX-variant">ℏ</mml:mi></mml:math>. The gate complexity in this case is reduced to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo maxsize="1.623em" minsize="1.623em">(</mml:mo></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:msup><mml:mi>ε</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">p</mml:mi><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">l</mml:mi><mml:mi mathvariant="normal">y</mml:mi><mml:mi mathvariant="normal">l</mml:mi><mml:mi mathvariant="normal">o</mml:mi><mml:mi mathvariant="normal">g</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>ε</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>−</mml:mo><mml:mfrac><mml:mn>3</mml:mn><mml:mrow><mml:mn>2</mml:mn><mml:mi>ℓ</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:msup><mml:mi class="MJX-variant">ℏ</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo maxsize="1.623em" minsize="1.623em">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ℓ</mml:mi></mml:math> again indicating the smoothness of the solution.</jats:p>