Recipes for the digital quantum simulation of lattice spin systems
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Abstract
<jats:p>We describe methods to construct digital quantum simulation algorithms for quantum spin systems on a regular lattice with local interactions. In addition to tools such as the Trotter-Suzuki expansion and graph coloring, we also discuss the efficiency gained by parallel execution of an extensive number of commuting terms. We provide resource estimates and quantum circuit elements for the most important cases and classes of spin systems. As resource estimates we indicate the total number of gates <jats:inline-formula><jats:alternatives><jats:tex-math>N</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>N</mml:mi></mml:math></jats:alternatives></jats:inline-formula> and simulation time <jats:inline-formula><jats:alternatives><jats:tex-math>T</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>T</mml:mi></mml:math></jats:alternatives></jats:inline-formula>, expressed in terms of the number <jats:inline-formula><jats:alternatives><jats:tex-math>n</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>n</mml:mi></mml:math></jats:alternatives></jats:inline-formula> of spin 1/2 lattice sites (qubits), target accuracy <jats:inline-formula><jats:alternatives><jats:tex-math>\epsilon</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>ϵ</mml:mi></mml:math></jats:alternatives></jats:inline-formula>, and simulated time <jats:inline-formula><jats:alternatives><jats:tex-math>t</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>t</mml:mi></mml:math></jats:alternatives></jats:inline-formula>. We provide circuit constructions that realize the simulation time <jats:inline-formula><jats:alternatives><jats:tex-math>T^{(1)}\propto nt^2/\epsilon</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mrow><mml:mo stretchy="false" form="prefix">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false" form="postfix">)</mml:mo></mml:mrow></mml:msup><mml:mo>∝</mml:mo><mml:mi>n</mml:mi><mml:msup><mml:mi>t</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi>/</mml:mi><mml:mi>ϵ</mml:mi></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>T^{(2q)}\propto t^{1+\eta}n^\eta/\epsilon^\eta</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mrow><mml:mo stretchy="false" form="prefix">(</mml:mo><mml:mn>2</mml:mn><mml:mi>q</mml:mi><mml:mo stretchy="false" form="postfix">)</mml:mo></mml:mrow></mml:msup><mml:mo>∝</mml:mo><mml:msup><mml:mi>t</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>η</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>n</mml:mi><mml:mi>η</mml:mi></mml:msup><mml:mi>/</mml:mi><mml:msup><mml:mi>ϵ</mml:mi><mml:mi>η</mml:mi></mml:msup></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> for arbitrarily small <jats:inline-formula><jats:alternatives><jats:tex-math>\eta = 1/2q</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>η</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mi>/</mml:mi><mml:mn>2</mml:mn><mml:mi>q</mml:mi></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> for the first-order and higher-order Trotter expansions. We also discuss the potential impact of scaled gates, which have not yet been fully explored.</jats:p>