Scattering Processes from Quantum Simulation Algorithms for Scalar Field Theories

We provide practical simulation methods for scalar field theories on a quantum computer that yield improved asymptotics as well as concrete gate estimates for the simulation and physical qubit estimates using the surface code. We achieve these improvements through two optimizations. First, we consider a finite volume approach for estimating the elements of the S-matrix. This approach is appropriate in general for 1+1D and for certain low-energy elastic collisions in higher dimensions. Second, we implement our approach using a series of different fault-tolerant simulation algorithms for Hamiltonians formulated both in the field occupation basis and field amplitude basis. Our algorithms are based on either second-order Trotterization or qubitization. The cost of Trotterization in occupation basis scales as <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>O</a:mi> <a:mo stretchy="false">(</a:mo> <a:mi>λ</a:mi> <a:msup> <a:mi>N</a:mi> <a:mn>7</a:mn> </a:msup> <a:mrow> <a:mo stretchy="false">|</a:mo> </a:mrow> <a:mi mathvariant="normal">Ω</a:mi> <a:msup> <a:mrow> <a:mo stretchy="false">|</a:mo> </a:mrow> <a:mn>3</a:mn> </a:msup> <a:mo>/</a:mo> <a:mo stretchy="false">(</a:mo> <a:msup> <a:mi>M</a:mi> <a:mrow> <a:mn>5</a:mn> <a:mo>/</a:mo> <a:mn>2</a:mn> </a:mrow> </a:msup> <a:msup> <a:mi>ϵ</a:mi> <a:mrow> <a:mn>3</a:mn> <a:mo>/</a:mo> <a:mn>2</a:mn> </a:mrow> </a:msup> <a:mo stretchy="false">)</a:mo> <a:mo stretchy="false">)</a:mo> </a:math> where <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:mi>λ</j:mi> </j:math> is the coupling strength, <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:mi>N</l:mi> </l:math> is the occupation cutoff, <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"> <n:mrow> <n:mo stretchy="false">|</n:mo> </n:mrow> <n:mi mathvariant="normal">Ω</n:mi> <n:mrow> <n:mo stretchy="false">|</n:mo> </n:mrow> </n:math> is the volume of the spatial lattice, <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mi>M</s:mi> </s:math> is the mass of the particles and <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:mi>ϵ</u:mi> </u:math> is the uncertainty in the energy calculation used for the <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"> <w:mi>S</w:mi> </w:math> -matrix determination. Qubitization in the field basis scales as <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:mi>O</y:mi> <y:mo stretchy="false">(</y:mo> <y:mrow> <y:mo stretchy="false">|</y:mo> </y:mrow> <y:mi mathvariant="normal">Ω</y:mi> <y:msup> <y:mrow> <y:mo stretchy="false">|</y:mo> </y:mrow> <y:mn>2</y:mn> </y:msup> <y:mo stretchy="false">(</y:mo> <y:msup> <y:mi>k</y:mi> <y:mn>2</y:mn> </y:msup> <y:mi mathvariant="normal">Λ</y:mi> <y:mo>+</y:mo> <y:mi>k</y:mi> <y:msup> <y:mi>M</y:mi> <y:mn>2</y:mn> </y:msup> <y:mo stretchy="false">)</y:mo> <y:mo>/</y:mo> <y:mi>ϵ</y:mi> <y:mo stretchy="false">)</y:mo> </y:math> , where <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"> <ib:mi>k</ib:mi> </ib:math> is the cutoff in the field and <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"> <kb:mi mathvariant="normal">Λ</kb:mi> </kb:math> is a scaled coupling constant. We find in both cases that the bounds suggest physically meaningful simulations can be performed using on the order of <nb:math xmlns:nb="http://www.w3.org/1998/Math/MathML" display="inline"> <nb:mn>4</nb:mn> <nb:mo>×</nb:mo> <nb:msup> <nb:mn>10</nb:mn> <nb:mn>6</nb:mn> </nb:msup> </nb:math> physical qubits and <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML" display="inline"> <pb:msup> <pb:mn>10</pb:mn> <pb:mn>12</pb:mn> </pb:msup> <pb:mspace width="0.2em"/> <pb:mi>T</pb:mi> </pb:math> -gates which corresponds to roughly one day on a superconducting quantum computer with surface code and a cycle time of 100 ns. This places the simulation of scalar field theory within striking distance of the gate counts for the best available chemistry simulation results.