Simulating Effective QED on Quantum Computers (2022)

In recent years simulations of chemistry and condensed materials has emerged as one of the preeminent applications of quantum computing, offering an exponential speedup for the solution of the electronic structure for certain strongly correlated electronic systems. To date, most treatments have ignored the question of whether relativistic effects, which are described most generally by quantum electrodynamics (QED), can also be simulated on a quantum computer in polynomial time. Here we show that effective QED, which is equivalent to QED to second order in perturbation theory, can be simulated in polynomial time under reasonable assumptions while properly treating all four components of the wavefunction of the fermionic field. In particular, we provide a detailed analysis of such simulations in position and momentum basis using Trotter-Suzuki formulas. We find that the number of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>T</mml:mi></mml:math>-gates needed to perform such simulations on a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>3</mml:mn><mml:mi>D</mml:mi></mml:math> lattice of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>n</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:math> sites scales at worst as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mi>n</mml:mi><mml:mi>s</mml:mi><mml:mn>3</mml:mn></mml:msubsup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mi>&amp;#x03F5;</mml:mi><mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>o</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> in the thermodynamic limit for position basis simulations and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mi>n</mml:mi><mml:mi>s</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>4</mml:mn><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msubsup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mi>&amp;#x03F5;</mml:mi><mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>o</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> in momentum basis. We also find that qubitization scales slightly better with a worst case scaling of <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>&amp;#x007E;</mml:mo></mml:mover></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mi>n</mml:mi><mml:mi>s</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msubsup><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> for lattice eQED and complications in the prepare circuit leads to a slightly worse scaling in momentum basis of <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>&amp;#x007E;</mml:mo></mml:mover></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msubsup><mml:mi>n</mml:mi><mml:mi>s</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>5</mml:mn><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msubsup><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>. We further provide concrete gate counts for simulating a relativistic version of the uniform electron gas that show challenging problems can be simulated using fewer than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mn>10</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>13</mml:mn></mml:mrow></mml:msup></mml:math> non-Clifford operations and also provide a detailed discussion of how to prepare multi-reference configuration interaction states in effective QED which can provide a reasonable initial guess for the ground state. Finally, we estimate the planewave cutoffs needed to accurately simulate heavy elements such as gold.