Solving lattice gauge theories using the quantum Krylov algorithm and qubitization

Computing vacuum states of lattice gauge theories (LGTs) containing fermionic degrees of freedom can present significant challenges for classical computation using Monte-Carlo methods. Quantum algorithms may offer a pathway towards more scalable computation of groundstate properties of LGTs. However, a comprehensive understanding of the quantum computational resources required for such a problem is thus far lacking. In this work, we investigate using the quantum subspace expansion (QSE) algorithm to compute the groundstate of the Schwinger model, an archetypal LGT describing quantum electrodynamics in one spatial dimension. We perform numerical simulations, including the effect of measurement noise, to extrapolate the resources required for the QSE algorithm to achieve a desired accuracy for a range of system sizes. Using this, we present a full analysis of the resources required to compute LGT vacuum states using a quantum algorithm using qubitization within a fault tolerant framework. We develop of a novel method for performing qubitization of a LGT Hamiltonian based on a 'linear combination of unitaries' (LCU) approach. The cost of the corresponding block encoding operation scales as <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:mo stretchy="false">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> with system size <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math>. Including the corresponding prefactors, our method reduces the gate cost by multiple orders of magnitude when compared to previous LCU methods for the QSE algorithm, which scales as <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:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>N</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> when applied to the Schwinger model. While the qubit and single circuit T-gate cost resulting from our resource analysis is appealing to early fault-tolerant implementation, we find that the number of shots required to avoid numerical instability within the QSE procedure must be significantly reduced in order to improve the feasibility of the methodology we consider and discuss how this might be achieved.