Lattice renormalization of quantum simulations

With advances in quantum computing, new opportunities arise to tackle challenging calculations in quantum field theory. We show that trotterized time-evolution operators can be related by analytic continuation to the Euclidean transfer matrix on an anisotropic lattice. In turn, trotterization entails renormalization of the temporal and spatial lattice spacings. Based on the tools of Euclidean lattice field theory, we propose two schemes to determine Minkowski lattice spacings, using Euclidean data and thereby overcoming the demands on quantum resources for scale setting. In addition, we advocate using a fixed-anisotropy approach to the continuum to reduce both circuit depth and number of independent simulations. We demonstrate these methods with qiskit noiseless simulators for a $2+1\mathrm{D}$ discrete non-Abelian $D_4$ gauge theory with two spatial plaquettes.

Figure 1

Schematic of the relations between the various lattice and continuum functions discussed in this work.

Figure 2

Ratio of the error bounds of Scheme B in Eq. (42) to that of Scheme A in Eq. (41). The black line indicates when the bounds are equal.

Figure 3

The lattice geometry used for the $D_4$ gauge simulation. The plaquettes are given by $U_2^{†}{U}_0^{†}{U}_3{U}_0$ and $U_3^{†}{U}_1^{†}{U}_2{U}_1$. Dash lines are used to indicate repeated links due to the periodic boundary conditions.

Figure 4

Expectation value of the plaquette $⟨{\mathcal{O}}_{1\times 1}(N{\delta }_t)⟩$ vs $N{\delta }_t$ for different ${\delta }_t$ with fixed $g_H^2=1.11$. The green line indicated the ${\delta }_t\to 0$ exact results.

Figure 5

$a_t{m}_{1\times 1}$ vs ${\delta }_t$ for $g_H^2=1.25$. The dashed (solid) lines indicate a linear (quadratic) fit to the data. The poor linear fit (${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}=40.2$) demonstrates that $a_t$ is not proportional to ${\delta }_t$ as might be naively expected. ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}=1.1$ for the quadratic fit.

Figure 6

$a_t{m}_{1\times 1}$ vs ${\delta }_t$ for different $g_H^2$. Notice that renormalization of $a_t$ is clearly $g_H^2$ dependent.

Figure 7

Errors with respect to the truth scale in a Minkowski calculation using Scheme A (left) and Scheme B (right), with the band showing the error from the $1\sigma$ region in the Euclidean fitting. We have assumed that the measurement precision in Euclidean spacetime could reach 1%. Together with fitting errors, we have $\epsilon \le 2\%$ at 68% C.L. for the $g_H^2$ and ${\delta }_{\tau }={\delta }_t$ used in this figure. For Scheme A we use $\mathrm{\Delta }{a}_t{m}_{1\times 1}$ from the fit, as $\epsilon$ is barely affected by the fitting procedure.

Figure 8

$a_t{m}_{2\times 1}/a_t{m}_{1\times 1}$ as a function of $a_t{m}_{1\times 1}$ for a variety of $g_H^2(a,a_t)$. The data points are extracted from fixed ${\delta }_t$ results from a qasm calculation. For each $g_H^2$, we have ${\delta }_t=\{0.1,0.25,0.5\}$ from left to right. The solid lines reflect the best fit for a fixed $g_H^2(a,a_t)$. The colored bands are the $1\sigma$ uncertainties on the fits.

Figure 9

$a_t{m}_{2\times 1}/a_t{m}_{1\times 1}$ as a function of $a_t{m}_{1\times 1}$ for a variety of $g_H^2(a,a_t)$. The data points are extracted from fixed ${\delta }_t$ results from a state_vector calculation. For each $g_H^2$, we have ${\delta }_t=\{0.01,0.05,0.1\}$ from left to right. The solid lines reflect the best fit for a fixed $g_H^2(a,a_t)$. The colored bands are the $1\sigma$ uncertainties on the fits.

Figure 10

$a{m}_{2\times 1}/a{m}_{1\times 1}$ vs $g_H^2(a,a_t)$ for three continuum trajectories where ${\xi }_t$ is approximately fixed. Closed and open symbols indicate qasm and state_vector results respectively. The black line is a best fit to ($\blacksquare$, $\square$) with a $1\sigma$ uncertainty band.