Chemical basis of Trotter-Suzuki errors in quantum chemistry simulation

Although the simulation of quantum chemistry is one of the most anticipated applications of quantum computing, the scaling of known upper bounds on the complexity of these algorithms is daunting. Prior work has bounded errors due to discretization of the time evolution (known as “Trotterization”) in terms of the norm of the error operator and analyzed scaling with respect to the number of spin orbitals. However, we find that these error bounds can be loose by up to 16 orders of magnitude for some molecules. Furthermore, numerical results for small systems fail to reveal any clear correlation between ground-state error and number of spin orbitals. We instead argue that chemical properties, such as the maximum nuclear charge in a molecule and the filling fraction of orbitals, can be decisive for determining the cost of a quantum simulation. Our analysis motivates several strategies to use classical processing to further reduce the required Trotter step size and estimate the necessary number of steps, without requiring additional quantum resources. Finally, we demonstrate improved methods for state preparation techniques which are asymptotically superior to proposals in the simulation literature.

Figure 1

Spin orbitals versus ground-state Trotter error for various molecular benchmarks in three different basis sets. Despite analytical predictions to the contrary (in prior works), it would appear that no clear relation holds between the Trotter error induced on the ground state and the number of spin orbitals for these benchmarks.

Figure 2

A comparison between the norm of the error operator and the error induced in the ground state. Notice that in many cases the basis of natural orbitals have the lowest Trotter error (especially for examples with large Trotter error).

Figure 3

We correlate the number of spin orbitals with the norm of the error operator in the local basis. The semblance of a positive slope appears to be a symptom of increasing nuclear charge as the number of spin orbitals increases. Red dots are atoms and blue dots are molecules.

Figure 4

The norm of the error operator appears extremely well correlated with the maximum nuclear charge in a molecule when using a local basis of atomic orbitals. The black line is the line of best fit for a $Z_{\max }^6$ scaling.

Figure 5

The norm of the error operator appears also well correlated with the maximum nuclear charge in a molecule when using the canonical basis of molecular orbitals. The black line is a least-squares fit to the data, which is roughly consistent with a $Z_{\max }^5$ scaling.

Figure 6

This plot shows the coefficients of normal-ordered terms in the error operator for water in a local basis as a function of the orbitals on which they act. The coefficients of the error terms are binned according to the orbitals involved in the term. This plot shows the marginal distribution of the magnitudes of those terms.

Figure 7

This plot shows the distribution of expectation values of the error operator for water in the local basis over its eigenstates and Haar random vectors. We see that the random vectors lead to substantially less error, on average, than do the Hamiltonian eigenstates. The Haar distribution of errors has a standard deviation of 4.82 while the Hamiltonian error distribution has standard deviation of 10.68.

Figure 8

These are histograms of the eigenspectra of the error operators for various molecular and atomic benchmarks in the local basis. Proceeding clockwise from top left, the molecules are water, hydrogen fluoride, methylene, atomic beryllium, atomic carbon, and atomic neon. Error operators for all of our benchmarks have surprisingly similar eigenspectra, regardless of the orbital basis. The source of this striking similarity and the reason for the particular structure is unknown.

Figure 9

Magnitude of the Trotter error in the exact ground state against the magnitude of the error induced on a classical ansatz for the ground state. Truncated CI computations are only performed when inexact; e.g., we have not computed HF using CISD because the calculation is exact in STO-6G.

Figure 10

Magnitude of the Trotter error induced in the exact ground state against percentage of the error that remains after subtracting the ansatz error from the exact error. This plot is intended to indicate the reduction in effective error when using a classical ansatz estimate. A black line is drawn at 100% remaining. In all benchmarks, using these ansatzae reduces effective error. Note that the quadruple calculation is so accurate for Be and LiH that the effective error appears to be exact to within double precision.

Figure 11

These plots show the coefficients of normal-ordered terms in the Hamiltonian and error operator as well as expectation values of the error operator terms for the ground state. The terms are binned according to the orbitals involved in the term. This plot shows the marginal distribution of the magnitudes of these terms.