Initial state preparation for quantum chemistry on quantum computers

TL;DR

The paper tackles the critical bottleneck of initial-state preparation for quantum algorithms in quantum chemistry by introducing an end-to-end workflow that starts from classical descriptions (SOS or MPS) and culminates in a high-quality quantum state with quantifiable energy-distribution-based metrics. It presents a groundbreaking SOS encoding with O(D log D) Toffoli complexity and discusses MPS-based implementations, providing detailed resource estimates and practical trade-offs. The core contribution is the energy distribution framework, which enables model-free state quality assessment, guides QPE-related decisions, and addresses the QPE leakage problem through quantum refining, notably favoring coarse QPE postselection over QETU in practice. Numerical demonstrations on hydrogen chains, N$_2$, Cr$_2$, and Fe$_4$S$_4$ illustrate how energy-distribution analysis can identify Goldilocks problems—where quantum advantage is plausible—and show substantial potential reductions in the overall energy-estimation cost when using advanced initial-state preparation.

Abstract

Quantum algorithms for ground-state energy estimation of chemical systems require a high-quality initial state. However, initial state preparation is commonly either neglected entirely, or assumed to be solved by a simple product state like Hartree-Fock. Even if a nontrivial state is prepared, strong correlations render ground state overlap inadequate for quality assessment. In this work, we address the initial state preparation problem with an end-to-end algorithm that prepares and quantifies the quality of initial states, accomplishing the latter with a new metric -- the energy distribution. To be able to prepare more complicated initial states, we introduce an implementation technique for states in the form of a sum of Slater determinants that exhibits significantly better scaling than all prior approaches. We also propose low-precision quantum phase estimation (QPE) for further state quality refinement. The complete algorithm is capable of generating high-quality states for energy estimation, and is shown in select cases to lower the overall estimation cost by several orders of magnitude when compared with the best single product state ansatz. More broadly, the energy distribution picture suggests that the goal of QPE should be reinterpreted as generating improvements compared to the energy of the initial state and other classical estimates, which can still be achieved even if QPE does not project directly onto the ground state. Finally, we show how the energy distribution can help in identifying potential quantum advantage.

Figures (17)

• Figure 1: The initial state preparation algorithm consists of the following steps: classical search for a low energy state, conversion of the state into a standardized form, i.e. SOS or MPS, quality assessment performed based on the energy distribution of the candidate state, implementation of the state on the quantum computer, and quantum refining. The end-to-end procedure results in a high quality state prepared on the quantum computer.
• Figure 2: Schematic representation of important steps in the SOS encoding algorithm. We consider an example with $2N=8$ orbitals and $D=4$ Slater determinants, for which the identification strings require $2 \log D -1=3$ bits. (a) Matrix of Slater determinant strings $\nu_i$. By selecting only rows 1,2, 5, and 8, we can construct substrings $\tilde{\nu}_i$ that form a matrix of full rank. (b) Using the result from \ref{['lem:unique_signature']}, we construct bitstrings $u_i$ that form a linear map transforming the substrings $\tilde{\nu}_i$ to the identification bitstrings $b_i$. (c) The encoding quantum algorithm applies a series of CNOT operations for each $u_i$, acting only on qubits 1, 2, 5 and 8 in the system register, in accordance to the choice of bitstrings. These are responsible for setting every individual qubit in the identification register to the desired value.
• Figure 3: Comparing the cost of implementing an SOS wavefunction with $D$ determinants between the algorithm proposed here and that of Ref. tubman2018postponing for a hypothetical system with $N = 100$ spatial orbitals. Inset: comparing the cost of preparing an SOS wavefunction of quality 0.2, as measured through overlap with the highest-fidelity reference, for specific molecules. In the inset, the superscript on the molecule formula indicates how much the bond is stretched relative to equilibrium. Note that system size $N$ varies between the molecules: $10$ for Cr$_2$, $32$ for Fe$_4$S$_4$, and $16$ for H$_{16}$. The number of determinants required to achieve quality of 0.2 also varies.
• Figure 4: Circuit for MPS implementation. Since the bond dimension can change as the circuit traverses the system, one can start with a number of ancillae equal to $\lceil \log \chi_{\text{max}} \rceil$ and use more or less of the available anciallae as the bond dimension dictates.
• Figure 5: Schematic depiction of an energy distribution for a particular initial state, illustrating the possibility of improvement over classically found energy estimates. If the best classical energy estimate is $E_{c,1}$, there is a high chance of obtaining better quantum estimates using quantum routines e.g. QPE. However, if the best classical estimate is $E_{c,2}$, the chance is quite slim. $\bar{E}$, the energy of the implemented state is also shown, but its value is irrelevant for predicting the likelihood of obtaining better energy estimates. Note that in practice, one must ensure that the weight to the left of each of these energy estimates is not due to the broadening of higher energy weight; such broadening ($\eta$ in \ref{['eq:def_energy_dist']}) is inevitable in any actual calculation, but one can examine the behavior of the tails as $\eta$ is varied to determine whether the weight in the tail is due to such broadening or not.

Theorems & Definitions (2)

• Lemma 3.1
• proof : Proof of \ref{['lem:unique_signature']}