Fast quantum simulation of electronic structure by spectrum amplification

TL;DR

This work introduces spectrum amplification (SA) for SOS-representable electronic-structure Hamiltonians to reduce ground-state energy estimation costs on fault-tolerant quantum computers. It develops a practical SA circuit framework with rectangular block-encodings and quantum walks, and combines this with a novel DFTHC-SOS factorization to compress two-electron tensors while controlling the SOS gap Δ_gap. Classical SDP relaxations and symmetry-shifting (BLISS) guide the construction of low-Λ SOS representations, and a unified DFTHC+BLISS+SA pipeline yields large speedups (up to 4–195×) over prior methods for systems like FeMoco and CO₂ catalysts, with realistic resource estimates. The results demonstrate that combining SA with compact SOS representations can make high-accuracy ground-state energy estimation feasible for chemically relevant, strongly correlated systems on future quantum hardware.

Abstract

The most advanced techniques using fault-tolerant quantum computers to estimate the ground-state energy of a chemical Hamiltonian involve compression of the Coulomb operator through tensor factorizations, enabling efficient block-encodings of the Hamiltonian. A natural challenge of these methods is the degree to which block-encoding costs can be reduced. We address this challenge through the technique of spectrum amplification, which magnifies the spectrum of the low-energy states of Hamiltonians that can be expressed as sums of squares. Spectrum amplification enables estimating ground-state energies with significantly improved cost scaling in the block encoding normalization factor $Λ$ to just $\sqrt{2ΛE_{\text{gap}}}$, where $E_{\text{gap}} \ll Λ$ is the lowest energy of the sum-of-squares Hamiltonian. To achieve this, we show that sum-of-squares representations of the electronic structure Hamiltonian are efficiently computable by a family of classical simulation techniques that approximate the ground-state energy from below. In order to further optimize, we also develop a novel factorization that provides a trade-off between the two leading Coulomb integral factorization schemes -- namely, double factorization and tensor hypercontraction -- that when combined with spectrum amplification yields a factor of 4 to 195 speedup over the state of the art in ground-state energy estimation for models of Iron-Sulfur complexes and a CO$_{2}$-fixation catalyst.

Figures (9)

• Figure 1: Controlled block encoding of the second Chebyshev polynomial of $O_{\alpha}$. The control register contains the control state, $\vert \alpha\rangle$, of Eq. \ref{['eq:rect_BE_primitives']} which provides the appropriate weighting to each $T_{2}$ in the LCU. For simplicity, $\lambda_\alpha=1$.
• Figure 2: The complete quantum circuit for the block-encoding. First, 'prep' is the state preparation on $x_{\text{o}}$. The operation 'eq' creates an equal superposition on the $b$ register with succ $b$ flagging success. Then the QROM on $x_{\text{o}}$ and $b$ is used to output alt and keep data for the coherent alias sampling, as well as $G,r,c$ (which is independent of $b$). The operation with 'In' on alt and keep, with 'prep' on $b$ indicates the inequality test and controlled swap used in coherent alias sampling. Then $b,r$ are used to output the data for the rotations as well as flipping a flag qubit to indicate if $b=B$. There are two spin qubits $\mathinner{|{\sigma_0}\rangle}$ and $\mathinner{|{\sigma_1}\rangle}$, with one used for SF (the inner sum) and the other for $\text{D}_1$ and $\text{Q}_1$ (the outer sum). The controlled operation copies it out into $\mathinner{|{\sigma}\rangle}$ depending on $G$, and that is used to control the swap between the spin up and spin down system registers. The operation 'Maj' indicates the circuit for performing the Majorana operator as shown in Fig. \ref{['fig:Majorana']}. The operations are all inverted, except for the Hadamard on $\mathinner{|{\sigma_1}\rangle}$ and prep on $x_{\text{o}}$, which are part of the outer sum. There is a reflection on the control qubits used in the inner block-encoding, then the entire procedure is performed again. Note that we apply the Hermitian conjugate of this operator using Maj$^\dagger$.
• Figure 3: FeMoCo-54 CCSD(T) correlation energy error vs. rank of DFTHC truncation (small dots). The dashed line at $1.6$mHa is the chemical accuracy threshold and the shaded region is the threshold $0.3$mHa used to select the rank $R^*$ (large dots) of good solutions.
• Figure 4: The quantum circuit used to apply the Majorana operators on the single qubit, with the target qubit at the bottom.
• Figure 5: For molecule a) Fe$_2$S$_2$, b) Fe$_4$S$_4$, c) FeMoCo-54, d) FeMoCo-76, e) Cpd1X-58, and f) XVIII-56: (left) CCSD(T) correlation energy error vs. rank of DFTHC truncation (small dots). Error bars are the standard deviation of the mean $\bar{\epsilon}_\text{corr}$ estimated with $64$ samples. Dashed lines at $\pm 1.6$mHa are at the chemical accuracy threshold. (right) $\bar{\epsilon}_\text{corr}$ and estimated standard deviation $\sigma_\text{corr}$ of $\epsilon_\text{corr}$. The shaded region is the threshold $\epsilon_\text{th}=0.7$mHa used to select the rank of good solution. (below) Additional fits of other parameters relevant to estimation quantum simulation costs.