Laura Clinton

The limiting constraint on simulating materials on near-term quantum hardware is the requisite circuit depths and qubit numbers, with current estimates placing them well beyond near-term capabilities. A critical subroutine of simulation algorithms is implementing a layer of unitary evolutions by each local term in the Hamiltonian. In this work we develop a new quantum algorithm which dramatically reduces the estimated cost of material simulations using this subroutine, improving circuit depths by up to 6 orders of magnitude for Strontium Vanadate, for example. We achieve this by introducing a fermionic encoding that leverages the locality of materials Hamiltonians describing an active space in the Wannier basis. This design generates quantum circuits whose depth is independent of the system’s size.

Estimating quantum time-series such as the Loschmidt echo $f(t)=\langle\psi|\mathrm{e}^{-\mathrm{i}Ht}|\psi\rangle$ is central to spectroscopy and Hamiltonian analysis. Direct estimation via the Hadamard test requires controlled implementations of $\mathrm{e}^{-\mathrm{i}Ht}$, and the depth of these controlled circuits grows with $t$, making long-time estimation challenging on near-term hardware. Inspired by the classical Phaselift approach to phase retrieval, we introduce Quantum Phaselift, a lifting-based framework that estimates the rank-one matrix $Z = f f^\dagger$ sampled at discrete times instead of $f$ directly. We propose quantum circuits for estimating the entries of $Z$ and prove that measuring only a narrow band of this matrix around the diagonal provides sufficient data for unique signal reconstruction. This reformulation reduces the depth of required controlled circuits to scale with the width of the measured band, rather than with the total evolution time. We then show that generic signals can be recovered from a band of width $O(1)$, providing a substantial savings in controlled operations compared to naïve algorithms. We develop three robust estimators to recover the signal from the noisy measurements of the entries on this narrow band: a block-by-block algebraic estimator, a block-by-block eigenvector estimator, and a least-squares estimator. We rigorously prove exact recovery for all three estimators in the noiseless setting and establish stability guarantees and sample complexity bounds for the block-by-block algebraic estimator in the presence of measurement noise. Finally, we numerically demonstrate that high-quality signal recovery is possible for the 2D Fermi-Hubbard and 2D transverse-field Ising model time-series with more than 100 points using only a few million samples and reasonable post-processing time, making our recovery techniques efficient and effective for near-term implementations.