Dhrumil Patel

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.

Simulating open quantum systems is essential for modeling realistic dynamics beyond closed-system Hamiltonian evolution. Such dynamics are described by the Lindblad master equation for Markovian systems and arise in fields ranging from condensed matter and quantum chemistry to quantum optics and noise analysis in quantum devices. Existing algorithms typically rely on sparse-access or linear-combination-of-unitaries input models. We propose an alternative framework, Wave Matrix Lindbladization, inspired by density matrix exponentiation for sample-based Hamiltonian simulation. In this model, Hamiltonians and Lindblad operators are encoded directly into program states, enabling sample-based Lindbladian simulation. We present algorithms for this task, analyze their sample and gate complexities, and demonstrate both efficiency and optimality. We also show that our algorithms achieve better sample complexity than any tomographic strategy for Lindbladian simulation. This further suggests a form of quantum copy-protection, where program states allow Lindbladian simulation without revealing the operators they encode.