Jumpei Kato

Density matrix exponentiation (DME) is a general procedure that converts an unknown quantum state into the Hamiltonian evolution. This enables state-dependent operations and can reveal nontrivial properties of the state, among other applications, without full tomography. However, it has been proven that for any physical process, the DME requires $\Theta(1/\varepsilon)$ state copies in error $\varepsilon$. In this work, we go beyond the lower bound and propose a procedure called the \textit{virtual} DME that achieves $\mathcal{O}(\log(1/\varepsilon))$ or $\mathcal{O}(1)$ state copies, by using non-physical processes. Using the virtual DME in place of its conventional counterpart realizes a general-purpose quantum algorithm for property estimation, that achieves \textit{exponential} circuit-depth reductions over existing protocols across tasks including quantum principal component analysis, quantum emulator, calculation of nonlinear functions such as entropy, and linear system solver with quantum precomputation. In such quantum algorithms, the non-physical process for virtual DME can be effectively simulated via simple classical post-processing while retaining a near-unity measurement overhead. We numerically verify this small constant overhead together with the exponential reduction of copy count in the quantum principal component analysis task. The number of state copies used in our algorithm essentially saturates the theoretical lower bound we proved.

Preparing low-energy eigenstates, such as excited states, in quantum systems is a central challenge. Previous method reduces this to a ground-state preparation problem by modifying the system Hamiltonian using an approximate energy of the target state. However, this approach results in a large Hamiltonian simulation cost. Here, by adding a penalty term based on a system-size-independent conserved quantity, we successfully reduce this cost by a square-root factor. We confirm this through numerical simulations on the transverse-field Ising model and the Hubbard model.