124 research outputs found
Simultaneous estimation of multiple eigenvalues with short-depth quantum circuit on early fault-tolerant quantum computers
We introduce a multi-modal, multi-level quantum complex exponential least
squares (MM-QCELS) method to simultaneously estimate multiple eigenvalues of a
quantum Hamiltonian. The circuit depth and the total cost exhibit
Heisenberg-limited scaling. The quantum circuit uses one ancilla qubit, and
under suitable initial state conditions, the circuit depth can be much shorter
than that of quantum phase estimation (QPE) type circuits. As a result, this
method is well-suited for early fault-tolerant quantum computers. Our approach
extends and refines the quantum complex exponential least squares (QCELS)
method, recently developed for estimating a single dominant eigenvalue [Ding
and Lin, arXiv:2211.11973]. Our theoretical analysis for estimating multiple
eigenvalues also tightens the bound for single dominant eigenvalue estimation.
Numerical results suggest that compared to QPE, the circuit depth can be
reduced by around two orders of magnitude under several settings for estimating
ground-state and excited-state energies of certain quantum systems
Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation
We develop a phase estimation method with a distinct feature: its maximal
runtime (which determines the circuit depth) is , where
is the target precision, and the preconstant can be
arbitrarily close to as the initial state approaches the target eigenstate.
The total cost of the algorithm satisfies the Heisenberg-limited scaling
. This is different from all previous
proposals, where is required even if the initial state is
an exact eigenstate. As a result, our algorithm may significantly reduce the
circuit depth for performing phase estimation tasks on early fault-tolerant
quantum computers. The key technique is a simple subroutine called quantum
complex exponential least squares (QCELS). Our algorithm can be readily applied
to reduce the circuit depth for estimating the ground-state energy of a quantum
Hamiltonian, when the overlap between the initial state and the ground state is
large. If this initial overlap is small, we can combine our method with the
Fourier filtering method developed in [Lin, Tong, PRX Quantum 3, 010318, 2022],
and the resulting algorithm provably reduces the circuit depth in the presence
of a large relative overlap compared to . The relative overlap
condition is similar to a spectral gap assumption, but it is aware of the
information in the initial state and is therefore applicable to certain
Hamiltonians with small spectral gaps. We observe that the circuit depth can be
reduced by around two orders of magnitude in numerical experiments under
various settings
Single-ancilla ground state preparation via Lindbladians
We design an early fault-tolerant quantum algorithm for ground state
preparation. As a Monte Carlo-style quantum algorithm, our method features a
Lindbladian where the target state is stationary, and its evolution can be
efficiently implemented using just one ancilla qubit. Our algorithm can prepare
the ground state even when the initial state has zero overlap with the ground
state, bypassing the most significant limitation of methods like quantum phase
estimation. As a variant, we also propose a discrete-time algorithm, which
demonstrates even better efficiency, providing a near-optimal simulation cost
for the simulation time and precision. Numerical simulation using Ising models
and Hubbard models demonstrates the efficacy and applicability of our method
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