First-quantized eigensolver for ground and excited states of electrons under a uniform magnetic field

Abstract

First-quantized eigensolver (FQE) is a recently proposed quantum computation framework for obtaining the ground state of an interacting electronic system based on probabilistic imaginary-time evolution. Here, we propose a method for introducing a uniform magnetic field to the FQE calculation. Our resource estimation demonstrates that the additional circuit responsible for the magnetic field can be implemented with a linear depth in terms of the number of qubits assigned to each electron. Hence, introduction of the magnetic field has no impact on the leading order of the entire computational cost. The proposed method is validated by numerical simulations of the ground and excited states employing filtration circuits for the energy eigenstates. We also provide a generic construction of the derivative circuits together with measurement-based formulae. We can obtain the electric-current density in an electronic system to gain insights into the microscopic origin of the magnetic response.

Non-unitary Trotter circuits for imaginary time evolution

“…By comparison, LCU PITE block encodes the Taylor expansion of the ITE operator and therefore incurs an additional first-order error ∆τ . As a result, it might be expected that Trotterised PITE produces shorter circuits than LCU PITE, at the cost of requiring more mid-circuit measurements: using a second quantised representation, LCU PITE uses L controlled Pauli gadgets and L (not controlled) Pauli gadgets per time step, where L is the number of Pauli strings comprising the Hamiltonian, followed by a single mid-circuit measurement [93]. On the other hand, Trotterised PITE uses L (not controlled) Pauli gadgets per time step, along with L mid-circuit measurements.…”