Mapping Molecular Hamiltonians into Hamiltonians of Modular cQED Processors

Abstract

We introduce a general method based on the operators of the Dyson-Masleev transformation to map the Hamiltonian of an arbitrary model system into the Hamiltonian of a circuit Quantum Electrodynamics (cQED) processor. Furthermore, we introduce a modular approach to programming a cQED processor with components corresponding to the mapping Hamiltonian. The method is illustrated as applied to quantum dynamics simulations of the Fenna-Matthews-Olson (FMO) complex and the spin-boson model of charge transfer. Beyond applications to molecular Hamiltonians, the mapping provides a general approach to implement any unitary operator in terms of a sequence of unitary transformations corresponding to powers of creation and annihilation operators of a single bosonic mode in a cQED processor.

Simulating Chemistry on Bosonic Quantum Devices

“…Within this equation, the generator of dynamics is the molecular Hamiltonian. It has recently been shown that the molecular Hamiltonian can be mapped onto a polynomial of bosonic creation and annihilation operators of a single bosonic mode (SBM) . Thus, using this so-called SBM mapping allows for the simulation of chemical dynamics on a cQED device.…”

“…Specifically, the SBM mapping makes it possible to transform an arbitrary k × k Hamiltonian matrix Ĥ = prefix∑ α = 0 k − 1 ∑ α ′ = 0 k − 1 H α α false′ | α ⟩ ⟨ α ′ | which is represented in terms of the basis set {|α⟩} of choice, into the following elementary bosonic operators of a single qumode ( â , â † ), as follows: Ĥ sbm = prefix∑ m = 0 k − 1 ∑ n = 0 k − 1 H n m P̂ n m where P̂ n m ≡ 1 ( k − 1 ) ! 2 m ! n ! false( â † false) n normalΓ̂ k k − 1 false( â † false) k − 1 − m …”

“…Similarly, any 2 × 2 Hamiltonian such as the Pauli operators can be mapped onto the Hamiltonian of a currently available bosonic quantum device known as driven superconducting nonlinear asymmetric inductive element (SNAIL) ,, Ĥ SNAIL = ℏ ω â † â + g 3 false( â + â † false) 3 Given that all 1-qubit gates correspond to unitary dynamics generated by a 2 × 2 Hamiltonian, they can therefore be generated by such SNAIL devices. Furthermore, 4 × 4 Hamiltonian matrices, which correspond to 2-qubit gates, can also be implemented by combining the SNAIL with a combination of Josephson junctions that gives rise to fourth-order terms in the device Hamiltonian …”

“…The SBM mapping has also been used to simulate quantum dynamics in several chemical systems, including energy transfer dynamics in the Fenna–Matthews–Olson (FMO) light-harvesting complex and charge-transfer (redox) chemical reactions within the spin-boson model . As an example, we consider the application of SBM mapping to the FMO complex (see Figure ).…”

“…An introduction to bosonic quantum computing is provided in Section . Recent progress in simulating molecular vibronic spectra , and chemical quantum dynamics on bosonic quantum devices is reviewed in Section . The potential for bosonic quantum devices to solve a wide range of timely problems in computational chemistry, including nonadiabatic chemical dynamics, the efficient solution of molecular graph theory problems and electronic structure calculations is highlighted in Section .…”