Theory of open quantum dynamics with hybrid noise

Abstract

We develop a theory to describe dynamics of a non-stationary open quantum system interacting with a hybrid environment, which includes high-frequency and low-frequency noise components. One part of the system-bath interaction is treated in a perturbative manner, whereas the other part is considered exactly. This approach allows us to derive a set of master equations where the relaxation rates are expressed as convolutions of the Bloch-Redfield and Marcus formulas. Our theory enables analysis of systems that have extremely small energy gaps in the presence of a realistic environment. As an illustration, we apply the theory to the 16 qubit quantum annealing problem with dangling qubits (Dickson et al 2013 Nat. Commun. 4 1903) and show qualitative agreement with experimental results.[28, 29] is not applicable. We provide an intuitively appealing and computationally convenient form for the transition rates in terms of a convolution between Redfield and Marcus formulas. As an example, we investigate a dissipative evolution of a 16 qubit system strongly interacting with LF noise and weakly coupled to a HF environment [33]. The problem is characterized by an extremely small gap in the energy spectrum of qubits in the middle of annealing. Solving the problem requires the right combination of Bloch-Redfield and Marcus approaches as well as a proper consideration of the calculation basis, which takes into account the nonstationary effects. The results of the present paper can also be applied to any other open quantum system within the validity range of the approximations.The paper is organized in the following way. Section 2 describes a single-qubit system to provide the necessary intuition before moving to more complicated multiqubit problems. Section 3 formulates the Hamiltonian and introduces important definitions and notations for the system and the bath. Master equations for the probability distribution of the quantum system are derived in section 4. Section 5 presents the relaxation rates as convolution integrals of the Bloch-Redfield and Marcus envelopes. In the section 6 we show that in equilibrium the master equations obey the detailed balance conditions. We also demonstrate that the convolution expression for the relaxation rates turns into the Marcus or to Bloch-Redfield formulas in the corresponding limits. Dissipative dynamics of a 16 qubit system with an extremely small energy gap is considered in section 7. A brief compilation of the commonly encountered notations is presented in appendix A. In other appendixes we provide a detailed derivation of many important formulas.

Benchmarking Hamiltonian Noise in the D-Wave Quantum Annealer

“…Similar to all architectures for quantum computing, QA hardware is sensitive to environmental noise. The theory of open quantum systems shows that noise during the annealing process leads to transitions out of the ground state [11]- [13]. However, under certain circumstances, noise can also help the system return to the ground state, leading to higher success probability [14], [15].…”

Single-Qubit Fidelity Assessment of Quantum Annealing Hardware

“…The adiabatic theorem states that if the interpolation is sufficiently slow and the quantum system is isolated, the proposed QA protocol will always find the ground state (i.e., optimal solution) to the H Ising problem [27], [28]. However, in existing QA hardware platforms, a wide variety of non-ideal properties can impact the results of a QA computation [29]- [31]. In particular, the D-Wave hardware documentation discusses five known sources of deviations from an ideal QA system called integrated control errors (ICE) [12], which include: background susceptibility; flux noise; DAC quantization; I/O system effects; and variable scale across qubits.…”

Anneal-path correction in flux qubits

“…The AME formalism breaks down if one tries to replace the Ohmic bath spectral function with a 1/f spectrum 27 . To handle this case other tools are needed, such as recent work on open-system evolution equations that can capture the effects of both fast and slow noise 34,35 . We have evidence (work in progress) that the polaron-transformed Redfield equation 36 with hybrid (slow and fast) environments yields linewidth-broadened features compared to the AME.…”