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Kraus map closed-form solution for general master equation dynamics

Shahrukh Chishti, Francisco Andrés Cárdenas-López, Felix Motzoi

Abstract

The Kraus representation of quantum channels allows for a precise emulation of the complex dynamics that take place on quantum processors, whether for benchmarking algorithms, predicting the performance of error correction and mitigation, or in the myriad other uses of compiled digital sequences. Nonetheless, starting from first principles to obtain continuous quantum master equations involves various approximations such as weak coupling to the environment. Further, converting these equations to Kraus operators cannot generally be obtained in closed-form due to the complicated commutator structure of the problem. In our work, we bridge this gap by providing a general closed form formulation for arbitrarily strong driving while remaining linear in the dissipator. The Kraus solution is expressed as a Riemann sum where higher terms can converge quickly to high precision, which we demonstrate numerically. Such a formulation is highly relevant to quantum computing and gate-based models, where effective models are highly sought for large rotation gate angles, even under the influence of underlying non-trivial noise mechanisms.

Kraus map closed-form solution for general master equation dynamics

Abstract

The Kraus representation of quantum channels allows for a precise emulation of the complex dynamics that take place on quantum processors, whether for benchmarking algorithms, predicting the performance of error correction and mitigation, or in the myriad other uses of compiled digital sequences. Nonetheless, starting from first principles to obtain continuous quantum master equations involves various approximations such as weak coupling to the environment. Further, converting these equations to Kraus operators cannot generally be obtained in closed-form due to the complicated commutator structure of the problem. In our work, we bridge this gap by providing a general closed form formulation for arbitrarily strong driving while remaining linear in the dissipator. The Kraus solution is expressed as a Riemann sum where higher terms can converge quickly to high precision, which we demonstrate numerically. Such a formulation is highly relevant to quantum computing and gate-based models, where effective models are highly sought for large rotation gate angles, even under the influence of underlying non-trivial noise mechanisms.
Paper Structure (7 sections, 11 equations, 2 figures)

This paper contains 7 sections, 11 equations, 2 figures.

Table of Contents

  1. Open quantum system dynamics
  2. Linear order dissipation dynamics
  3. Riemann Summation
  4. Radius of Convergence
  5. Example
  6. Discussion & Summary
  7. Acknowdedgements

Figures (2)

  • Figure 1: Comparison of accuracy of super-operator representation of various $1^{st}$ order approximations over total time of evolution (x-axis). The accuracy is calculated over $L_2$ norm against the difference of exact (exponential) evolution. $d\Phi$ is infinitesimal time evolution of the Liouvillian. Small error regulator is fixed at $\epsilon_0 = .01$ (horizontal dotted). Simulation parameters for Hamiltonian with leakage in Eq. \ref{['def:hamiltonian-3LS']}: $\Omega=\pi, \ \alpha = \Omega/20 , \ \eta = \sqrt{2}$ with a decay rate $\gamma = \epsilon_0 \Omega$.
  • Figure 2: Increasing accuracy of Kraus-approximation maps with number of quadratures (x-axis) describing the interaction ($R$). Accuracy comparison against exact simulation is normalized for three different times of simulation ($\tau = 1,.1,.01$ns). Simulation parameters are same as in Fig. \ref{['fig:error-L2-times']}.