  Decoherence between qubits is a major bottleneck in quantum computations.
Decoherence results from intrinsic quantum and thermal fluctuations as well as
noise in the external fields that perform the measurement and preparation
processes. With prescribed colored noise spectra for intrinsic and extrinsic
noise, we present a numerical method, Quantum Accelerated Stochastic Propagator
Evaluation (Q-ASPEN), to solve the time-dependent noise-averaged reduced
density matrix in the presence of intrinsic and extrinsic noise. Q-ASPEN is
arbitrarily accurate and can be applied to provide estimates for the resources
needed to error-correct quantum computations. We employ spectral tensor trains,
which combine the advantages of tensor networks and pseudospectral methods, as
a variational ansatz to the quantum relaxation problem and optimize the ansatz
using methods typically used to train neural networks. The spectral tensor
trains in Q-ASPEN make accurate calculations with tens of quantum levels
feasible. We present benchmarks for Q-ASPEN on the spin-boson model in the
presence of intrinsic noise and on a quantum chain of up to 32 sites in the
presence of extrinsic noise. In our benchmark, the memory cost of Q-ASPEN
scales linearly with the system size once the number of states is larger than
the number of basis functions.
