
This quantity, in general, becomes exponentially large in the system size due its denominator, so that the
coupling strength or the bath correlation time need to be sufficiently small for the use of Davies generators to
be justified, up to some upper system size limit. In practice, for convenience and also because it may apply
in a range that is larger than the rigorous but conservative bounds imply, the Lindblad master equation [5]
with Davies generators is often used outside its formal range of applicability in modeling of experiments
(e.g., Ref. [6–9]). In such cases it should be understood as a semi-empirical model of open system dynamics,
that possesses all the physical properties of the dynamics without reproducing them exactly. Unfortunately,
since we naturally wish to apply modeling tools to problems for which we do not know a priori what the
correct answer is, this approach cannot provide correctness guarantees. However, other methods are even
more susceptible to this problem, e.g., a path-integral (non-master equation) approach based on integrating
out the bath [10–12]: this involves integration over a very high-dimensional space, so methods [13] that bring
it back into the realm of numerical feasibility usually involve uncontrollable approximations.
One of the important requirements of open system dynamics is complete positivity [14], under which
density matrices whose eigenvalues are by definition non-negative are mapped to other such matrices, even
when applied to a subsystem of a larger system.
2
This property is equivalent to the Markovian master
equation being in Lindblad canonical form [5, 16]. There are numerous other master equations (e.g., Refs. [17–
39]), some in Lindblad form and some not, whose domains of validity and ranges of applicability were not
always studied in detail. One promising candidate is a coarse-grained master equation (CGME) [40], which
is in Lindblad form and thus automatically completely positive (CP). In this work we show that a suitably
generalized variant of this equation, that includes an arbitrarily time-dependent system Hamiltonian (hence
called the “time-dependent CGME”), has a much milder error than (1):
system-bath coupling strength ×bath correlation time , (2)
which does not diverge with the system size n for local observables, and has only a polynomial [O(n
α
) with
1/2 ≤ α ≤ 1] prefactor for nonlocal observables. We note that the full form of the error estimate (2) is not
uniform over the evolution time, unlike some of the error bounds developed elsewhere [39, 41]. We also show
that this new master equation is an improvement over the Lindblad equation with Davies generators in that
it has a local structure, compatible with approximate simulation methods with asymptotically smaller costs
[O(n) vs. O(exp(n))], and that the error estimate (2) is valid for an arbitrary time-dependence of the system
Hamiltonian. In particular, the time-dependent CGME is compatible with the assumption of arbitrarily fast
gates, often made in the circuit model of quantum computing, e.g., in the analysis of fault-tolerant quantum
error correction [42]. This assumption was shown to be incompatible with the derivation of the Davies master
equation [43].
This paper is structured as follows. We summarize our main results in Sec. 2, where we present the main
master equations involved in this work: Redfield and Davies-Lindblad (well known), and coarse-grained, both
time-independent and time-dependent (new). We also present bounds that provide the range of applicability
of each type of master equation (new). We show that these bounds can be expressed entirely in terms of
only two timescales, namely the fastest system decoherence timescale, and the characteristic timescale of the
decay of the bath correlation function. The reader interested only in the results and not in the details of
derivations can safely read only this section and then skip to the conclusions in Sec. 7. Derivations begin in
Sec. 3, where we present a simple, new derivation of the time-independent CGME. We express the equation
in CP (Lindblad) form, state the range of validity for different approximations made in the derivation, and
thus for the equation itself. We also compare the range of applicability with other master equations, and note
the spatial locality of the Lindblad generators of the CGME. At this point we are ready to address the case
of time-dependent Hamiltonians. The derivation of the equation for this case (which happens to result in
exactly the same form) is given in Sec. 4. In particular, this equation is well suited for the simulation of open
system adiabatic quantum computation, but also for dynamical decoupling, which involves the opposite limit
of very fast system dynamics. We note that while we will sometimes refer to qubits, none of the results we
discuss in this work are limited to qubits, and in fact any finite multi-level system, or interacting set of such
systems, is captured by the formalism. We study some applications and examples in Sec. 5, including error
suppression using a dynamical decoupling protocol, and numerical results of the comparison of our master
2
It should be noted that complete positivity can be relaxed without losing physicality; see, e.g., Ref. [15].
Accepted in Quantum 2020-01-19, click title to verify. Published under CC-BY 4.0. 2