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.
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.
For any master equation which is local in time, whether Markovian, non-Markovian, of Lindblad form or not, a general procedure is given for constructing the corresponding linear map from the initial state to the state at time t, including its Kraus-type representations. Formally, this is equivalent to solving the master equation. For an N-dimensional Hilbert space it requires (i) solving a first order N 2×N 2 matrix time evolution (to obtain the completely positive map), and (ii) diagonalizing a related N 2×N 2 matrix (to obtain a Kraus-type representation). Conversely, for a given time-dependent linear map, a necessary and sufficient condition is given for the existence of a corresponding master equation, where the (not necessarily unique) form of this equation is explicitly determined. It is shown that a “best possible” master equation may always be defined, for approximating the evolution in the case that no exact master equation exists. Examples involving qubits are given.
Quantum master equations are an invaluable and highly successful tool for the approximate, yet accurate, description of undriven open quantum systems weakly coupled to their environment. In contrast, results on the description of the equally important case of driven open quantum systems using quantum master equations remain scarce. Existing approaches typically impose restrictions on the driving protocol, including, for instance, Floquet master equations for periodic driving and adiabatic master equations for slow driving. However, no generally accepted formulation for arbitrary driving has yet been established. A promising candidate from which all existing approaches can be derived by approximations based on protocol constraints is the driven Redfield master equation. Strikingly, it has received very little attention so far. A main reason is that, in analogy to its undriven counterpart, the driven Redfield master equation notoriously violates complete positivity. In the undriven case, this problem can be successfully addressed using the rotating wave approximation (RWA). For approaches derived from the driven Redfield master equation for specific protocols, such as Floquet master equations and adiabatic master equations, a generalized form of the RWA can still be applied. However, a full generalization directly applicable to the driven Redfield master equation under arbitrary driving has not yet been developed. To address this issue, we propose a straightforward and natural generalization of the RWA based on the version of the RWA used for adiabatic master equations. We then apply the driven Redfield master equation to the paradigmatic, exactly solvable model of a driven quantum harmonic oscillator coupled to a bosonic bath and benchmark it against the exact solution. In particular, we compare the original driven Redfield master equation without an RWA to its generalized RWA version. We find that the driven Redfield master equation without an RWA exhibits excellent agreement with the exact solution across all driving speeds, while the violation of complete positivity shows no notable effect. By contrast, the RWA-based version of the driven Redfield master equation fails to reproduce the dynamics accurately beyond slow driving at short times. Additionally, we analyze known adiabatic master equations, both without and with an RWA. Remarkably, the non-RWA versions agree with the exact dynamics far beyond the strictly adiabatic regime, while the RWA-based versions already show noticeable deviations even for slow driving, contrasting previous observations in the literature. Taken together, these results indicate that the driven Redfield master equation and adiabatic master equations without RWA have been significantly underestimated in their practical usefulness, whereas RWA-based approaches in driven settings may have been overestimated. This strongly suggests that future work should focus on alternative approximation methods applied to the driven Redfield master equation that preserve its high numerical accuracy while restoring complete positivity.
The ubiquitous effects of the environment on quantum-mechanical systems generally cause temporally correlated fluctuations. This particularly holds for systems of interest for quantum computation where such effects lead to correlated errors. The Markovian approximation neglects these correlations and thus fails to accurately describe open-system dynamics where these correlations become relevant. In driven open systems, yet another approximation is persistently used, often unknowingly, in which one describes the decay effects independently from the time-dependent controlling fields acting on the system, thereby ignoring further temporally correlated effects. To overcome these shortcomings, we develop a quantum master equation for driven systems weakly coupled to quantum environments that avoids the aforementioned field-independent approximation, as well as the Markovian approximation. Our method makes it possible to track all occurring decay channels and their time-dependent generalized rates which we illustrate in the example of a generally driven two-level system. We also demonstrate that correlated and field-dependent dissipative effects can lead to an increase in the performance of single-qubit gate operations.
The primary aim of the thesis is to introduce and investigate an original protocol for the design and the optimization of parameterized noisy quantum channels, with the goal of mitigating hardware-induced noise. Significant attention is also devoted to the design of noise models, which are essential for emulation purposes. The protocol begins with the derivation of a noise model for a hardware gate, represented as a quantum channel, that can be simulated classically. This task is achieved through process tomography, a procedure that enables the characterization of the channel. It should be noted that such simulation is feasible on classical computers, since the optimization task concerns low-dimensional gates typically acting on 2–3 qubits. The noisy channel is then engineered by embedding it into a probabilistic ensemble of circuits, in which additional single-qubit parameterized gates — assumed to be noiseless — are introduced. Through this procedure, the noisy channel becomes parameterized and can be optimized according to an appropriate metric. Following a worst-case, agnostic perspective, we choose the diamond distance between the noisy channel and the target unitary channel as the cost function. This metric is generally considered the most suitable for distinguishing quantum channels. The resulting optimal ensemble of circuits can be implemented on real hardware under fixed calibration conditions, yielding a channel that approximates the target more closely than the original noisy channel. This approach can be interpreted as a noise mitigation strategy based on pre-processing design and optimization: rather than correcting errors after the computation, less noisy channels are employed during the computation itself. In this context, the protocol — within the noise-aware variational quantum compiling framework — shares with the randomized compiling the principle of circuit dressing to tailor noise during execution. The methodology is validated on two toy models of the single-qubit identity with amplitude damping noise, combining both analytical and numerical approaches. The analysis highlights the role of convexity in the space of quantum channels, with explicit demonstrations of optimal convex combinations and connections to encoding-decoding and dynamical decoupling strategies. The analysis of these simple models is intended as a proof-of-principle of the protocol in a controlled and well-characterized environment. In addition, this thesis introduces the design of random noise models that are constrained to maintain a fixed diamond distance from a target quantum channel. The controlled addition of random noise to a unitary channel represents a significant and — to our knowledge — previously unexplored approach within noise modeling. Indeed, in principle, the currently available built-in noise models may limit the expressivity of the emulation. The task was achieved by enriching the noise spectrum through the addition of random Kraus operators and small systematic rotations. To assess the effects of hardware-induced noise, we proposed an experimental design aimed at simulating the time evolution of a triangular Ising model in a transverse field. Our approach employs Trotterization, adapted to the topology and connectivity of a certain quantum processing unit (QPU) profile. Building on the noisy native gates of the QPU profile, we propose the emulation, engineering and optimization of a noisy CZ gate, subsequently employed within sequences of non-native gates. Using the designed noise models, we perform several emulations aimed at evaluating the effectiveness of the protocol in the presence of progressively dominant systematic rotations, and at comparing it with Pauli twirling, a technique of randomized compiling.