We demonstrate that the control protocols of quantum information devices can be simulated by assuming a low-rank ansatz for the density matrix. The rationale underlying this assumption is that quantum information protocols, by design, operate in a regime of nearly pure quantum states. Within the low-rank assumption, the simulation of these protocols is considerably faster than solving the full Lindblad master equation. This advantage can be used to increase the accuracy of the simulation by avoiding uncontrolled approximations and to streamline protocol optimization. We benchmark our approach on the optimization of the transmon qubit dispersive readout in a realistic transmon-resonator-filter model. With a Hilbert space dimension N = 2000 , assuming a rank as low as M = 20 , we achieve a nearly 100-fold speedup compared to full master equation integration while accurately reproducing all relevant observables. By combining the low-rank approximation with a compact pulse parametrization and gradient-free optimization, we obtain state-of-the-art readout assignment errors ɛ a ≈ 1.2 × 10 − 3 for a 40 ns readout pulse schedule, while comfortably running on a laptop and not relying on the rotating-wave approximation. Our approach is broadly applicable to most quantum control protocols, including quantum gates, state preparation, and fast reset operations. This establishes low-rank methods as a general tool for optimal control across diverse quantum platforms.
We demonstrate that the control protocols of quantum information devices can be simulated by assuming a low-rank ansatz for the density matrix. The rationale underlying this assumption is that quantum information protocols, by design, operate in a regime of nearly pure quantum states. Within the low-rank assumption, the simulation of these protocols is considerably faster than solving the full Lindblad master equation. This advantage can be used to increase the accuracy of the simulation by avoiding uncontrolled approximations and to streamline protocol optimization. We benchmark our approach on the optimization of the transmon qubit dispersive readout in a realistic transmon-resonator-filter model. With a Hilbert space dimension N = 2000 , assuming a rank as low as M = 20 , we achieve a nearly 100-fold speedup compared to full master equation integration while accurately reproducing all relevant observables. By combining the low-rank approximation with a compact pulse parametrization and gradient-free optimization, we obtain state-of-the-art readout assignment errors ɛ a ≈ 1.2 × 10 − 3 for a 40 ns readout pulse schedule, while comfortably running on a laptop and not relying on the rotating-wave approximation. Our approach is broadly applicable to most quantum control protocols, including quantum gates, state preparation, and fast reset operations. This establishes low-rank methods as a general tool for optimal control across diverse quantum platforms.
The simulation of many-body open quantum systems is key to solving numerous outstanding problems in physics, chemistry, material science, and in the development of quantum technologies. Near-term quantum computers may bring considerable advantage for the efficient simulation of their static and dynamical properties, thanks to hybrid quantum-classical variational algorithms to approximate the dynamics of the density matrix describing the quantum state in terms of an ensemble average. Here, a variational quantum algorithm is developed to simulate the real-time evolution of the density matrix governed by the Lindblad master equation, under the assumption that the quantum state has a bounded entropy along the dynamics, entailing a low-rank representation of its density matrix. The algorithm encodes each pure state of the statistical mixture as a parametrized quantum circuit, and the associated probabilities as additional variational parameters stored classically, thereby requiring a significantly lower number of qubits than algorithms where the full density matrix is encoded in the quantum memory. Two variational ansatze are proposed, and their effectiveness is assessed in the simulation of the dynamics of a 2D dissipative transverse field Ising model. The results underscore the algorithm's efficiency in simulating the dynamics of open quantum systems in the low-rank regime with limited quantum resources on a near-term quantum device.
We present a framework that combines the adjoint-state method together with reverse-time backpropagation to solve prohibitively large open-system quantum control problems. Our approach enables the optimization of arbitrary cost functions with fully general controls applied on large open quantum systems described by a Lindblad master equation. It is scalable, computationally efficient, and has a low-memory footprint. We apply this framework to optimize two inherently dissipative operations in superconducting qubits which lag behind in terms of fidelity and duration compared to other unitary operations: the dispersive readout and all-microwave reset of a transmon qubit. Our results show that while standard pulses for dispersive readout are nearly optimal, adding a transmon drive during the protocol can yield 2× improvements in fidelity and duration. We further demonstrate a 2× improvement in reset fidelity and duration through pulse shaping, indicating significant potential for enhancement in reset protocols. Our approach can readily be applied to optimize quantum controls in a vast range of applications such as reservoir engineering, autonomous quantum error correction, and leakage-reduction units.
We introduce low-rank optimal control (LROC), a method for designing control pulses in open quantum systems whose full density-matrix simulation is prohibitively expensive. The method exploits a feature of quantum computing itself: because protocols are designed to preserve purity, the density matrix is dominated by a few pure states and admits an accurate low-rank factorization. LROC propagates only this factorized form and, by deriving the corresponding adjoint equation, obtains the gradient of any differentiable objective at the same reduced cost as the simulation, leading to a quadratic improvement in time and memory compared to the full master equation. We illustrate the breadth of the method on four superconducting-circuit tasks: preparation of a five-qubit GHZ state, a CNOT gate, qubit readout, and an error correction primitive, modeled with realistic multilevel transmons, decay, and strong drives, in each case reaching fidelities consistent with the intrinsic dissipation limits. LROC thereby extends pulse-level optimization to system sizes beyond the reach of existing gradient-based methods.
Classical simulations of time-dependent quantum systems are widely used in quantum control research. In particular, these simulations are commonly used to host iterative optimal control algorithms. This is convenient for algorithms that are too onerous to run in the loop with current-day quantum hardware, as well as for researchers without consistent access to hardware. However, if the model used to represent the system is not selected carefully, an optimized control protocol may be rendered futile when applied to hardware. We present a series of models, ordered in a hierarchy of progressive approximation, which appear in quantum control literature. The validity of each model is characterized experimentally by designing and benchmarking control protocols for an IBMQ cloud quantum device. This result demonstrates error amplification induced by the application of a first-order perturbative approximation. Furthermore, the emergence of errors that cannot be corrected by simple amplitude scaling of control pulses is demonstrated in simulation, due to an underlying mistreatment of noncomputational dynamics. Finally, an evaluation of simulated control dynamics reveals that despite the substantial variance in numerical predictions across the proposed models, the complexity of discovering local optimal control protocols appears invariant in the simple control scheme setting.
In the current NISQ era, there is demand for functional quantum devices to solve relevant computational problems, which motivates a utilitarian perspective on device design: The goal is to create a device that is able to run a given algorithm with state-of-the-art performance. In this work, we use optimal control tools to derive the gate set required by a toy algorithm and, in tandem, explore the model space of superconducting quantum computer design, from dispersively coupled to stronger interacting qubits, to maximize gate fidelity. We employ perfect entangler theory to provide flexibility in the search for a two-qubit gate on a given platform and to compare designs with different entangling mechanisms, e.g., CPHASE and iSWAP. To ensure the applicability of our investigation, we limit ourselves to "simple" (i.e., sparse parametrization) pulses and quantify, where results differ from using the full complexity of piecewise constant controls.
In this work, we first consider standard quantum control problems for superconducting transmon qubits. Such quantum systems have been studied extensively both from an analytical and a numerical perspective [MSG&al11; TMW16]. Analytical pulses usually show dependencies from system parameters that are not easily found in numerical solutions. Therefore, our approach is then to employ neural networks to learn the functional dependence of optimal quantum control solutions from system parameters. We show that we can optimize such solutions analytically for single and two-qubit gates using either very few parameter samples or large numbers of pulse frequencies for large parameter ranges. Afterward, we move to a higher level of optimization. We consider variational gates in trapped-ion quantum computers [MMN&al16]. In this setting, we employ a hybrid scheme that uses both reinforcement learning and continuous optimization to optimize both circuit structure and variational angles concurrently [BDS&al18; SEL&al22]. We show that our reinforcement learning algorithm assisted by a continuous optimizer can construct effective solutions to the gate synthesis problem that matches and surpasses standard circuit compilers. Next, we consider variational optimization of non-linear maps [HCS&al23] acting on quantum states such as entanglement purification protocols. An entanglement purification protocol acts as a highly non-linear map, but it still outputs a faithful representation of a quantum system. Optimizing it for a specific family of states requires reducing the number of operations needed to retrieve the original state. This optimization has the potential to make purification protocols easier to implement on a quantum device. In fact, the exponential scaling of the purification protocol with the number of states [DB07] implies that such protocols may not be applied directly without careful engineering. We also show that the performance of standard purification protocols for arbitrary two-qubit input states leads to poor output values of the concurrence. Our optimized protocols prove instead able to increase the value of the concurrence above the maximum limit of traditional analytical protocols. We also show how the twirling operation becomes an obstacle to the performance of the protocol itself when considering random two-qubit states, although it is a useful tool in the design of entanglement purification protocols. Finally, we study parameter sampling in quantum circuits, focusing in particular on the LCU methods. Such problems are particularly interesting for meta-variational settings [CKA21] where we compute the average over observables evaluated at different points in the parameter space. They are also relevant for optimal quantum control algorithms because the computation of the fidelity with respect to a target operation is the basis of most optimal quantum control routines. In conclusion, we analyze several optimization problems that are relevant for quantum science and technology. We show that machine learning-assisted solutions can be applied successfully to engineer optimal quantum control pulses and compilation strategies based on variational angles. Also in the context of entanglement purification, we show how our optimized protocols can surpass current strategies for multiparametric families of states.