Quantum Optimal Control Research Articles

Integrating quantum processor device and control optimization in a gradient-based framework

In a quantum processor, the device design and external controls together contribute to the quality of the target quantum operations. As we continuously seek better alternative qubit platforms, we explore the increasingly large device and control design space. Thus, optimization becomes more and more challenging. In this work, we demonstrate that the figure of merit reflecting a design goal can be made differentiable with respect to the device and control parameters. In addition, we can compute the gradient of the design objective efficiently in a similar manner to the back-propagation algorithm and then utilize the gradient to optimize the device and the control parameters jointly and efficiently. Therefore, our work extends the scope of the quantum optimal control to device design and provides an efficient optimization method. We also demonstrate the viability of gradient-based joint optimization over the device and control parameters through a few examples based on the superconducting qubits.

Quantum Optimal Control: Practical Aspects and Diverse Methods

Quantum controls realize the unitary or nonunitary operations employed in quantum computers, quantum simulators, quantum communications, and other quantum information devices. They implement the desired quantum dynamics with the help of electric, magnetic, or electromagnetic control fields. Quantum optimal control (QOC) deals with designing an optimal control field modulation that most precisely implements a desired quantum operation with minimum energy consumption and maximum robustness against hardware imperfections as well as external noise. Over the last 2 decades, numerous QOC methods have been proposed. They include asymptotic methods, direct search, gradient methods, variational methods, machine learning methods, etc. In this review, we shall introduce the basic ideas of QOC, discuss practical challenges, and then take an overview of the diverse QOC methods.

Circuit Complexity from Supersymmetric Quantum Field Theory with Morse Function

Computation of circuit complexity has gained much attention in the theoretical physics community in recent times, to gain insights into the chaotic features and random fluctuations of fields in the quantum regime. Recent studies of circuit complexity take inspiration from Nielsen’s geometric approach, which is based on the idea of optimal quantum control in which a cost function is introduced for the various possible path to determine the optimum circuit. In this paper, we study the relationship between the circuit complexity and Morse theory within the framework of algebraic topology, which will then help us study circuit complexity in supersymmetric quantum field theory describing both simple and inverted harmonic oscillators up to higher orders of quantum corrections. We will restrict ourselves to N=1 supersymmetry with one fermionic generator Qα. The expression of circuit complexity in quantum regime would then be given by the Hessian of the Morse function in supersymmetric quantum field theory. We also provide technical proof of the well known universal connecting relation between quantum chaos and circuit complexity of the supersymmetric quantum field theories, using the general description of Morse theory.

Optimal Control of Families of Quantum Gates.

Quantum optimal control (QOC) enables the realization of accurate operations, such as quantum gates, and supports the development of quantum technologies. To date, many QOC frameworks have been developed, but those remain only naturally suited to optimize a single targeted operation at a time. We extend this concept to optimal control with a continuous family of targets, and demonstrate that an optimization based on neural networks can find families of time-dependent Hamiltonians realizing desired classes of quantum gates in minimal time.

Robust magnetometry with single nitrogen-vacancy centers via two-step optimization

Shallow Nitrogen-Vacancy (NV) centers are promising candidates for high-precision sensing applications; these defects, when positioned a few nanometers below the surface, provide an atomic-scale resolution along with substantial sensitivity. However, the dangling bonds and impurities on the diamond surface result in a complex environment which reduces the sensitivity and is unique to each shallow NV center. To avoid the environment's detrimental effect, we apply feedback-based quantum optimal control. We first show how a direct search can improve the initialization/readout process. In a second step, we optimize microwave pulses for pulsed Optically Detected Magnetic Resonance (ODMR) and Ramsey measurements. Throughout the sensitivity optimizations, we focus on robustness against errors in the control field amplitude. This feature not only protects the protocols' sensitivity from drifts but also enlarges the sensing volume. The resulting ODMR measurements produce sensitivities below 1$\mu$T\,Hz$^{-\frac{1}{2}}$ for an 83\% decrease in control power, increasing the robustness by approximately one third. The optimized Ramsey measurements produce sensitivities below 100\,nT\,Hz$^{-\frac{1}{2}}$ giving a two-fold sensitivity improvement. Being on par with typical sensitivities obtained via single NV magnetometry, the complementing robustness of the presented optimization strategy may provide an advantage for other NV-based applications.

Optimal Control of Molecular Spin Qudits

We demonstrate, numerically, the possibility of manipulating the spin states of molecular nanomagnets with shaped microwave pulses designed with quantum optimal control theory techniques. The state-to-state or full gate transformations can be performed in this way in shorter times than using simple monochromatic resonant pulses. This enhancement in the operation rates can therefore mitigate the effect of decoherence. The optimization protocols and their potential for practical implementations are illustrated by simulations performed for a simple molecular cluster hosting a single ${\mathrm{Gd}}^{3+}$ ion. Its eight accessible levels (corresponding to a total spin $S=7/2$) allow encoding an eight-level qudit or a system of three coupled qubits. All necessary gates required for universal operation can be obtained with optimal pulses using the intrinsic couplings present in this system. The application of optimal control techniques can facilitate the implementation of quantum technologies based on molecular spin qudits.

Time-Optimal Two- and Three-Qubit Gates for Rydberg Atoms

We identify time-optimal laser pulses to implement the controlled-Z gate and its three qubit generalization, the C2Z gate, for Rydberg atoms in the blockade regime. Pulses are optimized using a combination of numerical and semi-analytical quantum optimal control techniques that result in smooth Ansätze with just a few variational parameters. For the CZ gate, the time-optimal implementation corresponds to a global laser pulse that does not require single site addressability of the atoms, simplifying experimental implementation of the gate. We employ quantum optimal control techniques to mitigate errors arising due to the finite lifetime of Rydberg states and finite blockade strengths, while several other types of errors affecting the gates are directly mitigated by the short gate duration. For the considered error sources, we achieve theoretical gate fidelities compatible with error correction using reasonable experimental parameters for CZ and C2Z gates.

Speed Limits for Macroscopic Transitions

Speed of state transitions in macroscopic systems is a crucial concept for foundations of nonequilibrium statistical mechanics as well as various applications in quantum technology represented by optimal quantum control. While extensive studies have made efforts to obtain rigorous constraints on dynamical processes since Mandelstam and Tamm, speed limits that provide tight bounds for macroscopic transitions have remained elusive. Here, by employing the local conservation law of probability, the fundamental principle in physics, we develop a general framework for deriving qualitatively tighter speed limits for macroscopic systems than many conventional ones. We show for the first time that the speed of the expectation value of an observable defined on an arbitrary graph, which can describe general many-body systems, is bounded by the "gradient" of the observable, in contrast with conventional speed limits depending on the entire range of the observable. This framework enables us to derive novel quantum speed limits for macroscopic unitary dynamics. Unlike previous bounds, the speed limit decreases when the expectation value of the transition Hamiltonian increases; this intuitively describes a new tradeoff relation between time and quantum phase difference. Our bound is dependent on instantaneous quantum states and thus can achieve the equality condition, which is conceptually distinct from the Lieb-Robinson bound. Our work elucidates novel speed limits on the basis of local conservation law, providing fundamental limits to various types of nonequilibrium quantum macroscopic phenomena.

Fast, high precision dynamics in quantum optimal control theory

Quantum optimal control theory is becoming increasingly crucial as quantum devices become more precise, but the need to quickly optimize these systems classically remains a significant bottleneck in their operation. Here we present a new theoretical quantum control framework for much faster optimization than the state of the art by replacing standard time propagation with a product of short-time propagators, each calculated using the Magnus expansion. The derived formulas for exact series terms and their gradients, based on earlier approximate integrals in a simulation setting, allow us to subsume the high cost of calculating commutators and integrals as an initial overhead. This provides an order of magnitude speedup for quantum control optimization.

Engineering strong beamsplitter interaction between bosonic modes via quantum optimal control theory

In continuous-variable quantum computing with qubits encoded in the infinite-dimensional Hilbert space of bosonic modes, it is a difficult task to realize strong and on-demand interactions between the qubits. One option is to engineer a beamsplitter interaction for photons in two superconducting cavities by driving an intermediate superconducting circuit with two continuous-wave drives, as demonstrated in a recent experiment. Here, we show how quantum optimal control theory (OCT) can be used in a systematic way to improve the beamsplitter interaction between the two cavities. We find that replacing the two-tone protocol by a three-tone protocol accelerates the effective beamsplitter rate between the two cavities. The third tone's amplitude and frequency are determined by gradient-free optimization and make use of cavity-transmon sideband couplings. We show how to further improve the three-tone protocol via gradient-based optimization while keeping the optimized drives experimentally feasible. Our work exemplifies how to use OCT to systematically improve practical protocols in quantum information applications.

Quantum computing with two independent control functions: Optimal solutions to the teleportation protocol

We investigate the quantum computing paradigm consisted of obtaining a target state that encodes the solution of a certain computational task by evolving the system with a combination of the problem-Hamiltonian and the driving-Hamiltonian. We analyze this paradigm in the light of Optimal Control Theory considering each Hamiltonian modulated by an independent control function. In the case of short evolution times and bounded controls, we analytically demonstrate that an optimal solution consists of both controls tuned at their upper bound for the whole evolution time. This optimal solution is appealing because of its simplicity and experimental feasibility. To numerically solve the control problem, we propose the use of a quantum optimal control technique adapted to limit the amplitude of the controls. As an application, we consider a teleportation protocol and compare the fidelity of the teleported state obtained for the two-control functions with the usual single-control function scheme and with the quantum approximate optimization algorithm (QAOA). We also investigate the energetic cost and the robustness against systematic errors in the teleportation protocol, considering different time evolution schemes. We show that the scheme with two-control functions yields a higher fidelity than the other schemes for the same evolution time.

Model-Free Quantum Control with Reinforcement Learning

Model bias is an inherent limitation of the current dominant approach to optimal quantum control, which relies on a system simulation for optimization of control policies. To overcome this limitation, we propose a circuit-based approach for training a reinforcement learning agent on quantum control tasks in a model-free way. Given a continuously parameterized control circuit, the agent learns its parameters through trial-and-error interaction with the quantum system, using measurement outcomes as the only source of information about the quantum state. Focusing on control of a harmonic oscillator coupled to an ancilla qubit, we show how to reward the learning agent using measurements of experimentally available observables. We train the agent to prepare various non-classical states using both unitary control and control with adaptive measurement-based quantum feedback, and to execute logical gates on encoded qubits. This approach significantly outperforms widely used model-free methods in terms of sample efficiency. Our numerical work is of immediate relevance to superconducting circuits and trapped ions platforms where such training can be implemented in experiment, allowing complete elimination of model bias and the adaptation of quantum control policies to the specific system in which they are deployed.

Qopt: An Experiment-Oriented Software Package for Qubit Simulation and Quantum Optimal Control

Realistic modeling of qubit systems including noise and constraints imposed by control hardware is required for performance prediction and control optimization of quantum processors. We introduce qopt, a software framework for simulating qubit dynamics and robust quantum optimal control considering common experimental situations. To this end, we model open and closed qubit systems with a focus on the simulation of realistic noise characteristics and experimental constraints. Specifically, the influence of noise can be calculated using Monte Carlo methods, effective master equations or with the efficient filter function formalism, which enables the investigation and mitigation of autocorrelated noise. In addition, limitations of control electronics, including finite bandwidth effects as well as nonlinear transfer functions and drive-dependent noise, can be considered. The calculation of gradients based on analytic results is implemented to facilitate the efficient optimization of control pulses. The software easily interfaces with QuTiP, is published under an open-source license, is well tested, and features a detailed documentation.

Projection-operator-based Newton method for the trajectory optimization of closed quantum systems

Quantum optimal control is an important technology that enables fast state preparation and gate design. In the absence of an analytic solution, most quantum optimal control methods rely on an iterative scheme to update the solution estimate. At present, the convergence rate of existing solvers is at most superlinear. This paper develops a new general purpose solver for quantum optimal control based on the PRojection Operator Newton method for Trajectory Optimization, or PRONTO. Specifically, the proposed approach uses a projection operator to incorporate the Schr\"odinger equation directly into the cost function, which is then minimized using a quasi-Newton method. At each iteration, the descent direction is obtained by computing the analytic solution to a Linear-Quadratic trajectory optimization problem. The resulting method guarantees monotonic convergence at every iteration and quadratic convergence in proximity of the solution. To highlight the potential of PRONTO, we present an numerical example that employs it to solve the optimal state-to-state mapping problem for a qubit and compares its performance to a state-of-the-art quadratic optimal control method.

A tutorial on optimal control and reinforcement learning methods for quantum technologies

Quantum Optimal Control is an established field of research which is necessary for the development of Quantum Technologies. In recent years, Machine Learning techniques have been proved useful to tackle a variety of quantum problems. In particular, Reinforcement Learning has been employed to address typical problems of control of quantum systems. In this tutorial we introduce the methods of Quantum Optimal Control and Reinforcement Learning by applying them to the problem of three-level population transfer. The jupyter notebooks to reproduce some of our results are open-sourced and available on github1.

Super-Heisenberg scaling in Hamiltonian parameter estimation in the long-range Kitaev chain

In quantum metrology, nonlinear many-body interactions can enhance the precision of Hamiltonian parameter estimation to surpass the Heisenberg scaling. Here, we consider the the estimation of the interaction strength in linear systems with long-range interactions and using the Kitaev chains as a case study, we establish a transition from the Heisenberg to super-Heisenberg scaling in the quantum Fisher information by varying the interaction range. We further show that quantum control can improve the prefactor of the quantum Fisher information. Our results explore the advantage of optimal quantum control and long-range interactions in many-body quantum metrology.

Analytic Filter-Function Derivatives for Quantum Optimal Control

Autocorrelated noise appears in many solid-state qubit systems and hence needs to be taken into account when developing gate operations for quantum information processing. However, explicitly simulating this kind of noise is often less efficient than approximate methods. Here, we focus on the filter function formalism, which allows the computation of gate fidelities in the presence of autocorrelated classical noise. Hence, this formalism can be combined with optimal control algorithms to design control pulses, which optimally implement quantum gates. To enable the use of gradient-based algorithms with fast convergence, we present analytically derived filter function gradients with respect to control pulse amplitudes, and analyze the computational complexity of our results. When comparing pulse optimization using our derivatives to a gradient-free approach, we find that the gradient-based method is roughly 2 orders of magnitude faster for our test cases. We also provide a modular computational implementation compatible with quantum optimal control packages.

Optimizing quantum control pulses with complex constraints and few variables through autodifferentiation

Applying optimal control algorithms on realistic quantum systems confronts two key challenges: to efficiently adopt physical constraints in the optimization and to minimize the variables for the convenience of experimental tune-ups. In order to resolve these issues, we propose a novel algorithm by incorporating multiple constraints into the gradient optimization over piece-wise pulse constant values, which are transformed to contained numbers of the finite Fourier basis for bandwidth control. Such complex constraints and variable transformation involved in the optimization introduce extreme difficulty in calculating gradients. We resolve this issue efficiently utilizing auto-differentiation on Tensorflow. We test our algorithm by finding smooth control pulses to implement single-qubit and two-qubit gates for superconducting transmon qubits with always-on interaction, which remains a challenge of quantum control in various qubit systems. Our algorithm provides a promising optimal quantum control approach that is friendly to complex and optional physical constraints.

Robust Quantum Optimal Control with Trajectory Optimization

The ability to engineer high-fidelity gates on quantum processors in the presence of systematic errors remains the primary barrier to achieving quantum advantage. Quantum optimal control methods have proven effective in experimentally realizing high-fidelity gates, but they require exquisite calibration to be performant. We apply robust trajectory optimization techniques to suppress gate errors arising from system parameter uncertainty. We propose a derivative-based approach that maintains computational efficiency by using forward-mode differentiation. Additionally, the effect of depolarization on a gate is typically modeled by integrating the Lindblad master equation, which is computationally expensive. We employ a computationally efficient model and utilize time-optimal control to achieve high-fidelity gates in the presence of depolarization. We apply these techniques to a fluxonium qubit and suppress simulated gate errors due to parameter uncertainty below ${10}^{\ensuremath{-}7}$ for static parameter deviations of the order of 1%.

Robustly decorrelating errors with mixed quantum gates

Coherent errors in quantum operations are ubiquitous. Whether arising from spurious environmental couplings or errors in control fields, such errors can accumulate rapidly and degrade the performance of a quantum circuit significantly more than an average gate fidelity may indicate. As Hastings (2017 Quantum Inf. Comput. 17 488) and Campbell (2017 Phys. Rev. A 95 042306) have recently shown, by replacing the deterministic implementation of a quantum gate with a randomized ensemble of implementations, one can dramatically suppress coherent errors. Our work begins by reformulating the results of Hastings and Campbell as a quantum optimal control problem. We then discuss a family of convex programs able to solve this problem, as well as a set of secondary objectives designed to improve the performance, implementability, and robustness of the resulting mixed quantum gates. Finally, we implement these mixed quantum gates on a superconducting qubit and discuss randomized benchmarking results consistent with a marked reduction in the coherent error.