Session S38: Optimal Quantum Control

Variational quantum control for single- and two-qubit transmon gates

Transmon are among the most successful qubits in the superconducting world, enabling the largest gate-based quantum computations to-date. In this talk I will present a new set of techniques to design single-qubit and two-qubit gates for transmon qubits, which can be used to tune qubit frequencies, implement direct or cavity mediated CZ interactions [1], and √SWAP gates with tuneable couplers. The technique is based on variational estimates of the transmon qubit state dynamics, and provides semi-analytical controls that can be further tuned to develop more robust protocols. With minor efforts, methods provide various orders of magnitude improvement in fidelity over earlier techniques based on fast-quasi-adiabatic passages or conventional square or segmented pulses, as used in current experiments.

[1] L. DiCarlo et al, Nature 467, 574 (2010)
[2] R. Barends et al. Nature 508, 500 (2014)
[3] Y. Chen et al. Physical Review Letters 113,220502 (2014)   

Universal gates for protected superconducting qubits using optimal control

In this talk I will discuss the use of quantum optimal control theory to realize quantum gates for two protected superconducting circuits: the heavy-fluxonium qubit and the 0-π qubit. Utilizing automatic differentiation facilitates the simultaneous inclusion of multiple optimization targets, allowing one to obtain high-fidelity gates with realistic pulse shapes. For both qubits, disjoint support of low-lying wave functions prevents direct population transfer between the computational-basis states. Instead, optimal control favors dynamics involving higher-lying levels, effectively lifting the protection for a fraction of the gate duration. For the 0-π qubit, offset-charge dependence of matrix elements among higher levels poses an additional challenge for gate protocols. To mitigate this issue, we randomize the offset charge during the optimization process, steering the system towards pulse shapes insensitive to charge variations. Closed-system fidelities obtained are 99% or higher, and show slight reductions in open-system simulations.   

Uncomputability and complexity of quantum control

In laboratory and numerical experiments, physical quantities are known with a finite precision and described by rational numbers. Based on this, we deduce that quantum control problems both for open and closed systems are in general not algorithmically solvable, i.e., there exists no algorithm that can decide whether dynamics of an arbitrary quantum system can be manipulated by accessible external interactions (coherent and dissipative) such that a chosen target reaches a desired value. This conclusion holds even for the relaxed requirement of the target only approximately attaining the desired value. These findings does not preclude an algorithmic solvability for a particular class of quantum control problems. To arrive at these results, we develop a technique based on establishing the equivalence between quantum control problems and Diophantine equations, which are polynomial equations with integer coefficients and integer unknowns. In addition to proving uncomputability, this technique allows to construct quantum control problems belonging to different complexity classes. In particular, an example of the control problem involving a two-mode coherent field is shown to be NP-hard, contradicting a widely held believe that two-body problems are easy.   

Robust control for tight SWAP cold atom sensors

Cold atom sensors currently provide state-of-the-art performance under lab conditions. Deploying these in tight-SWAP conditions, however, presents a variety of challenges arising from laser instabilities, such as power, frequency, and phase fluctuations. We demonstrate that robust control techniques - related to dynamic decoupling and dynamically-corrected composite pulses - can significantly improve the sensitivity of cold atom sensors in tight SWAP conditions, achieving performance commensurate with state-of-the-art lab conditions. We derive these robust control pulses using a custom numerical-optimization package, producing control solutions robust to specific coloured noise spectrums, and with relevant experimental constraints including bandwidth limitations. Our robust protocol is benchmarked against standard atom interferometry protocols, showing comparable sensitivity under ideal conditions, and over an order or magnitude improvement in a simulated field-deployed environment including realistic laser fluctuations.   

Optimizing quantum circuits subject to cross coupling

We present a new optimization technique which creates time-optimized circuits capable of combating unwanted crosstalk in transmon circuits. The low anharmonicity of transmons allows for the performance of quantum computations with either qubits or qutrits realized using the three lowest energy levels of the device. Unfortunately when employed in circuits residual ZZ-type couplings can lead to error sources which degrade algorithmic performance. We demonstrate the optimization of complex Unitaries on up to five qutrits using SU(2) and SU(3) operations. Based on a custom GPU-compatible optimization toolkit, high-fidelity and robust circuits are generated by optimizing both applied SU(N) operations and their timing in the circuit. We further demonstrate the ability to find robust solutions optimized to account for additional time-varying noise processes. We compare numeric with analytic results and GPU-optimization speeds with CPU-limited methods. Overall this approach forms a new kind of optimized circuit compilation which trades-off algorithmic execution time for improved overall error performance in a deterministic fashion.   

High fidelity unitary evolution using constrained control fields.

The problem of coherent errors, where the quantum unitary evolution is corrupted by populating spurious states (leakage error) or accumulates the wrong phase (phase error), is generic to a variety of situations. This type of errors often arise because it is a difficult task to find control fields that would generate with high accuracy a desired unitary evolution. The task becomes even more complex when taking into account experimental constraints: unavailable control fields, bandwidth limitations, amplitude limitations, etc. Here we attack a class of problems where suitable controls are known in an idealized scenario, but unwanted perturbative interactions disrupt the desired unitary evolution. We show how a recently-proposed Magnus expansion based algorithm for quantum control [1] can be modified to find control fields that both suppress coherent errors and are compatible with experimental constraints. In this talk, we focus on presenting the general method and revisit the problem of single qubit gates in a transmon qubit.
[1] H. Ribeiro, A. Baksic, and A. A Clerk, Phys. Rev. X 7, 011021 (2017).   

Quantum optimal control in a ZZ-free qubit architecture

Quantum optimal control is a powerful technique for improving the performance of quantum gates. Numerical methods, such as GRAPE, have been demonstrated to reduce infidelity of single- and two-qubit gates by several orders of magnitude. In this talk, we demonstrate the application of quantum optimal control in a newly proposed architecture in which the ZZ-interaction is inherently suppressed. We explore the parameter regimes of this system to optimize the performance of a cross-resonance gate on this architecture.   

A comparative study for reinforcement learning and traditional algorithms on state transfer problem

While reinforcement learning (RL) has been widely used in quantum control problems, it remains unclear whether RL is the most suitable algorithm when the control has specific constraints. We perform a comparative study on the efficacy of three RL algorithms: tabular Q-learning, deep Q-learning and policy gradient, as well as stochastic gradient descent and Krotov algorithms, in the problem of quantum state preparation. Overall, the deep Q-learning and policy gradient algorithms outperform others when the problem is discretized, e.g. allowing discrete values of control and when the problem scales up. Moreover, the reinforcement learning algorithms can also adaptively reduce the complexity of the control sequences. Our work provides insights into the suitability of reinforcement learning in quantum control problems.   

A geometric approach to dynamically corrected gates

Future technologies such as quantum computing, sensing and communication demand the ability to control microscopic quantum systems with unprecedented accuracy. This task is particularly daunting due to unwanted and unavoidable interactions with noisy environments that destroy quantum information through decoherence. I will present recent progress in understanding and modeling the effects of noise on the dynamics of a qubit and show how this can be used to develop new ways to slow down decoherence. I will then describe a new general theory for dynamically combatting decoherence by driving qubits in such a way that noise effects destructively interfere and cancel out. This theory exploits a rich geometrical structure hidden within the time-dependent Schrödinger equation.
  

Application of Pontryagin's Minimum Principle to Grover's Quantum Search Problem

Grover's algorithm is one of the most famous algorithms which explicitly demonstrates how the quantum nature can be utilized to accelerate the searching process. In this work, Grover's quantum search problem is mapped to a time-optimal control problem. Resorting to Pontryagin's Minimum Principle we find that the time-optimal solution has the bang-singular-bang structure. This structure can be derived naturally, without integrating the differential equations, using the geometric control technique where Hamiltonians in the Schr\"odinger's equation are represented as vector fields. In view of optimal control, Grover's algorithm uses the bang-bang protocol to approximate the optimal protocol with a minimized number of bang-to-bang switchings to reduce the query complexity. Our work provides a concrete example how Pontryagin's Minimum Principle is connected to quantum computation, and offers insight into how a quantum algorithm can be designed.   

Variational quantum gate optimization on superconducting qubit system

Hybrid quantum-classical (HQC) algorithms aim at realizing the quantum advantage in shallow depth quantum circuits with an aid of classical computation. Recently, HQC algorithms have been extensively studied with the expectation that they may solve practical problems in the near future. However, the quantum gate fidelities directly limit the sizes of executable problems in quantum computers without quantum error correction. While HQC algorithms require fewer quantum gates, the state-of-the-art gate fidelities are still insufficient to deal with practical problems. In this presentation, we propose a gate optimization method, where high-fidelity multi-qubit gates are generated by optimizing parametrized quantum circuits consisting of tunable high-fidelity single-qubit gates and fixed multi-qubit gates with limited controllability. We call the method variational quantum gate optimization (VQGO) and demonstrate it on a superconducting qubit system.   

Using Algebra to Dissect Quantum Optical Evolution for Quantum Control

Quantum control is the ability to affect quantum systems through judicious selection of dynamical parameters and their time dependence. Wei-Norman factorization [Wei and Norman, Proc. Amer. Math.Soc.15, 327 (1964)] is a powerful method to parametrize the quantum mechanical evolution operator before applying it to specific states. We first illustrate this method on the dynamics of optomechanical oscillators. We then extend the method to the pseudo-dynamics associated with the squeeze operator and characterize its action on a variety of optical states. This leads to improved insight into quantum control at a level accessible to undergraduates.