New Algorithm Optimizes Quantum Gate Generation for Open Systems

The article presents a new monotonic numerical algorithm for generating quantum gates in open systems, governed by Lindblad master equations. The algorithm optimizes both the shape of the control input and the gate time, starting from an initial seed of the control input. It is particularly effective for cat-qubit gates, where Hilbert space dimensions are smaller than the physical Hilbert space. The algorithm’s stability is ensured by the stability of both the open-loop adjoint backward system and the forward closed-loop system. The research, funded by the European Research Council, is significant in the field of quantum computing.

What is the New Monotonic Numerical Algorithm for Quantum Gates?

The article discusses a new monotonic numerical algorithm that includes time optimization for generating quantum gates for open systems. These systems are assumed to be governed by Lindblad master equations for the density operators on a large Hilbert space, while the quantum gates are relative to a subspace of small dimension. The algorithm starts from an initial seed of the control input and consists of two steps: a backward integration of adjoint Lindblad Master equations from a set of final conditions encoding the quantum gate to generate, and a forward integration of Lindblad Master equations in a closed-loop where a Lyapunov based control produces the new control input.

The numerical stability of the algorithm is ensured by the stability of both the open-loop adjoint backward system and the forward closed-loop system. A clock-control input can be added to the usual control input. The algorithm allows for the optimization of not only the shape of the control input but also the gate time. Preliminary numerical implementations indicate that this algorithm is well suited for cat-qubit gates where Hilbert space dimensions 2 for the Z-gate and 4 for the CNOT-gate are much smaller than the dimension of the physical Hilbert space.

How Does the Algorithm Work?

The algorithm works by producing a new control input through the repetition of two steps. The first step involves a backward integration of adjoint Lindblad Master equations from a set of final conditions that encode the quantum gate to generate. The second step involves a forward integration of Lindblad Master equations in a closed-loop where a Lyapunov based control produces the new control input.

The algorithm’s numerical stability is ensured by the stability of both the open-loop adjoint backward system and the forward closed-loop system. A clock-control input can be added to the usual control input, allowing the algorithm to optimize not only the shape of the control input but also the gate time.

What are the Applications of the Algorithm?

The algorithm is particularly well-suited for cat-qubit gates where Hilbert space dimensions 2 for the Z-gate and 4 for the CNOT-gate are much smaller than the dimension of the physical Hilbert space. This makes it a valuable tool in the field of quantum computing, where the generation of quantum gates is a fundamental process.

The algorithm’s ability to optimize both the shape of the control input and the gate time simultaneously makes it a versatile tool for quantum gate generation. Its numerical stability, ensured by the stability of both the open-loop adjoint backward system and the forward closed-loop system, makes it a reliable tool for this purpose.

How Does the Algorithm Compare to Other Methods?

The algorithm presented in this work is very close to the Krotov method, at least in the case of the so-called sequential update of the control, which ensures a monotonic behavior of such method. To be monotonic in this case is a property that is analogous to the non-increasing property of the Lyapunov function in the context of the algorithm that is presented in this paper.

The algorithm generalizes such monotonic algorithms by considering the optimization of the shape of the control input and the gate time simultaneously. It also implements such generalization on physical case-studies of bosonic qubits where the dimension of the underlying Hilbert space is far much larger than the size of the orthonormal sets defining the gate.

What is the Significance of this Research?

This research is significant as it presents a new algorithm for generating quantum gates for open systems. The algorithm’s ability to optimize both the shape of the control input and the gate time simultaneously is a novel feature that sets it apart from other methods.

The algorithm’s suitability for cat-qubit gates, where Hilbert space dimensions are much smaller than the dimension of the physical Hilbert space, makes it a valuable tool in the field of quantum computing. The research also demonstrates the algorithm’s effectiveness through its implementation on physical case-studies of bosonic qubits.

The research was funded by the European Research Council under the European Union’s Horizon 2020 research and innovation programme.