Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Unitary Gate Synthesis via Polynomial Optimization

Llorenç Balada Gaggioli, Denys I. Bondar, Jiri Vala, Roman Ovsiannikov, Jakub Mareček

TL;DR

A polynomial optimization problem that allows us to find the global solution without resorting to approximations of the exponential, and provides a certificate of globality and lets us do single-shot optimization, which implies it is generally faster than local methods.

Abstract

Quantum optimal control plays a crucial role in the development of quantum technologies, particularly in the design and implementation of fast and accurate gates for quantum computing. Here, we present a method to synthesize gates using the Magnus expansion. In particular, we formulate a polynomial optimization problem that allows us to find the global solution without resorting to approximations of the exponential. The global method we use provides a certificate of globality and lets us do single-shot optimization, which implies it is generally faster than local methods. By optimizing over Hermitian matrices generating the unitaries, instead of the unitaries themselves, we can reduce the size of the polynomial to optimize, leading to fast convergence and scalability. Numerical experiments comparing our results with CRAB and GRAPE show that we maintain high accuracy while providing globality certificates.

Unitary Gate Synthesis via Polynomial Optimization

TL;DR

A polynomial optimization problem that allows us to find the global solution without resorting to approximations of the exponential, and provides a certificate of globality and lets us do single-shot optimization, which implies it is generally faster than local methods.

Abstract

Quantum optimal control plays a crucial role in the development of quantum technologies, particularly in the design and implementation of fast and accurate gates for quantum computing. Here, we present a method to synthesize gates using the Magnus expansion. In particular, we formulate a polynomial optimization problem that allows us to find the global solution without resorting to approximations of the exponential. The global method we use provides a certificate of globality and lets us do single-shot optimization, which implies it is generally faster than local methods. By optimizing over Hermitian matrices generating the unitaries, instead of the unitaries themselves, we can reduce the size of the polynomial to optimize, leading to fast convergence and scalability. Numerical experiments comparing our results with CRAB and GRAPE show that we maintain high accuracy while providing globality certificates.

Paper Structure

This paper contains 2 sections, 1 theorem, 14 equations, 4 figures.

Table of Contents

  1. Generalized BCH formula
  2. Numerical implementation

Key Result

Proposition 1

Let $X_U=\arg\min_{\mathbf{x}} D(U(T,\mathbf{x}),U^{\star})$ be the set of global minimisers with global optimum of 0, for some distance $D$, a unitary matrix $U(T,\mathbf{x})=e^{\Lambda}$ solving $\partial_tU(t)=A(t,\mathbf{x})U(t)$, and target unitary $U^{\star}=e^\Theta$, where $\|\Theta\|_2<\pi$

Figures (4)

  • Figure 1: Time efficiency. We study the time to evaluate the objective (left) and time it takes to solve the polynomial optimization problem (right) for both for unitary optimization and the more efficient gate synthesis with Hermitians. Based on 5 repetitions, solid lines present the means and shaded regions present $\pm$ one standard deviation.
  • Figure 2: Accuracy of quantum optimal control methods with number of qubits. Infidelity comparison between different quantum optimal control methods for the XX chain with controlled global X field for different number of qubits, tested with 100 random reachable unitary targets. The color shading is the range of infidelities and the width is the frequency. For 3 and 4 qubits, CRAB and GRAPE almost fully intersect, making them hard to distinguish.
  • Figure 3: Accuracy of quantum optimal control methods. Fidelity comparison between GRAPE, CRAB and the gate synthesis with Hermitians. We test the fidelity for a sample of 1000 random values of the control function, using an order 3 Magnus expansion.
  • Figure 4: Accuracy of quantum optimal control methods. Fidelity comparison between GRAPE, CRAB and the gate synthesis with Hermitians for a piecewise constant control. We test the fidelity for a sample of 100 random values of the control function, using an order 4 discrete Magnus expansion.

Theorems & Definitions (3)

  • Proposition 1
  • proof
  • Remark 1