# Effect of Phase Errors on a Quantum Control Protocol Using Fast Oscillations

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## Abstract

**:**

## 1. Introduction

## 2. Effective Counterdiabatic Driving

#### General Setup

## 3. System-Independent Expressions

#### 3.1. Magnus Terms

#### 3.2. Infidelity

#### 3.3. Real Hamiltonians

**Observation**

**1.**

- (i)
- $\u2329{\mathcal{M}}_{\mathrm{c}}^{(2n-1)}\u232a=0$ for $n\in \mathbb{N}$;
- (ii)
- $\u2329{\mathcal{M}}_{\mathrm{mix}}^{\left(n\right)}\left(\mathrm{even}\right)\u232a=0$.

**Proof.**

## 4. Effect of Phase Shifts: Two-Level System

#### 4.1. Application: Avoided Level Crossing

## 5. Beyond Two Levels: Two Atoms in a Cavity

## 6. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) The blue solid lines represent the time evolution of the two energy levels, measured in units of the coupling ${h}_{1}$, of the two-level Hamiltonian of Equation (27) with a sweep function of the form in Equation (28). The latter is represented, again in units of the coupling ${h}_{1}$, by the black dashed line. (

**b**) Infidelity as a function of time and representation on the Bloch sphere of the evolution given by the effective counterdiabatic driving for a two-level avoided crossing problem. The dashed black line shows the instantaneous ground state, which is initially close to the ${\widehat{\sigma}}_{3}$ eigenstate $|\downarrow \rangle $ while at the end is close to $|\uparrow \rangle $. The solid blue line shows the evolution given by the control method with Hamiltonian of Equation (30) and parameters $\Delta /{h}_{1}=10,\tau =0.1,\omega =2\pi \times 15$. The controlled dynamics oscillates around the target evolution.

**Figure 2.**General behaviour of the infidelity ${\mathbb{I}}_{\mathrm{T}}\left(\mathrm{\Phi}\right)$ at the end of one time step $T\sim 7\times {10}^{-4}$ as a function of the phase error $\mathrm{\Phi}$, in units of $\pi $, for the accelerated adiabatic protocol described by Equation (30). The blue circles indicate the result of numerical simulations, while the solid black line indicates the prediction given by the first term of Equation (26).

**Figure 3.**Behavior of the infidelity at the end of a time step T, for an avoided crossing problem with sweep function of the form in Equation (28), as a function of phase shifts. (

**a**) Dependence on the relative phase $\mathrm{\Phi}$ between the control functions of Equation (19) for different values of T (and so of $\mathrm{\Omega}$): $T={10}^{-4},7\times {10}^{-4},5\times {10}^{-3},4\times {10}^{-2}$. Symbols represent the result of numerical simulations, while the dashed blue curve represents the description given by the first term of Equation (26) and the solid red line is obtained by including both terms of Equation (26). Far from zero the first term is a sufficiently good description, and this is also true in the vicinity of zero when T decreases. The behavior near zero requires to take into account also the second term. The latter introduces asymmetries with respect to the sign of $\mathrm{\Phi}$, which are more pronounced for smaller T, as can be seen from panel (

**c**). The second term also introduces an explicit dependence on the specific phases ${\varphi}_{1},{\varphi}_{3}$: this is evident from panel (

**b**), where the infidelity for null relative phase $\mathrm{\Phi}=0$ is shown as a function of ${\varphi}_{1}={\varphi}_{3}$. This interestingly shows that, once the first term is made to vanish by choosing $\mathrm{\Phi}=0$, the second term can be minimized by a suitable choice of phases ${\varphi}_{1}={\varphi}_{3}$ in each time step. Again, the symbols represent numerical data while the solid green-blue line is the prediction of Equation (26).

**Figure 4.**Infidelity at the end of one time step T as a function of the phase error $\mathrm{\Phi}$. The blue symbols indicate the result of a numerical simulation, while the solid black line represents a fit according to the dependence $k{[1-\mathrm{cos}\left(\mathrm{\Phi}\right)]}^{2}$, suggested by the first term in Equation (26), with k a free parameter. The fit well describes the data far from the nodes of the fitting function, well reproducing their oscillatory pattern. Closer to $\pm 2n\pi $, higher order effects in T show up, inducing an asymmetry which can lead to smaller infidelities.

**Figure 5.**The left panel shows the behaviour of the infidelity at the end of one time step as a function of the phase offset $\mathrm{\Phi}$, for different values of the qubits-resonator couplings g. The values are rescaled in such a way that the highest value in the interval lies at ordinate 1 for all curves (all nonscaled maximal values lie below $2\times {10}^{-2}$). All curves present a general $\sim {(1-\mathrm{cos}\mathrm{\Phi})}^{2}$ pattern, as predicted using Equation (26). For small $g=$ 3 MHz, this pattern is dominant (black solid curve), while for larger values the contributions in the vicinity of $\mathrm{\Phi}=\pm 2k\pi $ become relevant, showing substantial asymmetries with respect to $\mathrm{\Phi}$. The right panel shows an inset for $\mathrm{\Phi}$ close to zero where asymmetries with respect to the sign of $\mathrm{\Phi}$ are evident.

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**MDPI and ACS Style**

Petiziol, F.; Wimberger, S.
Effect of Phase Errors on a Quantum Control Protocol Using Fast Oscillations. *Condens. Matter* **2019**, *4*, 34.
https://doi.org/10.3390/condmat4010034

**AMA Style**

Petiziol F, Wimberger S.
Effect of Phase Errors on a Quantum Control Protocol Using Fast Oscillations. *Condensed Matter*. 2019; 4(1):34.
https://doi.org/10.3390/condmat4010034

**Chicago/Turabian Style**

Petiziol, Francesco, and Sandro Wimberger.
2019. "Effect of Phase Errors on a Quantum Control Protocol Using Fast Oscillations" *Condensed Matter* 4, no. 1: 34.
https://doi.org/10.3390/condmat4010034