# Simulating Finite-Time Isothermal Processes with Superconducting Quantum Circuits

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## Abstract

**:**

## 1. Introduction

## 2. Discrete-Step Method to Quantum Isothermal Process

## 3. Simulation with Quantum Circuits

**Adiabatic process**. In the superconducting quantum computer, e.g., IBM Q system, the tuning of the physical energy levels of qubits is unavailable for the users. The physical parameters are fixed at the optimal values to possibly reduce noises and errors induced by decoherence and imperfect control.

**Isochoric process**. The dynamical evolution of the isochoric process can be simulated with the generalized amplitude damping channel (GADC)

#### 3.1. Hybrid Simulation of Isochoric Process with Classical Random Number Generator (CRNG)

#### 3.2. Fully Quantum Simulation of Isochoric Process

## 4. Testing $1/\tau $ Scaling of Extra Work

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Simulation of the isothermal process on the superconducting quantum computer. The finite-time isothermal process is divided into series of piecewise adiabatic and isochoric processes. In the adiabatic process, the energy of the two-level system is tuned with the switched-off interaction between the system and the thermal bath. In the isochoric process, the interaction is switched on with the unchanged energy spacing ${\omega}_{j}$. One qubit represents the simulated two-level system, and the ancillary qubits play the role of the thermal bath at the temperature T. After implementing the quantum circuit, the system qubit is measured to obtain the internal energy.

**Figure 2.**The quantum circuits in one elementary process. (

**a**) The amplitude damping (pumping) channel ${\mathcal{E}}_{\downarrow}^{\left(j\right)}$ (${\mathcal{E}}_{\uparrow}^{\left(j\right)}$) in the hybrid simulation. (

**b**) One elementary process in the hybrid simulation. The selection of the two sub-channels is realized by the classical random number generator. (

**c**) One elementary process in the fully quantum simulation. The selection of the two sub-channels is assisted by another ancillary qubit. (

**d**) Instruction of gates in the current simulation.

**Figure 3.**The circuit of the two-step isothermal process on ibmqx2. (

**a**) Excited state population-energy (${p}_{e}-E$) diagram. (

**b**) The circuit for the hybrid simulation. In each elementary process, the X gate is (or not) implemented for the sub-channel selected as the amplitude pumping (damping) channel according to the classical random number. Each elementary process requires another ancillary qubit. (

**c**) The circuit for the fully quantum simulation. Each elementary requires two ancillary qubits.

**Figure 4.**$1/N$ scaling of the extra work for the discrete isothermal process. The operation time of each isochoric process is set as $\delta \tau =0.5$ (blue dashed curve) or 10 (red solid curve). The ibmqx2 simulation results for $N=2,3$ and 4 are plotted. The empty squares present the results by the hybrid simulations, and the pentagrams for the fully quantum simulation. The $1/N$ scaling is shown by the solid black curve.

**Figure 5.**Comparison of the ibmqx2 simulation and the numerical results. (

**a**,

**b**) show the microscopic work in the hybrid simulation with the step number $N=2,\phantom{\rule{0.166667em}{0ex}}3$ and 4. The ibmqx2 simulation result (blue solid line) is compared with the numerical result (gray dashed line). (

**c**,

**d**) show the excited state population ${p}_{e}\left(t\right)$ at each step in the fully quantum simulation of the two-step isothermal process. The ibmqx2 simulation results (blue bar) are compared to the numerical results (gray bar).

**Table 1.**The discrete isothermal process to be simulated and the two simulation methods, the hybrid simulation and the fully quantum simulation

To be Simulated: | Simulation | ||
---|---|---|---|

Discrete Isothermal Process | Hybrid Simulation with CRNG | Fully Quantum Simulation | |

Adiabatic process | $U\left[R\right(t\left)\right]$, $t\in [{t}_{j-1},{t}_{j}]$ | The unitary evolution is realized with the virtual tuning on the system Hamiltonian. | |

Isochoric process | System relaxation in Equation (2) | Generalized amplitude damping channel ${\mathcal{E}}_{\mathrm{GAD}}^{\left(j\right)}$ with the classical random number generation | Generalized amplitude damping channel ${\mathcal{E}}_{\mathrm{GAD}}^{\left(j\right)}$ with an additional qubit at the state $cos({\alpha}_{j}/2)\left|0\right.\u232a+isin({\alpha}_{j}/2)\left|1\right.\u232a$ |

Parameters | Duration: $\delta \tau ={t}_{j}-{t}_{j-1}$ Temperature: T | $cos{\theta}_{j}=exp[-\frac{{\gamma}_{0}\delta \tau}{2}coth(\frac{\beta {\omega}_{j}}{2})]$ | $cos{\theta}_{j}=exp[-\frac{{\gamma}_{0}\delta \tau}{2}coth(\frac{\beta {\omega}_{j}}{2})]$$cos({\alpha}_{j}/2)={[{p}_{\downarrow}^{\left(j\right)}]}^{1/2}$ |

N | $\mathit{\delta}\mathit{\tau}=0.5$ | $\mathit{\delta}\mathit{\tau}=10$ | |||
---|---|---|---|---|---|

${\overline{W}}_{\mathrm{ibmqx}2}$ | ${\overline{W}}_{\mathrm{exact}}$ | ${\overline{W}}_{\mathrm{ibmqx}2}$ | ${\overline{W}}_{\mathrm{exact}}$ | ||

Hybrid simulation | 2 | 0.251 | 0.245 | 0.232 | 0.226 |

3 | 0.246 | 0.233 | 0.221 | 0.212 | |

4 | 0.243 | 0.224 | 0.217 | 0.206 | |

Fully quantum simulation | 2 | 0.251 | 0.245 | 0.238 | 0.226 |

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Chen, J.-F.; Li, Y.; Dong, H.
Simulating Finite-Time Isothermal Processes with Superconducting Quantum Circuits. *Entropy* **2021**, *23*, 353.
https://doi.org/10.3390/e23030353

**AMA Style**

Chen J-F, Li Y, Dong H.
Simulating Finite-Time Isothermal Processes with Superconducting Quantum Circuits. *Entropy*. 2021; 23(3):353.
https://doi.org/10.3390/e23030353

**Chicago/Turabian Style**

Chen, Jin-Fu, Ying Li, and Hui Dong.
2021. "Simulating Finite-Time Isothermal Processes with Superconducting Quantum Circuits" *Entropy* 23, no. 3: 353.
https://doi.org/10.3390/e23030353