Optimal realization of Yang-Baxter gate on quantum computers

TL;DR

The optimal realizations of two types of Yang–Baxter gates with a minimal number of controlled NOT (CNOT) or Rzz$R_{zz}$ gates are presented and the simulation of the Yang–Baxter equation on quantum computers is demonstrated.

Abstract

Quantum computers provide a promising method to study the dynamics of many-body systems beyond classical simulation. On the other hand, the analytical methods developed and results obtained from the integrable systems provide deep insights on the many-body system. Quantum simulation of the integrable system not only provides a valid benchmark for quantum computers but is also the first step in studying integrable-breaking systems. The building block for the simulation of an integrable system is the Yang-Baxter gate. It is vital to know how to optimally realize the Yang-Baxter gates on quantum computers. Based on the geometric picture of the Yang-Baxter gates, we present the optimal realizations of two types of Yang-Baxter gates with a minimal number of CNOT or $R_{zz}$ gates. We also show how to systematically realize the Yang-Baxter gates via the pulse control. We test and compare the different realizations on IBM quantum computers. We find that the pulse realizations of the Yang-Baxter gates always have a higher gate fidelity compared to the optimal CNOT or $R_{zz}$ realizations. On the basis of the above optimal realizations, we demonstrate the simulation of the Yang-Baxter equation on quantum computers. Our results provide a guideline and standard for further experimental studies based on the Yang-Baxter gate.

Figures (11)

• Figure 1: Yang-Baxter equation in the quantum circuit model. The crossing represents the solution of the Yang-Baxter equation. The box represents the two-qubit gate.
• Figure 2: Geometric representation of two-qubit gate $U=[a_1,a_2,a_3]$. The tetrahedron $OA_1A_2A_3$ has the vertexes $O=[0,0,0]$, $A_1=[\pi,0,0]$, $A_2=[\pi/2,\pi/2,0]$, and $A_3=[\pi/2,\pi/2,\pi/2]$. The Yang-Baxter gates $R_I(\theta)$ and $R_{II}(\phi)$ correspond to the blue and green edges respectively.
• Figure 3: Pulse realization of CNOT gate on qubit 0 and 1 of IBM ibmq_jakarta processor. The D0 and D1 are the DriveChannel for the single-qubit gate on qubit 0 and 1. The U0 is the ControlChannel for the two-qubit interaction. The circular arrow represents the virtual single-qubit $z$-axis rotation gate. Pulses with the bright and dark colors represent the in-phase and quadrature-phase components. The negative amplitude means that the pulse has a $\pi$-phase.
• Figure 4: The pulses of $R_{zz}(\pi/3)$ gate realized from (a) two CNOTs conjugated on the $R_z(\pi/3)$ gate as shown in Eq. (\ref{['eq:Rz_two_cnot']}); one echo cross-resonance operations with rescaled parameters of CNOT gate. The virtual $z$-axis rotation is omitted here for simplicity.
• Figure 5: Realizations of Yang-Baxter gate $\check R_I(\theta)$ on the quantum computer ibmq_montreal. Measured gate fidelity on (a) FakeMontreal ( ibmq_montreal simultor) or (b) ibmq_montreal. (c) The pulse durations of three different realizations. (d) The error reduction of pulse realization compared to the CNOT or $R_{zz}$ realization. The dashed line is the fitting. Each tomography circuit repeats four times and each time runs 4096 shots.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2