[Paper Review] Compiling Arbitrary Single-Qubit Gates Via the Phase-Shifts of Microwave Pulses

Jianxin Chen, Dawei Ding+2 more|arXiv (Cornell University)|May 6, 2021

Quantum Computing Algorithms and Architecture15 references4 citations

TL;DR

This paper presents a novel compilation scheme for arbitrary single-qubit quantum gates on superconducting qubits using phase-shifted microwave pulses (PMW), enabling continuous gate sets without infinite calibration. By leveraging tunable phase shifts to generate virtual rotations around any axis, the method achieves high-fidelity universal gate synthesis compatible with any two-qubit gate, including those incompatible with traditional virtual Z gates, and demonstrates state-of-the-art fidelities on fluxonium qubits.

ABSTRACT

We give an arbitrary single-qubit gate compilation scheme on superconducting processors that takes advantage of tuning the phase shift of microwave pulses to obtain a continuous gate set. This scheme is compatible with any two-qubit gate, and we only need to calibrate the $X_π$ and $X_{π/2}$ pulses. We implement this on fluxonium and obtain state-of-the-art fidelities. We give two other schemes: the first requires one $X_π$ pulse and one pulse with a variable rotation angle, and the second requires four $X_{π/2}$ pulses. We also find that if we can do virtual $Z$ gates, then we can also do virtual gates around any axis. Our results apply to any physical platform that natively supports virtual $Z$.

Motivation & Objective

• To develop a universal single-qubit gate compilation scheme for superconducting qubits that avoids infinite calibration overhead for arbitrary rotation angles.
• To overcome the incompatibility of certain two-qubit gate schemes (e.g., flux-tuned, tunable coupler gates) with conventional virtual Z gate techniques.
• To enable continuous gate sets via phase-shifted microwave pulses (PMW) that are experimentally precise and robust to pulse distortions.
• To generalize the virtual gate concept beyond Z-rotations to arbitrary axes using phase-shifted pulses.
• To demonstrate high-fidelity gate compilation on fluxonium qubits using minimal calibration of only $X_{\pi}$ and $X_{\pi/2}$ pulses.

Proposed method

• Utilizes phase-shifted microwave pulses (PMW) to generate a continuous set of single-qubit gates via conjugation: $X_{\sigma}(\phi) = Z_{-\phi}X_{\sigma}Z_{\phi}$, where $\phi$ is the phase shift.
• Employs the phase shift $\phi$ as a free parameter to generate arbitrary rotations around the Z-axis, with high experimental precision due to global frequency references.
• Proposes two new compilation schemes: one using one $X_{\pi}$ pulse and one variable-angle pulse, and another using four $X_{\pi/2}$ pulses, both avoiding extraneous Z rotations.
• Introduces the concept of virtual R gates (around any axis) by extending the virtual Z framework, using phase-shifted pulses to simulate arbitrary rotations.
• Demonstrates that leaky two-qubit gates (e.g., CNOT, DDCZ) are locally equivalent to phase carriers, enabling virtual R compilation even in the presence of leakage.
• Uses mathematical proofs involving unit quaternions and complex algebra to show that any single-qubit unitary can be exactly decomposed using the proposed PMW-based schemes.

Experimental results

Research questions

• RQ1 Can arbitrary single-qubit gates be compiled using only phase-shifted microwave pulses without infinite calibration of pulse parameters?
• RQ2 How can virtual gate compilation be extended beyond Z-rotations to arbitrary axes in superconducting qubit architectures?
• RQ3 Can the proposed PMW-based compilation schemes be made compatible with two-qubit gates that are incompatible with standard virtual Z gate techniques?
• RQ4 What is the mathematical condition under which a two-qubit gate allows for virtual R gate compilation, and how does this relate to gate leakage?
• RQ5 Can the proposed method achieve higher gate fidelities than existing approximate compilation schemes using finite gate sets like {H, S, T}?

Key findings

• The PMW-based compilation scheme enables exact compilation of arbitrary single-qubit gates using only calibration of $X_{\pi}$ and $X_{\pi/2}$ pulses, significantly reducing experimental overhead.
• The method achieves state-of-the-art gate fidelities on fluxonium qubits, demonstrating high-precision control via phase-shifted microwave pulses.
• The proposed scheme is compatible with any two-qubit gate, including those incompatible with virtual Z gates, such as flux-tuned iSWAP, fSim, and bSWAP gates.
• The paper proves that any two-qubit gate that is leaky on one qubit is locally equivalent to a phase carrier, enabling virtual R gate compilation even in the presence of leakage.
• A new virtual R gate compilation scheme is introduced that allows for arbitrary-axis rotations using phase-shifted pulses, generalizing the virtual Z framework.
• The mathematical proof shows that any single-qubit unitary can be exactly decomposed into a sequence of PMW pulses and a residual rotation, with the decomposition solvable via a quadratic equation.