[Paper Review] Implementing a Ternary Decomposition of the Toffoli Gate on Fixed-FrequencyTransmon Qutrits

Alexey Galda, Michael Cubeddu+3 more|arXiv (Cornell University)|Sep 1, 2021

Quantum Computing Algorithms and Architecture20 citations

TL;DR

This paper demonstrates an experimental implementation of a ternary decomposition of the Toffoli (CCNOT) gate on fixed-frequency superconducting transmon qutrits using Qiskit Pulse, achieving an average gate fidelity of 78.00% ± 1.93% via a four-two-transmon decomposition that outperforms the standard eight-CNOT binary decomposition in fidelity and variance. The approach leverages existing native gates and a just-in-time error mitigation technique to reduce charge noise effects without additional calibration.

ABSTRACT

Quantum computation is conventionally performed using quantum operations acting on two-level quantum bits, or qubits. Qubits in modern quantum computers suffer from inevitable detrimental interactions with the environment that cause errors during computation, with multi-qubit operations often being a primary limitation. Most quantum devices naturally have multiple accessible energy levels beyond the lowest two traditionally used to define a qubit. Qudits offer a larger state space to store and process quantum information, reducing complexity of quantum circuits and improving efficiency of quantum algorithms. Here, we experimentally demonstrate a ternary decomposition of a multi-qubit operation on cloud-enabled fixed-frequency superconducting transmons. Specifically, we realize an order-preserving Toffoli gate consisting of four two-transmon operations, whereas the optimal order-preserving binary decomposition uses eight \ exttt{CNOT}s on a linear transmon topology. Both decompositions are benchmarked via truth table fidelity where the ternary approach outperforms on most sets of transmons on \ exttt{ibmq\\_jakarta}, and is further benchmarked via quantum process tomography on one set of transmons to achieve an average gate fidelity of 78.00\\% $\\pm$ 1.93\\%.

Motivation & Objective

• To reduce circuit depth and improve fidelity in multi-qubit quantum operations by leveraging qudit-based logic instead of conventional qubit-only decompositions.
• To demonstrate that ternary decomposition of the Toffoli gate is feasible on fixed-frequency transmon architectures using existing hardware and software capabilities.
• To mitigate charge noise effects in higher-energy qutrit states through dynamical decoupling and just-in-time error correction techniques.
• To benchmark the ternary decomposition against standard qubit-based CCNOT decompositions in terms of truth table fidelity and quantum process tomography.
• To show that qutrit-based operations can achieve comparable or better performance than qubit-based decompositions with minimal calibration overhead.

Proposed method

• Extends the native gate set of IBM Quantum’s transmon devices using Qiskit Pulse to include single- and two-qutrit gates, enabling ternary logic operations.
• Constructs a four-two-transmon decomposition of the Toffoli gate using a controlled-controlled-NOT gate implemented via two consecutive CNOTs on a qutrit-controlled transmon.
• Applies a dynamical decoupling (DD) technique to stabilize the phase of the |2⟩-controlled NOT gate, reducing sensitivity to charge noise.
• Implements a just-in-time error mitigation technique to correct for local phase offsets, improving gate stability without additional calibration.
• Uses quantum process tomography (QPT) and truth table fidelity benchmarks to evaluate gate performance on ibmq_jakarta.
• Leverages existing ECR (echoed cross-resonance) gate implementations and avoids new calibration by reusing standard hardware control protocols.

Experimental results

Research questions

• RQ1 Can a ternary decomposition of the Toffoli gate be experimentally realized on fixed-frequency transmon qutrits using only existing native gate sets?
• RQ2 Does the ternary decomposition achieve higher fidelity and lower variance than the standard eight-CNOT qubit-based decomposition?
• RQ3 To what extent can dynamical decoupling and just-in-time error mitigation reduce phase instability in qutrit-based operations?
• RQ4 How does the gate fidelity of the ternary decomposition compare to the optimal qubit-based decomposition in terms of average fidelity and process tomography metrics?
• RQ5 What is the impact of gate duration and charge noise sensitivity on the performance of qutrit-based multi-controlled gates?

Key findings

• The ternary decomposition achieved an average gate fidelity of 78.00% ± 1.93% via quantum process tomography on ibmq_jakarta (qubits 6,5,4), outperforming the standard qubit-based decomposition in fidelity and variance.
• The truth table fidelity of the ternary decomposition outperformed the eight-CNOT decomposition on most transmon sets in the ibmq_jakarta device.
• The phase-stabilized ternary decomposition had a total gate time of 2.432 μs, slightly shorter than the eight-CNOT decomposition (2.510 μs), despite using a double-length DD sequence.
• The fidelity is limited primarily by total gate time and charge noise sensitivity, suggesting that hardware tuning (e.g., increasing Ej/Ec ratio) could further improve performance.
• The method requires no additional calibration experiments, as it reuses existing ECR gates and pulse-level control via Qiskit Pulse.
• The approach is broadly applicable to other qutrit realizations and quantum hardware platforms, as long as the noise characteristic time exceeds the gate duration.