[Paper Review] Universal qudit gate synthesis for transmons

Laurin E. Fischer, Alessandro Chiesa+4 more|arXiv (Cornell University)|Dec 8, 2022

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper proposes a universal gate set for superconducting qudit processors using transmon qubits extended to higher-excited levels (qudits), enabling efficient quantum computation with reduced gate counts. It demonstrates a two-qudit cross-resonance gate with predicted fidelities exceeding 99% for ququarts (d=4), and presents a decomposition routine that compiles arbitrary qudit unitaries more efficiently than qubit-based alternatives, enabling applications in circuit synthesis and embedded quantum error correction.

ABSTRACT

Gate-based quantum computers typically encode and process information in two-dimensional units called qubits. Using $d$-dimensional qudits instead may offer intrinsic advantages, including more efficient circuit synthesis, problem-tailored encodings and embedded error correction. In this work, we design a superconducting qudit-based quantum processor wherein the logical space of transmon qubits is extended to higher-excited levels. We propose a universal gate set featuring a two-qudit cross-resonance entangling gate, for which we predict fidelities beyond $99\%$ in the $d=4$ case of ququarts with realistic experimental parameters. Furthermore, we present a decomposition routine that compiles general qudit unitaries into these elementary gates, requiring fewer entangling gates than qubit alternatives. As proof-of-concept applications, we numerically demonstrate the synthesis of ${ m SU}(16)$ gates for noisy quantum hardware and an embedded error correction sequence that encodes a qubit memory in a transmon ququart to protect against pure dephasing noise. We conclude that universal qudit control -- a valuable extension to the operational toolbox of superconducting quantum information processing -- is within reach of current transmon-based architectures and has applications to near-term and long-term hardware.

Motivation & Objective

• To extend transmon qubit architectures to support universal qudit-based quantum computation using higher-excited levels.
• To design a universal two-qudit gate set, specifically a cross-resonance gate, with high fidelity under realistic experimental parameters.
• To develop a decomposition routine that compiles general qudit unitaries into elementary gates with fewer entangling gates than qubit equivalents.
• To demonstrate practical applications, including SU(16) gate synthesis and embedded quantum error correction for dephasing noise.
• To show that transmon qudits can serve as a viable platform for near-term and fault-tolerant quantum computing with reduced resource overhead.

Proposed method

• The authors extend the logical space of transmon qubits to d-level qudits by utilizing the lowest d energy levels, treating them as qudit states.
• They propose a two-qudit cross-resonance gate that implements a Z⊗X interaction, leveraging frequency detuning and microwave driving to achieve entanglement.
• The gate is analyzed using numerical simulations of the full qudit Hamiltonian, including anharmonicities and drive interactions, to predict gate fidelities.
• A decomposition algorithm is developed to compile arbitrary qudit unitaries into the universal gate set, minimizing the number of entangling gates.
• The method includes pulse shaping with Gaussian envelopes and accounts for unitary errors arising from frequency-dependent dressed states.
• The approach is validated through numerical simulations of SU(16) gate synthesis and a logical qubit memory encoded in a transmon ququart for dephasing protection.

Experimental results

Research questions

• RQ1 Can a universal set of two-qudit gates be implemented in transmon qudits with high fidelity using existing hardware control techniques?
• RQ2 How does the number of required entangling gates for qudit circuits compare to equivalent qubit circuits in terms of resource efficiency?
• RQ3 Can transmon qudits be used to implement embedded quantum error correction that protects logical qubits against pure dephasing with minimal overhead?
• RQ4 What are the dominant error sources in qudit cross-resonance gates, and how can they be mitigated under realistic experimental parameters?
• RQ5 To what extent can the full Hilbert space of transmons be exploited for qudit operations before decoherence and frequency crowding limit performance?

Key findings

• The proposed two-qudit cross-resonance gate achieves predicted average gate fidelities above 99% for the d=4 ququart case under realistic experimental parameters.
• The decomposition routine compiles general qudit unitaries using fewer entangling gates than equivalent qubit-based decompositions, improving circuit efficiency.
• Numerical simulations demonstrate successful synthesis of SU(16) gates on noisy quantum hardware, validating the gate set's utility for near-term applications.
• A proof-of-concept embedded quantum error correction sequence is implemented, encoding a logical qubit in a transmon ququart to protect against pure dephasing noise.
• The study shows that transmon qudits can support high-fidelity multi-ququart operations, with potential for scalable quantum information processing.
• The work establishes that universal qudit control is feasible in current transmon architectures, offering a path toward reduced-error, resource-efficient quantum computing.