Universal Qudit Gate Synthesis for Transmons

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 $99\mathrm{\%}$ 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 $\mathrm{SU}(16)$ 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.

Popular Summary

Quantum computers typically store information in qubits, where each qubit is a quantum system with two states. Superconducting transmons are a leading technology platform to build a quantum computer. They build qubits out of the two energetically lowest states of a large quantum space. Qudits, that is, systems with more than two levels, are an intriguing alternative to qubits. Qudits make the best use of the available computational resources, for instance, by synthesizing quantum circuits more efficiently than qubits can. We design a superconducting qudit processor with a logical space extended to higher excited transmon levels. We propose a universal set of gates, present a systematic procedure that synthesizes arbitrary gates, and showcase the qudit advantage over regular qubits.

Our universal gate set includes arbitrary single-qudit gates implemented by driving transitions between neighboring transmon levels and an entangling gate based on a qudit generalization of the popular cross-resonance gate. Our numerical simulations predict fidelities beyond $99\mathrm{\%}$ in the four-level case (ququarts) in a realistic experimental setting. In principle, a logical qubit—protected from environmental noise and faulty operations—could be embedded into a qudit, realizing the coveted goal of quantum error correction (QEC). We further demonstrate that ququarts can implement qudit-based QEC protocols. Our code protects against pure dephasing noise and could lower the error rates in qubit computations. Alongside recent experimental advances in qudit control of individual transmons, our work offers a blueprint to realize multivalued quantum logic in transmons, with applications for both current noisy and future error-corrected hardware.

Figure 1

Transition frequencies of a two-transmon model. Blue and red lines denote the control and target, with bare frequencies of ${\omega }_c/(2\pi )=6.3\mathrm{GHz}$ and ${\omega }_t/(2\pi )=6.1\mathrm{GHz}$, respectively. The enlarged inset shows the dressing of the bare target frequency due to the capacitive coupling with the average transition frequency of ${\overline{\omega }}_t/(2\pi )$ in dashed green.

Figure 2

Qudit space action of the echoed cross-resonance gate. (a) The ecr pulse sequence applied to the control qudit consists of two cross-resonance tones (gray) played at the target qudit frequency ${\overline{\omega }}_t$ with opposite amplitudes, each followed by a $\pi$ pulse on the control. (b) Action of each pulse on the initial target state ${|0⟩}_t$ depending on the control state ${|\psi ⟩}_c$ on the Bloch sphere spanned by $\{{|0⟩}_t,{|1⟩}_t\}$. By construction of the echo sequence, a $\theta$ ($-\theta$) rotation is applied in the ${|1⟩}_c$ (${|0⟩}_c$) case. (c) Evolution of the populations starting from ${|0⟩}_t$ depending on the control state ${|\psi ⟩}_c$ for the pulse sequence shown in (a). Color denotes the state of the control while line style denotes the state of the target. Pulse durations are calibrated to $\theta =\pi$.

Figure 3

Gate decompositions to implement a general singly controlled two-qudit gate $C^m[U]$. (a) Diagonalization of $C^m[U]$ leads to a sequence of controlled $z$ rotations $R_z^{0j}$ between the $0\mathrm{th}$ and $j\mathrm{th}$ level. (b) Each $C^m[R_z^{0j}]$ is implemented through two $C^m[R_x^{0j}]$ rotations and local phase gates. (c) Decomposition of $C^m[R_x^{0j}]$ into local permutations and a single $C^0[R_x^{01}]$ gate. Gray shaded gates cancel between consecutive $C^m[R_x^{0j}]$ gates as they arise in (b). (d) Realization of the $C^0[R_x^{01}(\theta )]$ gate through echoed cross-resonance gates.

Figure 4

Example application of the general qudit transpiler decomposing a $|2⟩$-controlled $i\mathcal{S}\mathcal{W}\mathcal{A}\mathcal{P}$ gate between two ququarts. Colors indicate the different steps of the transpilation shown in Fig. 3, while two-qudit boxes denote $\mathcal{E}\mathcal{C}\mathcal{R}$ gates.

Figure 5

Benchmark study of the qudit transpiler on the Hamiltonian exponentiation of a lithium hydride molecule. Two ququarts require only a third of the entangling gates that four linearly connected qubits require, at the expense of 3.4 times as many single-qudit $\sqrt{X}$ gates.

Figure 6

Sequence to correct pure dephasing errors in a transmon. The initial qubit state ${|\psi ⟩}_0$ is encoded into a ququart logical state $|{\psi }_L⟩$. After dephasing, a decoding unitary $U_D$ maps the ideal and erroneous cases to orthogonal states. The error is detected by applying a $U_{\mathrm{ECR}}(\pi )$ gate where the target is an ancilla qubit prepared in $|1⟩$. In the ideal case where the ancilla is de-excited to 0, the state is re-encoded, while in the error case, where the ancilla is measured in 1, a recovery operation $U_R$ restores $|{\psi }_L⟩$.

Figure 7

Numerical simulations of a single correction cycle applied after dephasing for a duration of ${\tau }_{\varphi }$. (a) Qubit error $1-\overline{\mathcal{F}}$. (b) Error reduction compared to the bare qubit case.

Figure 8

Numerical simulations of the error-correction pulse sequence with repeated cycles where $\mathrm{\Delta }{t}_{\mathrm{corr}}$ is the time between each cycle. (a) Exponential decay of the fidelity $\overline{\mathcal{F}}$ for the bare qubit case and two different correction cycle times. (b) Time constant of the exponential decay $T_{2,\mathrm{eff}}$ as a function of the cycle time $\mathrm{\Delta }{t}_{\mathrm{corr}}$. (c) Asymptotic fidelity for $t\to \mathrm{\infty }$.

Figure 9

Effect of amplitude damping on the error reduction of one correction cycle. ${\tau }_{\varphi }=0.12T_2$ and $T_2=200\text{μ}s$ are fixed while the strength of $T_1$ is varied. For the noisy simulation, the code breaks even for $T_1>6T_2$. With idealized gates (no unitary error), this decreases to $T_1>3.3T_2$.

Figure 10

Numerical simulation of the unitary matrix ${\tilde{R}}_{\mathrm{CR}}$ of a single CR pulse in the two-transmon dressed basis. The resulting operation is well described by an $R_x^{01}$ rotation on the target with a direction and angle that depend on the state of the control.

Figure 11

Numerical simulation of the unitary matrix of the full echoed CR sequence in the two-transmon dressed basis. The resulting operation is well described by an $U_{\mathrm{ECR}}(\pi )$ gate.

Figure 12

Evolution of the populations in the two-transmon dressed basis during the echoed CR sequence for different initial states. Color indicates the state of the control, while the linestyle indicates the state of the target. Vertical dashed black lines form four regions that correspond to the four total pulses of the ECR sequence.