Dancing the Quantum Waltz: Compiling Three-Qubit Gates on Four Level Architectures

TL;DR

This work explores encoding two qubits into a single four-level system (a ququart) to enable native three-qubit gates on superconducting hardware. By synthesizing direct-to-pulse gates through quantum optimal control and developing a compiler (the Quantum Waltz) that leverages mixed-radix and full-ququart configurations, the study demonstrates notable fidelity gains (up to $\approx$2–3×) compared to qubit-only strategies, particularly for circuits rich in three-qubit interactions. The results combine experimental demonstrations of single-ququart control with extensive simulations that show substantial improvements in three-qubit gate performance and circuit EPS, while highlighting tradeoffs from higher-level decoherence and leakage. The findings suggest ququart-based architectures provide a flexible and scalable route to accelerate near-term quantum algorithms by expanding effective connectivity and reducing decomposition overhead in multi-qubit gates.

Abstract

Superconducting quantum devices are a leading technology for quantum computation, but they suffer from several challenges. Gate errors, coherence errors and a lack of connectivity all contribute to low fidelity results. In particular, connectivity restrictions enforce a gate set that requires three-qubit gates to be decomposed into one- or two-qubit gates. This substantially increases the number of two-qubit gates that need to be executed. However, many quantum devices have access to higher energy levels. We can expand the qubit abstraction of $|0\rangle$ and $|1\rangle$ to a ququart which has access to the $|2\rangle$ and $|3\rangle$ state, but with shorter coherence times. This allows for two qubits to be encoded in one ququart, enabling increased virtual connectivity between physical units from two adjacent qubits to four fully connected qubits. This connectivity scheme allows us to more efficiently execute three-qubit gates natively between two physical devices. We present direct-to-pulse implementations of several three-qubit gates, synthesized via optimal control, for compilation of three-qubit gates onto a superconducting-based architecture with access to four-level devices with the first experimental demonstration of four-level ququart gates designed through optimal control. We demonstrate strategies that temporarily use higher level states to perform Toffoli gates and always use higher level states to improve fidelities for quantum circuits. We find that these methods improve expected fidelities with increases of 2x across circuit sizes using intermediate encoding, and increases of 3x for fully-encoded ququart compilation.

Figures (9)

• Figure 1: A comparison of a Toffoli gate execution on a three-qubit-only system versus a Toffoli gate execution on a ququart and qubit in a mixed-radix system. In a qubit-only system, we must use a decomposition that uses eight two-qubit gates that can be reduced to one two-qudit gate that has a shorter duration.
• Figure 2: Interleaved Randomized Benchmarking for an optimal control $H \otimes H$ pulse on a superconducting transmon ququart following our qubit encoding. We use two-qubit Clifford sequences of gate depth up to 100 and average each data point over 10 samples. Error bars show the standard deviation of the mean but they are smaller than the mean markers. Red: Standard two-qubit Randomized Benchmarking to estimate the average Clifford gate fidelity to be $F_\mathrm{RB} \approx 95.8\%$. Blue: Interleaving the $H \otimes H$ pulse between the RB Cliffords yields a combined per-operation fidelity of $F_\mathrm{IRB} \approx 92.1\%$, resulting in an $H \otimes H$ fidelity $F_{H \! H} \approx 96.0\%$.
• Figure 3: Visualization of connectivity advantages in qubit-ququart systems. Encoding qubits in ququarts (light blue) enables triangle connectivity between triples of qubits, where two of which are encoded in the same ququart and one appears either in a bare qubit or encoded in a neighboring ququart.
• Figure 4: Visualization comparing the evolution of a $\ket{3}$-controlled $X$ gate in a mixed-radix environment for a CCX gate in (a) and a CX gate in (b).
• Figure 5: Examples of mixed-radix two-control and two-target gates. a) A configuration where both controls are encoded in the ququart and the target is mapped to a qubit. b) A configuration where the controls are split across the qubit and the ququart and the target is encoded in the ququart. c) A configuration where both targets are encoded in the ququart and the control is mapped to the qubit. d) A configuration where the targets are split across the qubit and the ququart and the control is encoded in the ququart.