Qutrit Randomized Benchmarking

Ternary quantum processors offer significant potential computational advantages over conventional qubit technologies, leveraging the encoding and processing of quantum information in qutrits (three-level systems). To evaluate and compare the performance of such emerging quantum hardware it is essential to have robust benchmarking methods suitable for a higher-dimensional Hilbert space. We demonstrate extensions of industry standard randomized benchmarking (RB) protocols, developed and used extensively for qubits, suitable for ternary quantum logic. Using a superconducting five-qutrit processor, we find an average single-qutrit process infidelity of $3.8\times {10}^{-3}$. Through interleaved RB, we characterize a few relevant gates, and employ simultaneous RB to fully characterize crosstalk errors. Finally, we apply cycle benchmarking to a two-qutrit CSUM gate and obtain a two-qutrit process fidelity of 0.85. Our results present and demonstrate RB-based tools to characterize the performance of a qutrit processor, and a general approach to diagnose control errors in future qudit hardware.

Figure 1

Randomized benchmarking of a single qutrit. For depths up to 400, we generate 20 Clifford-random gate sequences (with a final inversion gate) and measure (a) the resulting $Z$-basis state occupancies and (b) the observable $⟨Z⟩$. Each measurement is repeated 1024 times. Individual points correspond to each of the 20 randomizations, with dashed lines showing fits to a single exponential decay. As qutrit Clifford twirling effectively maps gate errors into a depolarizing channel, for long sequences the populations all tend to $1/3$ and the sequence-averaged qutrit state tends to the maximally mixed state $\rho =\frac{1}{3}{\mathbb{1}}_3$. Panel (b) illustrates the utility of measuring the expectation value of the Pauli operator $Z$ and taking its real part. The imaginary part of the expectation value remains at zero throughout, which means that the RB sequence is effectively depolarizing the noise.

Figure 2

Interleaved RB of three single-qutrit gates: (i) $\pi$ rotation in the $\{|0⟩,|1⟩\}$ subspace (blue), (ii) $\pi$ rotation in the $\{|0⟩,|1⟩\}$ subspace (orange), and (iii) the qutrit Hadamard gate $H$. (a) Depolarization decay for standard (no interleaving) qutrit RB (gray) as a reference, and interleaved sequences for each of the three gates under test. As in Fig. 1, each point represents a single random sequence measured 1024 times. (b) Process infidelities (dots) of each gate calculated from the data in (a). Also plotted for comparison (stars) is an estimation of the process infidelity from the Clifford-averaged value computed in the previous section: Our decomposition used an average of 3.325 native gates per Clifford, so a simple estimate for gate error for an $n$-pulse gate is $n/3.325$ times the Clifford average. For a Hadamard, $n=4$; for the $\pi$ rotations, $n=1$.

Figure 3

Average error per Clifford while simultaneously running RB sequences on two qutrits. The solid bars indicate the average error per Clifford with the crosstalk cancellation applied to the system; transparent bars are corresponding errors without cancellation. The gray bars indicate isolated qutrit results. Note that the large uncertainties are for noncancelled crosstalk.

Figure 4

Cycle benchmarking of the entangling controlled-SUM (CSUM) gate. For comparison, we have included the same experiment with (orange) only Pauli twirling, i.e., using no interleaved gate (for a control), and (blue) with the interleaved CSUM. We present the exponential decay obtained for each Pauli channel (we have omitted the complex conjugate as its value is identical). The process fidelity is calculated by averaging all the Pauli channels, yielding a process infidelity of $2.22\times {10}^{-2}$ for the control (or average over the $9^2$ 2-qutrit Pauli that compose the twirling)—and $1.83\times {10}^{-1}$ for the CSUM.