Reducing Unitary and Spectator Errors in Cross Resonance with Optimized Rotary Echoes

We present an improvement to the cross resonance gate realized with the addition of resonant, target rotary pulses. These pulses, applied directly to the target qubit, are simultaneous to and in phase with the echoed cross resonance pulses. Using specialized Hamiltonian error amplifying tomography, we confirm a reduction of error terms with target rotary—directly translating to improved two-qubit gate fidelity. Beyond improvement in the control-target subspace, the target rotary reduces entanglement between target and target spectators caused by residual quantum interactions. We further characterize multiqubit performance improvement enabled by target rotary pulsing using unitarity benchmarking and quantum volume measurements, achieving a new record quantum volume for a superconducting qubit system.

Popular Summary

Quantum technologies promise the ability to solve problems that are difficult on classical systems. Quantum information processors of moderate sizes have recently been realized; however, achieving large-scale quantum processors requires overcoming various obstacles such as improving physical gates and reducing unwanted interactions between system components. Here we introduce an improved understanding and implementation of quantum gates that provide a significant step forward in overcoming these issues.

Superconducting circuit-based quantum processors are one of the leading candidates for achieving large-scale quantum computers. The overall performance of the processor can be quantified by the quantum volume, which encapsulates important system features such as physical gate error and unwanted subsystem interactions. Here we present a detailed theoretical understanding as well as an experimental implementation of an improved entangling quantum gate that reduces both the gate error as well as unwanted interactions with nearby subsystems. The system improvement from using this gate is verified by the first demonstration of quantum volume 32 in a superconducting circuit-based processor.

The theoretical and experimental techniques developed here will be necessary for continuing to understand and design improved quantum gates with minimized unwanted interactions. Overall the methods will be important as quantum systems scale to larger sizes with larger quantum volume.

Figure 1

(a) Diagram of the echoed CR gate with target rotary pulse sequence, where $R_{\pm }$ denotes a fixed $\pm X$ rotation on target. (b) Hamiltonian error amplifying tomography (HEAT): example of four gate sequences used to accurately reconstruct echoed CR Hamiltonian error terms. (c) Hamiltonian errors with target rotary on a pair of qubits (inset: f₀₁ = qubit frequency, δ = anharmonicity). Solid lines are numerical fits to the data described in the Appendix pp8. (d) Two-qubit error per gate (EPG) from randomized benchmarking tracks the error is blue estimated by only considering the contribution from the four error terms measured in (c) added to the coherence limit set by the measured $T_1$s and $T_2$s for a gate time of 484 ns (dashed line).

Figure 2

(a) Quantile distribution of measured static $ZZ$ between coupled qubits on $ibmq\mathrm{\_}johannesburg$. Median $ZZ$ denoted by gray dashed line. (b) Modified HEAT sequence to measure dominant target-target spectator entangling Hamiltonian terms. (c) Hamiltonian errors with target rotary [numerical fits (solid line) and experiment (circles)] on a pair of qubits (inset).

Figure 3

Unitarity of the target spectator (red) and control-target (blue) subspaces extracted from purity RB on each subspace while performing Clifford gates in the opposite subspace. The product unitarity [Eq. (14)] is shown in green, as well as the estimated unitarity in each case including only measured $T_1$ s and $T_2$ s (dashed lines) and also unitary errors from fits above (solid curves). Error bars are the standard deviation of measurements sampled over approximately 1 week to represent drift in unitarities. Black solid line (right axis) shows the numerically simulated unitary entanglement over the same range of rotary amplitudes.

Figure 4

HEAT sequences for reconstructing error terms.

Figure 5

(a) Typical parameters and performance of $ibmq\mathrm{\_}johannesburg$. QV32 HOP on subset $A$ (blue dashed line) with and without rotary is presented in Table 1 in the main text. (b) To more rigorously study the impact of target rotary on quantum volume circuits, we executed the same set of circuits on all possible 5Q linear subsets of $ibmq\mathrm{\_}johannesburg$ for the same two cross-resonance gate conditions—with and without rotary. In this comparison plot of the HOP we see that, when using target rotary, the HOP is almost always greater than that obtained without for this experiment. As a guide to the eye, the black line denotes equal HOPs and the dashed black lines mark the $\frac{2}{3}$ QV success threshold. As the intent of this study is to estimate an average change in the HOP, we used only a limited number of circuits such that the experiment could run in a reasonable time frame. Thus, these experiments did not pass the QV32 threshold with confidence as was done on subset $A$.