Calibrating single-qubit gates by a two-dimensional Rabi oscillation

Recent advances of quantum technologies, such as atom interferometric sensors,¹ superconducting circuits,² trapped ions,³ and quantum information processors,^4,5 rely on high-fidelity unitary quantum gates throughout extended sequences of operations, allowing for precise manipulation of quantum states. In practice, however, the quantum systems that we are attempting to control suffer from unavoidable environmental noise and systematic errors caused by the experimental apparatus. Precise characterization of these errors in specific architecture is essential for improving performance.

In comparison with the errors from decoherence, correcting systematic errors is regarded as an easier task. Techniques, such as composite pulses⁶ and optimal control,^7,8 can provide resilience against them. Even where resource constraints preclude the use of these techniques, systematic errors can, in principle, be completely removed by tuning the parameters of the experimental apparatus. However, the process of calibrating gates is delicate and time-consuming, especially without resources for perfect state preparation and measurement.^9,10 Our work is a contribution to this calibration problem.

While the systematic errors of any single qubit gate can, in principle, be fully characterized by quantum process tomography¹¹ (QPT), this scheme requires fiducial state preparation and measurements. To circumvent this rigorous requirement, alternatives have been developed. Gate-set tomography (GST)^12,13 extracts a full tomographic characterization of multiple quantum gates from concatenated sequences of them, even without perfect state preparation and measurement. Similarly, randomized benchmarking (RB)^14–18 characterizes the error of Clifford gates from random concatenations. It is computationally more efficient than GST but, in general, only provides an average gate fidelity rather than tomographic information.

A conceptually similar approach has been developed in nuclear magnetic resonance.^19–21 Here, too, long concatenated sequences of gates have been employed for gate calibration. In contrast to GST and RB, however, these sequences have been handcrafted to reveal specific errors, such as a mismatch in pulse duration or phase, and allow for quick human interpretation and correction.

Here, we analyze a maximally simple scheme targeting the same goal. Composed of only two control pulses, it is especially suited for systems where extended concatenated sequences are not available, e.g., due to inefficient readout or a short coherence time. The latter can arise, for instance, in sensing applications where a qubit has to sit close to a material surface. Our scheme requires more measurements than approaches that explicitly aim at gate characterization from a minimal number of data points, such as bootstrap tomography.²² In turn, it allows for a straightforward graphical interpretation of the measurement.

As in NMR, we start from the hypothesis that the dominant gate errors arise from a few deviant parameters that we aim to characterize. These are the Rabi frequencies Ω_x and Ω_y of a quadrature pair of drives, the deviation Δϕ of their relative phase from ϕ = π/2, and the detuning Δ in the rotating-wave Hamiltonian

Our focus on these parameters is motivated by a typical practical implementation of this Hamiltonian. Frequently, a single microwave (MW) source is split by a quadrature splitter, and the resulting outputs are employed as the drive pair. While downstream components, such as switches, can affect amplitude and phase separately for each line of the drive pair, the detuning remains common mode for both quadratures.

The key idea of our work is shown schematically in Fig. 1. We exploit a sequence that we term a 2D Rabi oscillation to estimate the source of systematic errors [Fig. 1(a)]. This 2D Rabi oscillation pulse sequence consists of two consecutive MW pulses along different axes, followed by a measurement of the $σ̂z$ component of the spin state (quantified as the probability |⟨ψ|0⟩|²). We first discuss the result of this sequence in the absence of systematic errors [Figs. 1(b)–1(d)]. In this case, the rotation axes of the MW pulses are perpendicular to each other on the Bloch sphere. Figure 1(b) presents the corresponding evolution paths for the two 2D Rabi oscillation sequences of τ_x–τ_y and τ_y–τ_x, and Figs. 1(c) and 1(d) present a simulation of the measurement result based on the time evolution of a spin-1/2 Hamiltonian without decoherence (see the supplementary material for details). Three key features of this result deserve discussion. First, fixing the duration of either one of the two MW pulses [i.e., taking a horizontal or vertical line cut across Figs. 1(c) and 1(d)], the evolution is a one-dimensional Rabi oscillation with reduced amplitude and no phase shift. Second, the sequence is symmetric, in the sense that τ_x–τ_y and τ_y–τ_x induce an identical result. Third, if the first pulse is tuned to a duration of τ_x,y = π/(2Ω_x,y) and thus implements a perfect π/2 pulse, the second pulse will not alter the state of the spin. The axis of the second pulse here coincides with the state of the spin, implementing a spin lock. In general, each of these features is lost if pulse errors are present.

In order to investigate the behavior of the 2D Rabi oscillation under different systematic errors, we introduce amplitude (ΔΩ), phase (Δϕ), and detuning (Δ) errors in the simulation. An amplitude error encodes a difference in the Rabi oscillation frequency, e.g., due to a difference in the MW power in the two axes. The phase error (Δϕ) is the deviation of the angle between the two MW rotational axes on the Bloch sphere equator from π/2. A nonzero detuning (Δ) tilts both MW rotational axes out of the equatorial plane. Through the contribution of these errors, the 2D Rabi oscillation for τ_x–τ_y and τ_y–τ_x will evolve asymmetrically. The exact evolution is characteristic of the systematic error involved and hence allows for its determination [Figs. 2(a)–2(d)]. An amplitude error will result in a mere rescaling of either axis of the 2D measurement result [Fig. 2(a)]. A phase error [Fig. 2(b)] destroys the spin lock. In the presence of a phase error, no choice of τ can align the spin state with the second pulse axis, destroying the spin lock condition and creating a Rabi oscillation of non-vanishing amplitude in every line cut of Fig. 2(b). The sequences τ_x–τ_y and τ_y–τ_x remain symmetric, in the sense that they induce the same signal on the spin. In contrast, a nonzero detuning creates a response that is mirror-symmetric under the exchange of τ_x–τ_y and τ_y–τ_x [Fig. 2(c)]. When only one of the two sequences is considered, the result appears superficially similar to a phase error. Crucially, a phase error and a detuning can thus compensate for each other due to their opposite symmetry, leading to an apparently error-free result when only one of the sequences is recorded [lower plot of Fig. 2(d)].

The mechanism of this cancellation is presented in detail in Fig. 2(e), which models cancellation in the τ_y–τ_x sequence, i.e., the lower plot of Fig. 2(d). We focus on the case where the first pulse of the 2D Rabi sequence is close to a π/2 pulse (Ω_yτ_y ≈ π/2) for two reasons. First, it can be regarded as a worst-case example. In this setting, the second pulse should implement a spin lock (see above), and departure from this result is an easily visible signature. Second, it is a relevant test case, as optimization of orthogonal π/2 pulses is a key overall goal of pulse calibration.

To understand the compensation, we first note that a perfect spin lock will occur whenever the state of the spin after the first pulse coincides with the rotation axis of the second pulse, even if the latter is not located on a coordinate axis of the Bloch sphere. Second, we also note that, to first order in Δ, a detuning Δ has the same effect as a phase error. Rotating around a small circle around the tilted Ω_y axis, the spin state will still pass the equator of the Bloch sphere, but it will be slightly offset to the left or the right of the x-axis. This shift of the state can thus be compensated by a shift of the second pulse axis, which is provided by a phase error Δϕ. Interestingly, perfect cancellation can be reached despite the fact that the detuning also acts on the second axis because a slight lengthening of the first pulse will be sufficient to move the spin state slightly below the equator. To see in more detail why the effect of a detuning to first order in Δ resembles the effect of a phase error Δϕ, we expand the rotation operator of a θ-pulse by the Zassenhaus formula. The spin state manipulation by the pulse can be described as a unitary rotation. Taking the τ_y–τ_x sequence as an example, a θ rotation manipulation around the y-axis (with negative detuning error Δ) can be written as²² 

Expanding it by using the Zassenhaus formula, we obtain

The details of the calculation are presented in the supplementary material. Figure 2(f) illustrates this evolution trajectory in the Bloch sphere for an approximate π/2 pulse along y when taking into account both phase and detuning errors. According to the result of Eq. (3), the evolutionary trajectory of the spin state on the Bloch sphere can be decomposed into three consecutive error-free manipulations. First, the NV center spin state rotates around the x-axis for Δ(cos θ − 1) value [the orange curve in Fig. 2(f)]. Second, a Δsin θ rotation manipulation around the z-axis is applied to the spin state [the red curve in Fig. 2(f)]. Third, the spin state will evolve around the y-axis for an angle θ ≈ π/2. The slight offset caused by the first step (rotation around the x-axis) causes the spin to cross the equator on the left of the x-axis, mimicking a phase error Δϕ. For a rotation angle slightly larger than π/2, the state will reach the detuned second rotation $Ω⃗x$ [orange axis in Fig. 2(e)].

We experimentally verified the protocol on an ensemble of around 23 NV centers in a Chemical Vapor Deposition (CVD) synthesis process diamond (ElementSix, [100] orientation, $N≲1$ ppm, $NV≈$ 0.13 ppb) (Fig. 3). The experiment was conducted on a home-built confocal microscope with green laser excitation by optically detected magnetic resonance (ODMR) under ambient conditions. Microwave pulses from a quadrature pair of drives are generated by splitting the signal of a signal generator (SMIQ06B) with a quadrature power splitter (ZX10Q-2-34-S+) into two paths, which can be switched separately. After recombination in a second power splitter, the signal is amplified (ZHL-16W-43-S+) and sent to the sample. We note that the axis of the microwave pulse can be changed in the rotating frame of the qubit by changing the phase of the pulse. No change in the physical orientation of the driving field is required. The fluorescence is normalized to a one-dimensional Rabi oscillation to obtain the probability P_|0⟩.

We recorded both τ_x–τ_y and τ_y–τ_x for varying detuning Δ. For a large (|Δ/Ω| ≫ |Δϕ|) negative detuning [Fig. 3(a)], a mirror-symmetric response emerges between both sequences, as predicted by the simulation [Fig. 2(c)]. As the detuning is moved to a value that is small, but still negative, a cancellation of the detuning error and a phase error present in our microwave path appears [Fig. 3(b)]. The observed experimental response closely matches the simulation of [Fig. 2(d)]. The cancellation can be shifted from the τ_y–τ_x sequence to the τ_x–τ_y sequences by moving to a positive detuning [Fig. 3(c)], where the slope of the fringes is caused by the detuning changes sign. Here, too, the result remains asymmetric. While a detuning error can be compensated by a phase error for one sequence, it is not feasible to achieve the compensation for τ_x–τ_y and τ_y–τ_x sequences synchronously.

We finally turn to an analysis of the effects of a large detuning and the waiting time between the two Rabi pulses (Fig. 4). Here, we fix the 2D Rabi sequence at the spin lock condition [Fig. 4(a), first pulse close to Ωτ = π/2], while we vary the detuning continuously over a large range (vertical axis of the plots) and the pulse spacing dt in three discrete steps [columns of Figs. 4(b)–4(g)]. This measurement can be understood as a frequency-domain Ramsey experiment. Accordingly, the Ramsey fringes appear along the detuning axis, visible as horizontal stripes in [Figs. 4(b) and 4(c)]. Their spacing is inversely proportional to the pulse spacing dt. The spin response is expected to be perfectly anti-symmetric in Δ with respect to the point Δ = 0 [Figs. 4(d) and 4(e)]. At this point, a spin-lock condition occurs and no oscillation can be observed for varying the length of the second pulse. Surprisingly, we frequently observed a result that instead superficially appears symmetric around a negative value of Δ, which depends on dt [black dotted axis in Figs. 4(b) and 4(c), red dotted axis denoting Δ = 0]. This behavior can be caused by a miscalibration of the π/2 pulse, as can be verified by a simulation [Figs. 4(f)–4(g)]. A miscalibration of this parameter tends to wash out the ridge between areas of high and low spin signal (blue and red regions in Fig. 4), causing the signal to appear symmetric instead of antisymmetric.

In summary, we have demonstrated a simple technique to calibrate systematic errors of single-qubit gates, based on a two-dimensional Rabi oscillation. Requiring only two pulses without long waiting times, the scheme is amenable to systems with a short coherence time or inefficient readout. It requires more measurements than calibration schemes, such as gate set tomography or randomized benchmarking, but in turn provides a graphical result from which the errors can be read off qualitatively without further analysis. While the scheme is conceptually simple, care is required in its implementation. Different systematic errors can partially compensate for each other, which requires recording the measurement twice with alternating pulse axes.

See the supplementary material for details of the simulation of 2D Rabi oscillation and the calculation process of the Zassenhaus formula.

This work was supported by the Deutsche Forschungsgemeinschaft [DFG, Grant Nos. RE3606/1-1, RE3606/1-2, RE3606/2-1, RE3606/3-1, EXC-2111–390814868, SFB 1477 “Light–Matter Interactions at Interfaces” (Project No. 441234705)] from the European Union’s Horizon 2020 Research and Innovation Program under Grant Agreement No. 820394 (ASTERIQS) of the European Union, the National Key R&D Program of China (Grant No. 2018YFA0306600), the National Natural Science Foundation of China (Grant No. T2125011), and the Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0302200). Y.H. acknowledges the financial support from the China Scholarship Council. The authors acknowledge P. F. Wang and F. Kong for their helpful discussion.