Doubly Geometric Quantum Control

In holonomic quantum computation, quantum gates are performed using driving protocols that trace out closed loops on the Bloch sphere, making them robust to certain pulse errors. However, dephasing noise that is transverse to the drive, which is significant in many qubit platforms, lies outside the family of correctable errors. Here, we present a general procedure that combines two types of geometry—holonomy loops on the Bloch sphere and geometric space curves in three dimensions—to design gates that simultaneously suppress pulse errors and transverse noise errors. We demonstrate this doubly geometric control technique by designing explicit examples of single-qubit and two-qubit dynamically corrected holonomic gates.

Popular Summary

The ability to control qubits with unprecedented precision is essential to quantum computers, devices that leverage quantum mechanics to perform tasks beyond the reach of present-day computers. Achieving the requisite precision is daunting due to both the complex, noisy environment of qubits and to unavoidable control imperfections. Most existing control methods focus on combating only one of these types of error. A general approach to designing control waveforms that suppress both environmental noise and control errors at the same time is presented.

Our approach exploits two types of geometric curves that can be associated with the evolution of a qubit. One curve describes how the state of the qubit evolves in time, while the other characterizes the error incurred by environmental noise. We show that, by imposing certain constraints on these curves, both control and noise errors can be reduced. Once a suitable set of curves is identified, the corresponding control waveforms that perform the desired operation can be extracted from their shapes. This “doubly geometric quantum control” method thus provides a general, systematic way to design control waveforms that perform arbitrary qubit operations with high precision despite the presence of multiple error sources. Moreover, this method provides a global, analytical understanding of the solution space of robust control waveforms, as well as a broad family of user-defined waveforms that respect experimental constraints while performing a target operation. More broadly, these findings also shed further light on the geometry underlying quantum dynamics.

Figure 1

(a) Schematic depiction of how the Aharonov-Anandan geometric phase is defined for a two-level system. States in the full Hilbert space (open dark magenta curve) are projected onto the Bloch sphere (blue arrows), and the projected states (closed red curve) enclose an area that determines the geometric phase. (b) Different Bloch sphere paths that realize holonomic gates, including the orange-slice model (top) and the triangle-cap model (bottom).

Figure 2

Error curve $\mathbf{r}(t)$ for $H_c(t)=\mathrm{\Omega }(t){\sigma }_x$ where $\mathrm{\Omega }(t)$ is a square pulse (orange) or a hyperbolic secant pulse (blue). In both cases, the pulses have area $\pi$. The sech pulse is given by $\mathrm{\Omega }(t)=\mathrm{sech}(t)$, where $t\in [-10,10]$.

Figure 3

The Bloch sphere representations (left) and error curves $\mathbf{r}(t)$ (right) of the standard orange-slice model holonomic gate, generated by the control Hamiltonian $H_{\mathrm{OS}}(t)$ [Eq. (13)]. The upper (lower) plots correspond to ${\mathrm{\Phi }}_0=0$ (${\mathrm{\Phi }}_0=\pi /2$). The curves are colored to indicate which part of the Bloch sphere trajectory corresponds to which part of the error curve, starting from red at $t=0$ and ending with violet at $t=T$.

Figure 4

The two-field control (top) and error curves (bottom) of the extended orange-slice DoG gate for three different gate angles ${\mathrm{\Phi }}_0$. Each DoG gate consists of two pole-to-pole evolutions along different geodesic lines on the Bloch sphere. Each pole-to-pole evolution is generated by four sech pulses that collectively implement a $\pi$ pulse. Each closed $\mathbf{r}(t)$ consists of two closed planar lobes that correspond to each pole-to-pole evolution. The colors indicate which part of the curve corresponds to which part of the pulse.

Figure 5

Twisted error curves ${\mathbf{r}}_{\xi }(t)$ for three values of the twist parameter $\xi$: $\pi /1000$ (left), $\pi /3000$ (middle), and $\pi /5000$ (right). As $\xi$ is reduced to zero, the curve flattens out and lies in a vertical plane. Note that ${\dot{\mathbf{r}}}_{\xi }(T)={\dot{\mathbf{r}}}_{\xi }(0)=\hat{z}$ (corresponding to the point where the red and purple segments meet) for all values of $\xi$.

Figure 6

3D DoG control fields $\mathrm{\Omega }(t)$, $\mathrm{\Phi }(t)$, $\mathrm{\Delta }(t)$ generated from a twisted error curve ${\mathbf{r}}_{\xi }(t)$ with $\xi =\pi /20000$ (left) and $\xi =\pi /2000$ (right). Time is shifted such that the zero points on the abscissas are centered in all plots. Here $\mathrm{\Omega }(t)$ and $\mathrm{\Delta }(t)$ both have near-sech envelopes.

Figure 7

Gate fidelities of two 3D DoG gates constructed from twisted error curves ${\mathbf{r}}_{\xi }(t)$ versus the detuning error rate (the ratio of the detuning error ${\delta }_z$ to the time-averaged driving strength $\overline{\mathrm{\Omega }}$). Results for the standard orange-slice model-based holonomic gate (HG) are shown for comparison. The top (bottom) plot corresponds to the target gate $U(\xi =\pi /2000)$ [$U(\xi =\pi /20000)$].

Figure 8

Comparing the performance of 3D and 2D DoG gates. Both panels show gate fidelity versus the pulse amplitude error rate [$ϵ$ in ${\mathrm{\Omega }}^{\prime }(t)=\mathrm{\Omega }(t)(1+ϵ)$]. The 3D DoG gates are obtained from twisted 3D error curves ${\mathbf{r}}_{\xi }(t)$. The top (bottom) panel shows results for the target gate $U(\xi =\pi /2000)$ [$U(\xi =\pi /20000)$].

Figure 9

Two-qubit DoG gate by tuning a tunable coupler and by driving qubit 2. (Left) A pair of dual curves $r_1(t)={\mathbf{r}}_{\xi }(t)$ and $r_2(t)={\mathbf{r}}_{-\xi }(t)$ with $\xi =\pi /2000$ that have same curve length and curvature and opposite torsion. (Right) Control fields that yield the two-qubit DoG gate: ${\lambda }_z(t)$ and ${\lambda }_{\perp }(t)$ in interqubit coupling (blue) and $\mathrm{\Omega }(t)$ in qubit 2 driving (red).

Figure 10

The error curve generated by the control field ${\mathrm{\Omega }}_1(t)$ described in Eq. (F1). This corresponds to one of the planar lobes of the 3D error curves plotted in Fig. 4.

Figure 11

Error curve ${\mathbf{r}}_{\xi }(t)$ with $\xi =\pi /20000$, and the corresponding tantrix ${\dot{\mathbf{r}}}_{\xi }(t)$ and the evolution trajectory on the Bloch sphere $\mathbf{h}(t)$.

Figure 12

Error curve ${\mathbf{r}}_{\xi }(t)$ with $\xi =\pi /2000$, and the corresponding tantrix ${\dot{\mathbf{r}}}_{\xi }(t)$ and the evolution trajectory on the Bloch sphere $\mathbf{h}(t)$.

Figure 13

The geometric phase ${\beta }_g$ of a 3D DoG gate versus the scaled twist parameter $1000\xi$ of the error curve.