Methodology of Resonant Equiangular Composite Quantum Gates

The creation of composite quantum gates that implement quantum response functions $\widehat{U}(\theta )$ dependent on some parameter of interest $\theta$ is often more of an art than a science. Through inspired design, a sequence of $L$ primitive gates also depending on $\theta$ can engineer a highly nontrivial $\widehat{U}(\theta )$ that enables myriad precision metrology, spectroscopy, and control techniques. However, discovering new, useful examples of $\widehat{U}(\theta )$ requires great intuition to perceive the possibilities, and often brute force to find optimal implementations. We present a systematic and efficient methodology for composite gate design of arbitrary length, where phase-controlled primitive gates all rotating by $\theta$ act on a single spin. We fully characterize the realizable family of $\widehat{U}(\theta )$, provide an efficient algorithm that decomposes a choice of $\widehat{U}(\theta )$ into its shortest sequence of gates, and show how to efficiently choose an achievable $\widehat{U}(\theta )$ that, for fixed $L$, is an optimal approximation to objective functions on its quadratures. A strong connection is forged with classical discrete-time signal processing, allowing us to swiftly construct, as examples, compensated gates with optimal bandwidth that implement arbitrary single-spin rotations with subwavelength spatial selectivity.

Popular Summary

Many important existing and next-generation technologies rely on quantum systems sensitive to weak signals. However, these systems are vulnerable to noise, and extracting signals encoded in quantum states is hampered as a result of destruction through their very measurement. Obtaining maximum information in a single measurement is essential and requires a quantum form of signal processing, such as through composite quantum gates. Unfortunately, a full understanding of this approach is lacking, and determining how to implement useful composite gates is difficult. Here, we theoretically characterize the possibilities allowed by a common class of composite gates, and we suggest several efficient design algorithms and the associated quantum response functions.

Composite quantum gates can be thought of as a sequence of more primitive quantum gates. Compared with a single quantum gate, they exhibit an altered functional dependence on the amplitude of the incoming signal. Considering a system of a resonantly driven single spin, we demonstrate all achievable quantum response functions, and we suggest an efficient methodology for finding composite gates that implement an optimal approximation to some arbitrary objective (e.g., subwavelength spatial selectivity). Our work makes a close connection with classical discrete-time signal processing, which allows us to translate some of its tools and insights into this decidedly nonclassical quantum problem.

Our methodology is applicable to designing certain quantum algorithms, some of which may advance applications such as magnetic resonance imaging, quantum metrology, and quantum error suppression.

Figure 1

${\mathrm{DC}}_{L,\mathcal{I}}$ (black lines) and ${\mathrm{M}}_{L,\mathcal{I}}$ (teal lines) polynomials plotted for $L=9$ and target worst-case infidelity $\mathcal{I}={10}^{-2}$ (solid lines) and $\mathcal{I}\to 0$ (dashed lines), indexed by f. The observed ripples are a generic feature of bandwidth-optimized polynomials, unlike those optimized for maximal flatness ${\mathrm{DC}}_{\mathrm{f}}$, ${\mathrm{M}}_{\mathrm{f}}$. The inset plots their squares and defines the bandwidth $\mathcal{B}$ in $x$ coordinates.

Figure 2

Worst-case infidelity $\mathcal{I}$ of NOT gates $\mathrm{OB}n=[·,0,M_{2n+1,\mathcal{I}}(\sin (\theta /2)),·]$ [solid line; Eq. (37)] optimized for target bandwidth $\theta \in \mathcal{B}$ compared to the flatness-optimized NOT gate $\mathrm{BB}n=[·,0,M_{2n+1,\mathrm{f}}\mathbf{(}\sin (\theta /2)\mathbf{)},·]$ [dashed line; Eq. (35)], plotted for $L=2n+1=5,9,\dots ,25$ (from top to bottom). Observe that $\mathcal{I}$ for $\mathrm{OB}n$ is exponentially smaller by a factor $\approx 4^n$ than $\mathrm{BB}n$. Alternatively, an $\mathrm{OB}n$ gate can approximate NOT with infidelity of, at most, $\mathcal{I}$ over a much wider bandwidth than $\mathrm{BB}n$. The table provides examples of $\overrightarrow{\varphi }$ for $\mathrm{OB}n$ rounded to three decimal places.

Figure 3

Infidelity of spatially selective composite gates $(M_{L,{10}^{-4}}\left\{\cos [({\theta }_0/2)e^{-r^2/{2\lambda }^2}]\right\},0,·,·)$ plotted for ${\theta }_0=\pi$ and $L=1,\dots ,25$ (solid lines, from right to left). The effective beam width ${\overline{\mathcal{B}}}_{\text{space}}=\mathcal{O}(L^{-1/2})$ (inset) beyond which the identity gate is well approximated is dramatically reduced over that of a single gate ${\overline{\mathcal{B}}}_1$. By varying ${\theta }_0$, arbitrary unitary gates can be applied at $r=0$ with high beam-pointing stability. Poorer scaling ${\overline{\mathcal{B}}}_{\text{space}}=\mathcal{O}(L^{-1/4})$ results from using the flat $(M_{L,\mathrm{f}},0,·,·)$ (dashed lines). The table provides examples of $\overrightarrow{\varphi }$ to three decimal places.