The methodology of resonant equiangular composite quantum gates

TL;DR

The paper tackles the challenge of designing parameter-dependent single-qubit gates by introducing equiangular composite gates built from resonant primitive rotations and mapping the problem to polynomial representations $A,B,C,D$. It provides a complete characterization of achievable quantum response functions, an efficient $\mathcal{O}(\mathrm{poly}(L))$ compilation algorithm, and DSP-inspired selection methods to optimize gate performance under various objective criteria. By connecting quantum gate design to classical finite-impulse-response and Chebyshev minimax techniques, it delivers a practical, scalable methodology demonstrated on broadband/narrowband population inversion, broadband NOT, and sub-wavelength spatially selective gates. The work enables systematic, high-performance quantum control with wide bandwidths and precise spatial addressing, with implications for metrology, quantum computing, and quantum sensing, and lays a foundation for extensions to more complex or open quantum systems.

Abstract

The creation of composite quantum gates that implement quantum response functions $\hat{U}(θ)$ dependent on some parameter of interest $θ$ is often more of an art than a science. Through inspired design, a sequence of $L$ primitive gates also depending on $θ$ can engineer a highly nontrivial $\hat{U}(θ)$ that enables myriad precision metrology, spectroscopy, and control techniques. However, discovering new, useful examples of $\hat{U}(θ)$ 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 $θ$ act on a single spin. We fully characterize the realizable family of $\hat{U}(θ)$, provide an efficient algorithm that decomposes a choice of $\hat{U}(θ)$ into its shortest sequence of gates, and show how to efficiently choose an achievable $\hat{U}(θ)$ that for fixed $L$, is an optimal approximation to objective functions on its quadratures. A strong connection is forged with \emph{classical} discrete-time signal processing, allowing us to swiftly construct, as examples, compensated gates with optimal bandwidth that implement arbitrary single spin rotations with sub-wavelength spatial selectivity.

Figures (3)

• Figure 1: (color) $\text{DC}_{L,\mathcal{I}}$ (black), $\text{M}_{L,\mathcal{I}}$ (teal) polynomials plotted for $L=9$ and target worst-case infidelity $\mathcal{I}=10^{-2}$ (solid) and $\mathcal{I}\rightarrow 0$ (dashed), indexed by $_\text{f}$. The observed ripples are a generic feature of bandwidth optimized polynomials, unlike those optimized for maximal flatness $\text{DC}_{\text{f}},\text{M}_{\text{f}}$. The inset plots their squares and defines the bandwidth $\mathcal{B}$ in $x$ coordinates.
• Figure 2: (color) Worst-case infidelity $\mathcal{I}$ of NOT gates $\text{OB}n=(\cdot,0,M_{2n+1,\mathcal{I}}(\sin{(\theta/2)}),\cdot)$ (solid, Eq. \ref{['eq:OBInfidelity']}) optimized for target bandwidth $\theta\in\mathcal{B}$ compared to flatness-optimized NOT gate $\text{BB}n=(\cdot,0,M_{{2n+1},\text{f}}(\sin{(\theta/2)}),\cdot)$ (dashed, Eq. \ref{['eq:BBInfidelity']}), plotted for $L=2n+1=5,9,...,25$ (from top). Observe that $\mathcal{I}$ for OB$n$ is exponentially smaller by factor $\approx 4^{n}$ than BB$n$. Alternatively, an OB$n$ gate can approximate NOT with infidelity at most $\mathcal{I}$ over a much wider bandwidth than BB$n$. The table provides examples of $\vec{\phi}$ for OB$n$ rounded to $3$ decimal places.
• Figure 3: (color) Infidelity of spatially selective composite gates $(M_{L,10^{-4}}(\cos{(\frac{\theta_0}{2} e^{- r^2/\lambda^2})}),0,\cdot,\cdot)$ plotted for $\theta_0=\pi$ and $L=1,...,25$ (solid, from right). The effective beam width $\bar{\mathcal{B}}_{\text{space}}=\mathcal{O}(L^{-1/2})$(inset) beyond which the identity gate is well-approximated is dramatically reduced over that of a single gate $\bar{\mathcal{B}}_1$. By varying $\theta_0$, arbitrary unitary gates can be applied at $r=0$ with high beam-pointing stability. Poorer scaling $\bar{\mathcal{B}}_{\text{space}}=\mathcal{O}(L^{-1/4})$ results from using the flat $(M_{L,\text{f}},0,\cdot,\cdot)$ (dashed). The table provides examples of $\vec{\phi}$ to $3$ decimal places.

Theorems & Definitions (13)

• Definition 1: Achievable polynomial tuples
• Theorem 1: Achievable tuples
• proof
• Theorem 2: Achievable $2$-partial tuples
• proof
• Theorem 3: Achievable $3$-partial tuples
• proof
• Lemma 1: Optimal quantum response compilation
• proof
• Lemma 2: Transition probability sum-of-squares