Resource state generation for a multispin register in a hybrid matter-photon quantum information processor

TL;DR

The paper tackles the challenge of generating high-fidelity resource states for fusion-based, hybrid matter–photon quantum computing in solid-state spin registers. It introduces a pulsed-control framework that nonuniformly modulates inter-spin couplings to preserve nearest-neighbor ($NN$) interactions while canceling long-range $zz$ couplings, using composite pulses, shaped pulses, and optimal control to realize broadband and selective gates. The approach is demonstrated in solvable four- and six-spin spin rings and extended to larger rings via collective-pulse schemes, with detailed NV-center implementations including robust broadband $\pi_x$ and selective $\alpha_x$ pulses validated by numerical simulations. The results offer a scalable path toward deterministic resource-state generation and integration with photonic links, advancing the practicality of hybrid FBQC in solid-state platforms. Key mathematical elements include engineering phases via $\theta_{ij}=g_{ij}\sum_k f_{ij}(k)\tau_k$, targeting $\theta_{ij}=\pi$ for NN couplings and $\theta_{ij}=0$ for long-range terms, and solving linear systems $M\cdot\vec{\tau}=\vec{\alpha}$ or $\tilde{M}\cdot\vec{\tau}=\vec{\alpha}$ to determine pulse timings. The framework is poised for other spin-based solid-state systems and molecular qubits, offering a robust method to realize deterministic cluster-state resources for scalable quantum computation.

Abstract

Hybrid quantum architectures that integrate matter and photonic degrees of freedom present a promising pathway toward scalable, fault-tolerant quantum computing. This approach needs to combine well-established entangling operations between distant registers using photonic degrees of freedom with direct interactions between matter qubits within a solid-state register. The high-fidelity control of such a register, however, poses significant challenges. In this work, we address these challenges with pulsed control sequences which modulate all inter-spin interactions to preserve the nearest-neighbor couplings while eliminating unwanted long-range interactions. We derive pulse sequences, including broadband and selective gates, using composite pulse and shaped pulse techniques as well as optimal control methods. This ensures a general pulse sequence in the presence of spin-position bias, and robustness against static offset detunings, and Rabi frequency fluctuations of the control fields. The control techniques developed here apply well beyond the present setting to a broad range of physical platforms. We demonstrate the efficacy of our methods for the resource state generation for fusion-based quantum computing in four- and six-spin systems encoded in the electronic ground states of nitrogen-vacancy centers or other molecular solid-state qubits. We also outline other elements of the proposed architecture, highlighting its potential for advancing quantum computing technology.

Figures (11)

• Figure 1: (a) Scheme of the pulsed cluster preparation in a quantum multi-spin system, where all spins are driven by the global field (light green). The solid lines indicate the NN couplings that have to be preserved, while the dashed lines represent the unwanted long-range couplings. $\omega_i$ indicates the unique energy splittings of spin $i$. (b) Pulsed dynamical sequence to eliminate long-range couplings. The whole process is divided into $m$ segments, and each segment consists of a free evolution time $\tau_k$ sandwiched by two selective $\pi_x$ and $\pi_x^{\dagger}$ gates denoted as $R_\pi^x(\textbf{i}^{(k)})$ and $R_\pi^{-x}(\textbf{i}^{(k)})$. The index-vector $\textbf{i}^{(k)}$ indicates the spins flipped by the $\pi_x$ pulses in $k$-th segment. The foundation of the scheme lies in a selective single $\pi_x$ pulse, which selectively flips the resonant spin while leaving the remainder unchanged. As we discuss in Sec. \ref{['subsec_broad-band']} and Sec. \ref{['subsec:selective_gate']}, the selective $\pi_x$ and $\pi_x^\dagger$ pulses could be implemented by a sequence of two shaped $(\pi/2)_x$ pulses (purple curves) and two composite $\pi_x$ pulses (yellow and blue blocks), depicted within the dashed box. The generation of all pulses is attributed to the global driving field due to limitations in the spatial resolution of the control field.
• Figure 2: The preparation of a cluster state in a four-spin system, where the square lattice without and with position errors are represented by green dots and black circles in (a), respectively. The corresponding coupling strengths $g_{ij}$ are represented by green and purple bars in (b). $g_1$ is the NN coupling strength in the ideal case. (c) The evolution of the accumulated phase $\theta_{ij}$ with respect to the NN couplings $g_{12,14}$ and one NNN coupling $g_{13}$ are represented. In each segment, the derivatives of the phases with respect to time are determined by the corresponding modulation factors. The sequence of the selective $\pi$ pulses employed are $\mathcal{S}_4=${(0), (1), (1, 2), (2), (2, 3), (3), (4)}, resulting in a pulse sequence [$\tau_1$-$\pi_1$-$\tau_2$-$\pi_2$-$\tau_3$-$\pi_1^\dagger$-$\tau_4$-$\pi_3$-$\tau_5$-$\pi_2^\dagger$-$\tau_6$-$\pi_3^\dagger\pi_4$-$\tau_7$-$\pi_4^\dagger$], the notation and meaning of $\tau_i$ and $\pi_i$ are introduced in main text. The parameters are identical to those presented in Sec. \ref{['subsec:numerical']} within a solid-state system.
• Figure 3: The preparation of cluster states in the larger spin-ring system. (a) The fidelity of the state preparation of the solutions obtained within a ten-spin system, along with the corresponding number of single-spin $\pi_x$ pulses and duration of the control field, $T_c$. (b) The minimum number of single-spin $\pi_x$ pulses required to prepare a cluster state in a multi-spin system, which increases with system size.
• Figure 4: Simulated infidelities between the optimal composite pulses and the target pulses. (a) Infidelity of the composite pulse $\pi_x$ versus static detuning and control amplitude error with $n_{\phi}=10$. (b) The vertical and horizontal cut along $\delta=0$ and $\varepsilon=0$, respectively. (c) and (d) are the same with (a) and (b) but for an optimal $(\pi/2)_x$ gate and the corresponding parameter $n_{\phi}=8$. The integers $m$ in (a, c) indicate the infidelity $10^{-m}$. Additionally, two regions of high fidelity ($F>1-10^{-5}$) are identified by red dashed boxes, bounded by $|\delta|\leq0.35$ and $|\varepsilon|\leq0.03$.
• Figure 5: The scheme of the composite selective pulse $\alpha_x$. (a) Time-domain of the selective pulse $\alpha_x$. The pulse is composed of two sequential shaped $(\alpha/2)_x$ (purple curves) and two broadband $\pi_x$ and $\pi_x^\dagger$ pulses (yellow and blue blocks) applied at $t=T_s/2, T_s$ respectively. (b) Frequency domain of the pulse $\alpha_x$. The first two diagrams represent the performances of two pulses in the time domain with the corresponding unitary $(\alpha/2)_x$ and $\pi_x^\dagger(\alpha/2)_x\pi_x$, respectively. In the far detuning region (DR), defined as $|\delta|\geq\delta_d$, the large detuning term $\delta\Omega_0\sigma_z/2$ dominates and results in two nearly pure $z$ pulses $\pm\theta_z$ with $\theta = \delta\Omega_0 T_s$. As shown in the third diagram, the two $z$ pulses ultimately cancel, yielding an identity operator Meanwhile, in the resonant region (RR), $|\delta|\leq\delta_r$, both of the two unitary operators remain $(\alpha/2)_x$ (purple blocks), resulting in a desired $\alpha_x$ pulse. The transition region (TR) is given by $|\delta|\in(\delta_s, \delta_d)$, where the evolution (green blocks) is usually quite different from the desired gate. (c) Gaussian-shaped pulses $\Omega_{x(y)}$ to implement a single $(\alpha/2)_x$ pulse. (d) Contour plots of the simulated infidelity of the selective $\pi_x$ pulse versus $\delta$ and $\varepsilon$. The high fidelity area is highlighted by the red box bounded by $|\delta|=0.05$ and $|\varepsilon|=0.03$. (e) The fidelity between the composite pulse and ideal $\pi_x$ gate as a function of detuning, and the insert shows the fidelity in RR and $f_5=0.99999$. The resonant region is indicated by two red dotted lines $\delta = \pm 0.05$. (f) The infidelity between the composite pulse and identity gate as a function of detuning, and the DRs are defined as $|\delta|>\delta_d=1.75$ here.