Quantum logic between remote quantum registers

We consider two approaches to dark-spin-mediated quantum computing in hybrid solid-state spin architectures. First, we review the notion of eigenmode-mediated unpolarized spin-chain state transfer and extend the analysis to various experimentally relevant imperfections: quenched disorder, dynamical decoherence, and uncompensated long-range coupling. In finite-length chains, the interplay between disorder-induced localization and decoherence yields a natural optimal channel fidelity, which we calculate. Long-range dipolar couplings induce a finite intrinsic lifetime for the mediating eigenmode; extensive numerical simulations of dipolar chains of lengths up to $L=12$ show remarkably high fidelity despite these decay processes. We further briefly consider the extension of the protocol to bosonic systems of coupled oscillators. Second, we introduce a quantum mirror based architecture for universal quantum computing that exploits all of the dark spins in the system as potential qubits. While this dramatically increases the number of qubits available, the composite operations required to manipulate dark-spin qubits significantly raise the error threshold for robust operation. Finally, we demonstrate that eigenmode-mediated state transfer can enable robust long-range logic between spatially separated nitrogen-vacancy registers in diamond; disorder-averaged numerics confirm that high-fidelity gates are achievable even in the presence of moderate disorder.

Figure 1

Quantum state transfer is achieved by tuning the left and right quantum registers (blue) to a particular eigenmode (red) of the intermediate data bus. By ensuring that the coupling $g$ between the registers and the chain is sufficiently weak relative to the spacing of adjacent eigenmodes, it is possible to consider evolution in an effective three-mode picture. Such eigenmode-mediated QST is applicable in a variety of contexts, ranging from solid-state spin chains to coupled bosonic degrees of freedom (e.g., pendulums or cavity arrays).

Figure 2

Schematic circuit diagram depicting eigenmode-mediated state transfer between the quantum registers (QR) 0 and $N+1$. Controlled-phase gates are represented as circle-ending dumbbells while cross-ending dumbbells depict a swap gate. In addition to the desired state transfer, each register is cp entangled with all intermediate spins owing to the Wigner strings associated with fermionization. This additional entanglement can be canceled by utilizing a simple two-qubit encoding.

Figure 3

(a) Contour plots for $N=11$ characterizing the average achievable fidelity as a function of the NV depolarization time $T_1$ and the coupling strength disorder induced by imperfect implantation. Numerics utilize an average intrachain spin spacing of $d=10$ nm corresponding to a $\kappa =50$ kHz dipole-dipole interaction strength. This intrachain spacing is assumed to be Gaussian distributed and the implantation deviation represents the standard deviation ${\sigma }_d$. For each ${\sigma }_d$, 1000 realizations were averaged to obtain the plotted fidelity and a smooth contour plot is generated via a third-order spline interpolation. The register-chain coupling strengths $g_L$ and $g_R$ [Eq. (29)] are assumed to be fully tunable via control of the three-level NV ground-state manifold [3]. (b) Analogous contour plots for $N=51$. In this case, the NV registers are separated by an order optical wavelength enabling individual laser manipulation without the need for subwavelength techniques. (c) Participation ratio for eigenmodes ($N=51$). Each eigenmode is indexed by its PR and the number of states within a certain PR bin is shown. For each disorder (which are represented as fractions of the bare coupling strength $\kappa =50$ kHz), 1000 realizations are averaged.

Figure 4

Coupling pattern $\left\{J_i\right\}$ between spins for two different cases: (i) engineered couplings (circles) as in Ref. [31] and (ii) the strong-coupling regime (squares). The left $y$ axis characterizes the coupling strength for each case and is associated with solid symbols; the couplings are plotted between spin numbers (e.g., $J_0$ is plotted between spin numbers 0 and 1). The right $y$ axis characterizes the fermionic spectrum (in this case, the $x$ axis is simply an index) and is associated with the open symbols. The open red circles depict the exactly linear spectrum of the engineered chain, while the open green squares depict the quasilinear spectrum of the strong-coupling case with uniform interchain couplings $\kappa =1$ and optimized $g\approx 0.7$.

Figure 5

Strong-coupling regime: By tuning $g/\kappa \sim N^{-1/6}$, we obtain high-fidelity QST utilizing an unpolarized chain with two-qubit encoding (paired protocol). The transfer time scales linearly with $N$ (Lieb-Robinson bound) [76] and high fidelities greater than $90\%$ can be maintained for chain lengths up to $N=100$.

Figure 6

Infidelity of QST for a strongly coupled chain with long-range interactions ($1/T_1$ is assumed to be negligible). Encoded state-transfer (paired-protocol) fidelities are shown for dipolar (diamonds), NNN-canceled-dipolar (squares), and NN interaction (circles) models.

Figure 7

(a) In a qubit chain of length $N$, a mirror swap operation is defined as the pairwise swap between the $(1,N),(2,N-1),...$ qubits. This can be achieved via global control in the form of single-qubit gates (Hadamards) and controlled-phase gates. Regardless of the initial state, a mirror swap occurs after a $N+1$ cycles of $Q=\tilde{H}\widetilde{cp}$ [43], where $\tilde{H}$ represents a global Hadamard operation and $\widetilde{cp}$ denotes a global controlled-phase operation. (b) Coupling the central NV register to the NV on the left (right) requires the ability to perform a swap directed to a neighboring nitrogen spin (gray). These directed swap operations are made possible by using combinations of the mirror sequences $Q_M$ (swaps a pair of impurities directly surrounding a NV) and $Q_L$ (mirror swaps an individual impurity chain). (c) Utilizing a fast echo pulse on the NV register (in combination with $Q_M$ and $Q_L$) allows one to generate selective interactions between the NV and any outlined nitrogen.

Figure 8

Schematic diagram of the 2D computational lattice showing limitations imposed by missing nitrogen implantations (stars) and imperfect nitrogen-to-NV conversions. Coherent coupling of distant NV registers in a faulty 2D array can be achieved via global pulsed control of a spin-chain quantum data bus. A combination of optical beams and a magnetic-field gradient allows for individual control of NV registers; combined with global single-qubit operations on nitrogen impurities in any given row (orthogonal to the field gradient) [3], directed swap operations (e.g., dark green path) can be achieved, which allow for quantum information transfer along arbitrary paths. This field gradient enables a swap gate to be performed between two NV registers in adjacent rows, which occupy the same column. Moreover, it in fact also enables any pair of rows to be swapped provided the intrarow interactions refocus.

Figure 9

Individual control of any given NV register (row 1) enables a swap operation between any two neighboring qubits along the same row. We illustrate the specific example of a four-qubit chain. The depicted gate sequence achieves a swap gate (up to individual qubit rotations $A$ and $B$) between the second and third qubit by only applying a local gate $X$ on the first qubit (NV register) and global operations elsewhere.

Figure 10

Comparison between the perturbative (dotted) and exact (solid) calculations for the matrix element $M_{0,N+1}$ for $N=205$. Here $1-|M_{0,N+1}{|}^2$ is plotted as a function of $g$, the register-chain coupling strength ($\kappa =1$). As expected, perturbation theory breaks down as $g\sim \kappa /\sqrt{N}$. The inset depicts a close-up of the region for $0.005<g<0.04$, where the perturbative expression is in excellent agreement with the exact calculation.