Generation of $n$-qubit $W$ states using spin torque

The $W$ state is a symmetrically entangled multipartite state where a single excitation is shared by all parties. It is an important resource for various quantum algorithms and communication systems, and hence, its preparation is of immense interest to the quantum information community. We examine here a deterministic scheme to prepare a $W$ state of an $n$-qubit system with all-to-all pairwise exchange interaction between $n$ qubits. This relies on sharing superposed excitations of a smaller number of $q$ qubits among others. We present a bound on the maximal jumps from $q$ to $n$ and formalize a scheme to generate the $W_n$ state in $O(logn)$ stages. We demonstrate this scheme in the context of spin-torque-based quantum computing architecture that is characterized by repeated interactions between static and flying qubits.

Figure 1

$W_n$-state generation protocol. (a) $W_q$ to $W_n$ generator block: $n$-qubit entangling operation followed by phase-correction operations (single-qubit gates) on $q$ qubits to generate a $W_n$ state from a $W_q$ state. (b) Circuit to generate the $W_n$ state: Several $W_q$ to $W_n$ generator blocks are arranged successively in $\lceil {log}_{4}n\rceil$ to prepare a $W_n$ state starting from ${|0\rangle }^{\otimes n}$ with one initial bit-flip gate required to excite a single qubit.

Figure 2

Schematic of the system. $n$ static qubits (colored red) in a spin-coherent channel (shaded yellow). There are barrier gates (colored black) to facilitate creation of standing waves and a reservoir (colored purple) to inject and extract spin-polarized carriers. Additional barrier gates (shaded light green) between the qubits are used to isolate the qubits when not in use. The distance between two successive qubits is $d$, while that between a qubit and a barrier gate is $d_0$. Individual qubits act as spin-dependent scatterers with reflection and transmission denoted by $\left[r\right]$ and $\left[t\right]$ matrices. Reflection matrices looking into the cascade of scatterers are also shown. In addition, each qubit is separately connected to spin reservoirs through transverse channels to enable single-qubit operations.

Figure 3

Number of electrons required for evolution. Simulated number of electrons required for entangling evolution for the ${\overline{W}}_n$ state and for phase correction to get the $W_n$ state starting from a $W_3$ state in the spin-torque setting for $3<n\le 12$. Corresponding estimations shown as data circles match well with the simulations. Also, variation of $J_{\text{eff}}\delta t$ as a function of the number of qubits shown on the right axis.

Figure 4

Fidelity of obtained states. Fidelity of the $n$-qubit $W$ state obtained using a single-qubit sharing procedure vs sharing a superposition of three singly excited qubits (a $W_{q=3}$ state) as a function of $n$.

Figure 5

Average fidelity obtained and the number of electrons required for preparation (entangling evolution plus phase-correction evolution) of the $W_8$ state from the $W_2$ state on introduction of nonidealities. Nonidealities are introduced assuming a normal distribution with mean $\mu$ and standard deviation $\sigma$. For each $x$ data point, i.e., the ($\mu ,\sigma$) pair, 100 sample sets were simulated, followed by averaging corresponding data. Here, one sample set refers to the set of parameters sampled at each site. (a) Effect of nonideal transmission of barriers in between the qubits. $\mathrm{\Gamma }$ for the barriers is chosen from a distribution with $\mu =0$. (b) Effect of uncertainty in the exchange interaction of flying and static qubits. The distribution from which $\mathrm{\Omega }$'s were sampled had a mean of 0.0001. (c) Effect of nonideal distances between successive pairs of qubits. The parameters $kd$ and $k{d}_0$ were respectively sampled from distributions with $\mu =\pi$ and $\pi /2$, and each set was adjusted uniformly to have a constant sum (explained in the main text).

Figure 6

Schematic of the proposed architecture. Static qubits (red) embedded in the columns of a spin-coherent channel (bright yellow) grid. Hard barriers $G_{i,\widehat{k}}$ (dark turquoise) guide the electrons along a path (transparent and green) shown for three-qubit entanglement. The partial barrier which is also a part for implementing the scheme is colored pink. The distance between a qubit and the gates facing it in each column is $d_0$, while the distance of the farther gates is roughly $d/2$. The Manhattan distance between any two nodes on a horizontal level is an integer multiple of $\pi /k$. This ensures any two qubits are separated effectively by a distance $d$ up to certain tolerances governed by optimization of parameters.

Figure 7

${\mathcal{R}}_B$ for three qubits in the channel. Nonzero elements of the overall reflection matrix ${\mathcal{R}}_B$ are colored red, while the zero elements are colored blue independent of the parameters. The Kraus operators $M_0$ and $M_1$ are partitions of the full ${\mathcal{R}}_B$, highlighted in yellow and green. The terms relevant to the evolution are $\alpha$ and $\beta$ as labeled, while $\gamma , {\gamma }_1$, and ${\gamma }_2$ give undesired superpositions in evolution.

Figure 8

${\mathrm{P}}_{\mathit{3}}$ as a function of $\mathit{kd}$ and ${\mathit{kd}}_{\mathit{0}}$ for different values of $(\mathrm{\Gamma },\mathrm{\Omega })$ about $(1000,0.0001)$. Good choices of $kd$ and $k{d}_0$ occur around a negative sloped line about $(\pi ,\pi /2)$.

Figure 9

${\mathrm{P}}_{\mathit{3}}$ as a function of $\mathit{\Gamma }$ and $\mathit{\Omega }$ for different choices of $(kd,k{d}_0)$ about $(\pi ,\pi /2)$. $\mathrm{\Delta }kd$ and $\mathrm{\Delta }k{d}_0$ represent the offsets $kd-\pi$ and $k{d}_0-\pi /2$.

Figure 10

Quantum circuit for generation of the $W_3$ state.

Figure 11

Three-qubit evolution. Evolution of diagonal entries (left axis) and their product (right axis) of the system's density matrix represented in ${\mathcal{B}}_1^3$. The vertical line (black) marks the point when the product reaches the maximum, supposedly 1/27, in this case indicating the creation of the ${\overline{W}}_3$ state. The horizontal line (black) has an ordinate equal to 1/3 and indicates that the evolved desired density-matrix components intersect when close to ideal value 1/3.

Figure 12

Density-matrix components. (a) and (c) Real and (b) and (d) imaginary parts of the obtained density matrix before and after phase correction, respectively. The blue translucent and slightly thicker bars correspond to the density matrix of the desired $W_3$ state. There is a good match between the obtained and expected density matrices. The blue bars sheath the bars of the real part of the obtained density matrix after phase correction with negligible error in the imaginary part, as also reflected by the obtained fidelity of $99.9\%$.