Quantifying entanglement in cluster states built with error-prone interactions

Measurement-based quantum computing is an alternative paradigm to the circuit-based model. This approach can be advantageous in certain scenarios, such as when read-out is fast and accurate, but two-qubit gates realized via inter-particle interactions are slow and can be parallelized to efficiently create a cluster state. However, understanding how two-qubit errors impact algorithm accuracy and developing experimentally viable approaches to characterize cluster-state fidelity are outstanding challenges. Here, we consider one-dimensional cluster states built from controlled phase, Ising, and XY interactions with slow two-qubit error in the interaction strength, consistent with error models of interactions found in a variety of qubit architectures. We detail an experimentally viable teleportation fidelity that offers a measure of the impact of these errors on the cluster state. Our fidelity calculations show that the error has a distinctly different impact depending on the underlying interaction used for the two-qubit entangling gate. In particular, the Ising and XY interactions can allow perfect teleportation through the cluster state even with large errors, but the controlled phase interaction does not. Nonetheless, we find that teleportation through cluster state chains of size $N$ has a maximum two-qubit error for teleportation along a quantum channel that decreases as $N^{-1/2}$. To enable the construction of larger cluster states, we design lowest-order refocusing pulses for correcting these slow errors in the interaction strength. Our work generalizes to higher-dimensional cluster states and sets the stage for experiments to monitor the growth of entanglement in cluster states built from error-prone interactions.

Figure 1

Schematic of two stages of the fidelity measurement shown for five qubits. The first column depicts preparation of qubit 1 defining the input state, $|{\psi }_{\text{I}}^{\widehat{r}}\rangle$, where information is encoded in the qubit orientation, $\widehat{r}$. This qubit is entangled with the remaining qubits (2–5) that define a cluster state $|{\mathrm{\Phi }}_C\rangle$. The resulting state $|{\psi }_{\text{I}}^{\widehat{r}},{\mathrm{\Phi }}_{\text{C}}\rangle$ is then measured. The second column depicts the measurement stage where measurement along the qubit-$x$ direction for qubits 1 through 4 effectively moves the information encoded in qubit 1 to qubit 5. Quantum state tomography on qubit 5 allows reconstruction of an output state, $|{\psi }_{\text{O}}\rangle$, that is, in the absence of error, identical to the input, $|{\psi }_{\text{I}}^{\widehat{r}}\rangle$, up to a matrix, $U_{\mathrm{\Sigma }}$, defined from measurement results of qubits 1 to 4. An error-free cluster state teleports information along a quantum channel from qubit 1 to qubit 5. Sufficiently strong gate errors will destroy entanglement in the cluster state which can be detected in degraded teleportation. The information recorded in the measurements is used offline to construct the fidelity ${\mathcal{F}}_{\widehat{r}}$ of the cluster state, where ${\mathcal{F}}_{\widehat{r}}>2/3$ guarantees a quantum channel.

Figure 2

Circuit diagram depicting the construction of a four-qubit cluster state. The initial kets are entangled with the imperfect controlled phase interaction between qubits: $U_{\text{CP}}^{\pi /4,\varepsilon }$, where $\varepsilon$ denotes inclusion of a dimensionless error parameter, Eq. (10). A fifth qubit, in state $|{\psi }_{\text{I}}^{\widehat{r}}\rangle$, initializes information at one end of the cluster state. The build and initialization protocols here consist of commuting operations and can therefore be performed in a different time sequence than the one shown, e.g., all at the same time. Single-qubit projective measurements along the qubit-$x$ direction, $\mathcal{P}\left({\text{M}}_x\right)$, are recorded and used offline in $U_{\mathrm{\Sigma }}$ [See Eqs. (2) and (3)]. The measurements teleport the information defined in $|{\psi }_{\text{I}}^{\widehat{r}}\rangle$ along the chain to the last qubit where quantum state tomography yields a fidelity, Eq. (3), which tends to unity as $\varepsilon \to 0$.

Figure 3

A cross section of the fidelity plotted for the $N=5$ cluster state built from the error-prone controlled phase gate $U_{\text{CP}}^{\pi /4,\varepsilon }$ as a function of error strength computed by assuming the same fixed error in all two-qubit gates (${\varepsilon }_1={\varepsilon }_2={\varepsilon }_3={\varepsilon }_4$ in Fig. 2). The fidelity, Eq. (3), is computed for the process described in Fig. 2. We set a fixed interaction time using $\theta =\pi /4$ in Eq. (11), so that the two-qubit interaction returns the desired controlled-Z gate (and therefore a perfect cluster state) in the absence of error. The error is chosen as a static perturbation of the controlled phase interaction strength: $\pi /4(1+\varepsilon )$, so that $\varepsilon$ is the fractional change in the interaction strength. The initial state orientation $\widehat{r}$ is chosen from a uniform distribution on the Bloch sphere to display benchmarking over varied initial states. Here the $x, y$, and $z$ axes correspond to three respective directions in qubit space for the input qubit, $\widehat{r}$. The distance of the plotted points from the origin corresponds to the output fidelity, Eq. (3). For low error, e.g., $\varepsilon =1/\left(2\pi \right)$ (red circles), the output qubit is nearly identical to the input qubit leading to an approximate map of the Bloch sphere with nearly unity radius. The blue circles correspond to a larger error, $\varepsilon =1/\pi$, where the fidelity is well below unity. The resulting map of the Bloch sphere is a smaller shape. The resulting shape approximates a sphere but with weak anisotropy.

Figure 4

The minimum fidelity of the cluster state built from the error-prone controlled phase gate $U_{\text{CP}}^{\pi /4,\varepsilon }$ as a function of error strength computed by assuming the same fixed error in all two-qubit gates (${\varepsilon }_1={\varepsilon }_2=\cdots ={\varepsilon }_N$ in Fig. 2). The fidelity, Eq. (3), is computed by the process described in Fig. 3. The minimum fidelity found from the initial state sampling distribution is chosen to show the worst case scenario. Fidelities above the dashed line (2/3) teleport along a quantum channel. As we increase the chain length from $N=3$ to $N=9$, the fidelity degrades. However, we see that even the worst performing initial states allow teleportation through a quantum channel in cluster states defined with fractional errors in two-qubit gate strength as large as $15\%$.

Figure 5

Fidelity averaged over randomized input qubit orientation and two-qubit disorder strengths that varies from bond-to-bond along the chain. (Inset) Histogram of fidelities obtained by constructing cluster states using the error-prone controlled phase interaction, $U_{\text{CP}}^{\pi /4,\varepsilon }$, with $\varepsilon$ a random variable chosen from a normal distribution of standard deviation $\sigma =0.1$ that disorders the two-qubit gate along the chain (${\varepsilon }_1,{\varepsilon }_2,\cdots ,{\varepsilon }_N$ in Fig. 2 are separate random variables). The input qubit orientation is also randomly chosen, but from a uniform distribution of angles on the Bloch sphere. The process depicted in Fig. 2 is repeated for 7 qubits. The histogram is truncated because the distribution of errors is still small enough to allow cases near the maximum fidelity ($\mathcal{F}=1$) for certain input states (See discussion in Sec. 3). The star indicates the minimum half width at half maximum. (Main) Minimum half widths at half maximum of fidelities plotted as a function of the number of qubits for various error distribution widths, $\sigma$. The red star plots the same point as in the inset. Fidelities above the dashed line are guaranteed to have used a quantum channel for teleportation. We see that rather large errors are needed to pull the minimum fidelities below the dashed line for these small clusters states.

Figure 6

Circuit diagram depicting the construction of a four-qubit cluster state using the Ising interaction followed by an entanglement measure. The build and initialization protocols here consist of commuting operations and can therefore be performed in a different time sequence than the one shown here, e.g., all at the same time. The initial kets are entangled with the imperfect Ising interaction between two qubits, $U_{\text{ZZ}}^{\pi /4,\varepsilon }$, where $\varepsilon$ is a dimensionless error parameter, Eq. (19). $R_{\mathrm{Z}}^{\alpha }$ denotes single-qubit error-free rotation about the $z$ axis. The series of measurements used to teleport the input state $|{\psi }_{\text{I}}^{\widehat{r}}\rangle$ to the end of the chain with output $|{\psi }_{\text{O}}\rangle$ is the same as Fig. 2. The single-qubit rotation gates can also be implemented by changing the measurement angles.

Figure 7

The same as Fig. 3 but for the Ising interaction. Red circles: $\varepsilon =1/\pi$, Blue circles: $\varepsilon =2/\pi$. Here we also see that error in the two-qubit gate used to form the cluster state diminishes the fidelity. However, for initial qubit orientations $\widehat{r}$ along the $\pm \widehat{y}$ direction, we see that the error does not impact the output result. The corresponding fidelity for these initial states is unity and corresponds to perfect transmission along the cluster state in spite of two-qubit error. The resulting fidelity map is considerably distorted. The minimum fidelity occurs for initial qubits oriented anywhere in the $x-z$ plane, i.e., $\widehat{r}·\widehat{y}=0$. The perfect transmission along $\pm \widehat{y}$ can be used to diagnose the relative strength of Ising errors to all other errors.

Figure 8

The same as Fig. 4 but for the Ising interaction protocol described in Fig. 6. Here we set the two-qubit gate time to be $J^{\text{Z}}t=\pi /4$ which leaves a perfect cluster state constructed from Ising gates in the absence of error. The minimum fidelity for the Ising interaction occurs for input qubits oriented in the $x-z$ plane, i.e., $\widehat{r}·\widehat{y}=0$ in Fig. 7.

Figure 9

The same as Fig. 5 but for the Ising interaction protocol described in Fig. 6 and with the star chosen for $\sigma =0.2$. Here the truncation of the Gaussian is due to the perfect transmission of two initial states through the disordered cluster state. See Sec. 4 for a discussion of perfect transmission.

Figure 10

Circuit diagram depicting the construction of a four-qubit twisted cluster state using the XY interaction followed by an entanglement measure. The twisted cluster state, Eq. (27), is the same as the cluster state but with neighboring qubit labels on every other bond swapped. The initial kets denote four qubits prepared as eigenstates of ${\sigma }_y, {|+\rangle }_y$. These four qubits are entangled with the imperfect XY interaction between two qubits, $U_{\text{XY}}^{\pi /4,\varepsilon }$, where $\varepsilon$ is a dimensionless error parameter, Eq. (31). The $U_{\text{XY}}^{\pi /4,\varepsilon }$ gates do not commute on bonds sharing a qubit and the order of execution must be respected, e.g., the XY interaction on all even bonds are executed simultaneously, followed by the simultaneous execution of the XY interaction on all odd bonds. The series of measurements used to teleport the input state $|{\psi }_{\text{I}}^{\widehat{r}}\rangle$ to the end of the chain with output $|{\psi }_{\text{O}}\rangle$ is the same as Fig. 2. The single-qubit rotation gates can also be implemented by changing the measurement angles.

Figure 11

The same as Fig. 4 but for the XY interaction protocol described in Fig. 10. Here we set the two-qubit gate time to be $J^{\text{XY}}t=\pi /4$ which leaves a perfect cluster state constructed from iSWAP gates in the absence of error.

Figure 12

The same as Fig. 5 but for the XY interaction protocol described in Fig. 10 and with the star chosen for $\sigma =0.15$. Here, as in Fig. 9, the truncation of the Gaussian is due to the perfect transmission of two initial states through the disordered cluster state.

Figure 13

Circuit diagrams depicting refocusing schemes used to improve faulty two-qubit gates under the assumption of perfect single-qubit gates. The top, middle, and bottom rows depict refocusing pulses for the error-prone Ising gate, Eq. (37), the XY gate, Eq. (45), and the controlled phase gate, Eq. (51), respectively. The controlled phase gate refocusing sequence is a composite sequence using $U_{\text{EX}}^{\theta ,\varepsilon }$ with a further reduction depicted in Fig. 14.

Figure 14

Full circuit diagram depicting the refocusing scheme, Eqs. (50) and (51), used to improve the faulty controlled phase gate.

Figure 15

Schematic depicting qubit refreshing of a five-qubit cluster state as we teleport $|{\psi }_{\widehat{r}}\rangle$ using measurements on at most three qubits. The bold boxes contain the information regarding $|{\psi }_{\widehat{r}}\rangle$. The greyed-out qubits are not needed. The first column depicts the initial qubit prepared in the state $|{\psi }_{\widehat{r}}\rangle$ and entangled with two other qubits in a cluster state. The second column depicts measurements that move $|{\psi }_{\widehat{r}}\rangle$ from the first to the third qubit. The third column depicts the entangling of two new qubits into the cluster state after the first two qubits are discarded. The final row depicts the final measurements that moves $|{\psi }_{\widehat{r}}\rangle$ to the last qubit. The entire refreshing process depicted here uses at most three qubits at the same time while effectively teleporting along a five-qubit cluster state.