Efficient Quantum Circuits for One-Way Quantum Computing

While Ising-type interactions are ideal for implementing controlled phase flip gates in one-way quantum computing, natural interactions between solid-state qubits are most often described by either the $XY$ or the Heisenberg models. We show an efficient way of generating cluster states directly using either the imaginary SWAP ($i\mathrm{SWAP}$) gate for the $XY$ model, or the $\sqrt{\mathrm{SWAP}}$ gate for the Heisenberg model. Our approach thus makes one-way quantum computing more feasible for solid-state devices.

Figure 1

Illustration of how to generate cluster states with $i\mathrm{SWAP}$ gates. Each circle represents a qubit. Each solid line represents a bond by cluster states. (a) A three-qubit cluster state from a two-qubit cluster state. (b) Creation of a chain cluster state. The first step is to create separated two-qubit cluster states. The second step is to apply an $i\mathrm{SWAP}$ gate between the 3 two-qubit cluster states. (c) The two-chain cluster states produced in (b) are vertically connected by $i\mathrm{SWAP}$s, thus producing a ladder cluster state. Afterwards, several of these can be connected to produce 2-D cluster states.

Figure 2

Replacing CPF gates by $i\mathrm{SWAP}$ gates provides higher fidelity. The fidelity of a chain cluster state is plotted versus both the number of qubits and pulse phase errors $\delta$ (i.e., $Jt\to Jt+\delta$, and $\theta \to \theta +\delta$) assuming Gaussian distribution with a variance $\sigma$ of phase error $\delta$. (a) Fidelity $F^{\mathrm{new}}$ of this proposal. (b) Fidelity increase $\Delta F\equiv (F^{\mathrm{new}}-F^{\mathrm{old}})/[(F^{\mathrm{new}}+F^{\mathrm{old}})/2]$, where $F^{\mathrm{old}}$ is the fidelity of the previous method [19]. Note that $F^{\mathrm{old}}$ is almost zero when $\sigma \gtrsim 10°$ where the fidelity increase is $\sim 200\%$.

Figure 3

Connecting two half-infinite cluster states via an $i\mathrm{SWAP}$ gate and measurement. After applying an $i\mathrm{SWAP}$ gate between qubits “3” and “4,” qubit “3” is measured on the ${\sigma }^x$ basis, following appropriate rotations in qubits “4” and “5.” Qubit “3” is discarded after the measurement.