Measurement-Based Quantum Computation with Trapped Ions

Measurement-based quantum computation represents a powerful and flexible framework for quantum information processing, based on the notion of entangled quantum states as computational resources. The most prominent application is the one-way quantum computer, with the cluster state as its universal resource. Here we demonstrate the principles of measurement-based quantum computation using deterministically generated cluster states, in a system of trapped calcium ions. First we implement a universal set of operations for quantum computing. Second we demonstrate a family of measurement-based quantum error correction codes and show their improved performance as the code length is increased. The methods presented can be directly scaled up to generate graph states of several tens of qubits.

Figure 1

Demonstration of one-qubit gates via measurement-based quantum computing. (a) Cluster state $|L{C}_4⟩$. Qubits 1, 2, and 3 are measured consecutively in the basis $B_1(\alpha )$, $B_2(\pm \beta )$, and $B_3(\pm \gamma )$. (b) Equivalent quantum circuit. Rotation angles $\alpha$, $\beta$, and $\gamma$. Hadamard ($H$) [1]. (c) Ideal output states on the Bloch sphere for $[\alpha ,\beta ,\gamma ]=[\pi /2,0,0]$ (red, $|{-}_y⟩$), [0, 0, $-\pi /2$] (green, $|{+}_y⟩$), [$\pi /2$, $-\pi /2$, 0] (blue, $|{-}_x⟩$), [$\pi /2$, 0, $-\pi /2$] (cyan, $|0⟩$), [$\pi /4$, 0, 0] [magenta]. (d) Experimentally measured states; the average fidelity with the ideal cases is $0.92\pm 0.01$. Gray extrapolations show nearest pure state.

Figure 2

Demonstration of two-qubit gates via measurement-based quantum computing. (a) Qubits 1 and 4 of cluster state $|L{C}_4⟩$ are measured in the basis $B_1(\alpha )$ and $B_4(\beta )$, respectively. Qubits 2 and 3 encode the output state. (b). Equivalent quantum circuit, with angles determined by $\alpha$ and $\beta$. Final element is a CP gate. (c)–(d). Experimental output state density matrices (left and right; real and imaginary parts, respectively) in two cases: (c) an entangled state for $\alpha =\pi /2$, $\beta =-\pi /2$, with fidelity $0.88\pm 0.02$, and tangle $0.59\pm 0.05$, and (d) an ideally separable state for $\alpha =0$, $\beta =0$, with fidelity $0.83\pm 0.01$ and tangle $0.02\pm 0.01$.

Figure 3

Graph state $|E{C}_n⟩$ and its use in measurement-based quantum error correction. (a) ($n+2$)-qubit graph state $|E{C}_n⟩$ and protocol, which can correct for phase-flip errors ($Z$) occurring on up to $(n-1)/2$ of qubits $C_1$ to $C_n$. Ideally an encoded one-qubit state is perfectly teleported across the cluster, from $A$ to $B$. (b) Conceptually equivalent quantum logic circuit.

Figure 4

Quantum error correction performance against errors on subsets of codeword qubits. (a) to (c). Graph states $|E{C}_n⟩$ for $n=1$, 3 and 5, respectively. Errors are applied to qubits in blue. (d) Solid blue line: ideal case in (a). Experimental results for cases (a)–(c) are shown as blue diamonds, red squares and grey inverted triangles (two errors), respectively. Black circles show case (c) for only a single error applied to $C_1$. For more details see supplementary material. Errors are one standard deviation and derived from quantum projection noise.

Figure 5

Quantum error correction performance against errors on all code word qubits. (a)–(c). Graph states $|E{C}_n⟩$ for $n=1$, 3, and 5, respectively. Errors are applied to qubits in blue. (d) Ideal performance for cases (a)–(c) shown as solid blue, red, and black lines, respectively. Increasing the code word length ($n$) improves performance for $p<0.5$. (e) Experimental results for cases (a)–(c) shown as blue diamonds, red squares, and black triangles, respectively. Errors are one standard deviation and derived from quantum projection noise.