Efficient measurement-based quantum computing with continuous-variable systems

We present strictly efficient schemes for scalable measurement-based quantum computing using continuous-variable systems: These schemes are based on suitable non-Gaussian resource states, ones that can be prepared using interactions of light with matter systems or even purely optically. Merely Gaussian measurements such as optical homodyning as well as photon counting measurements are required, on individual sites. These schemes overcome limitations posed by Gaussian cluster states, which are known not to be universal for quantum computations of unbounded length, unless one is willing to scale the degree of squeezing with the total system size. We establish a framework derived from tensor networks and matrix product states with infinite physical dimension and finite auxiliary dimension general enough to provide a framework for such schemes. Since in the discussed schemes the logical encoding is finite dimensional, tools of error correction are applicable. We also identify some further limitations for any continuous-variable computing scheme from which one can argue that no substantially easier ways of continuous-variable measurement-based computing than the presented one can exist.

Figure 1

Interpretation of single-qudit MBQC as sequential interaction with an ancilla. The physical sites are initially in the state vector $|\Psi \rangle$ and interact with the correlation system through $\widehat{U}$. Depending on the measurement results in $M_1$ and $M_2$, the unitary gate $U$ is applied on $|\phi \rangle$.

Figure 2

Coupling of two quantum wires. The interaction of two physical systems through $\widehat{W}$ and the subsequent measurements ($M_1$ and $M_2$) induces a entangling gate on the two correlation systems.

Figure 3

Nonadaptive coupling of four quantum wires where the wiggly lines denote the probabilistic measurement-based $C_Z$ gate. By scaling the distance between the single couplings polylogarithmically in the length of the computation, the total probability of failure can be made arbitrarily small.