Measurement-based quantum computation beyond the one-way model

We introduce schemes for quantum computing based on local measurements on entangled resource states. This work elaborates on the framework established in Gross and Eisert [Phys. Rev. Lett. 98, 220503 (2007); quant-ph/0609149]. Our method makes use of tools from many-body physics—matrix product states, finitely correlated states, or projected entangled pairs states—to show how measurements on entangled states can be viewed as processing quantum information. This work hence constitutes an instance where a quantum information problem—how to realize quantum computation—was approached using tools from many-body theory and not vice versa. We give a more detailed description of the setting and present a large number of examples. We find computational schemes, which differ from the original one-way computer, for example, in the way the randomness of measurement outcomes is handled. Also, schemes are presented where the logical qubits are no longer strictly localized on the resource state. Notably, we find a great flexibility in the properties of the universal resource states: They may, for example, exhibit nonvanishing long-range correlation functions or be locally arbitrarily close to a pure state. We discuss variants of Kitaev’s toric code states as universal resources, and contrast this with situations where they can be efficiently classically simulated. This framework opens up a way of thinking of tailoring resource states to specific physical systems, such as cold atoms in optical lattices or linear optical systems.

Figure 1

(Color online) Measurement-based quantum computing as a generalization of the one-way model as being considered in this work. Initially, an entangled resource state is available, different from the cluster state, followed by local projective measurements on all individual constituents in the regular not necessarily cubic lattice. In all figures, dark gray circles denote individual physical systems.

Figure 2

(Color online) A universal resource deriving from the AKLT model.

Figure 3

Implementation of single-qubit and two-qubit operations in the first toric code model. (a) The measurement pattern for single-qubit operations and (b) the corresponding circuit. (c) Pattern for a two-qubit gate between logical qubits, (d) the corresponding circuit, and (e) the circuit after some simplifications.

Figure 4

Interpretation of the first toric code scheme in terms of parity encoded qubits. The boxed parts of the circuit decode and encode the system. (a) $Z$ rotations result in $Z$ rotations in the encoded system. (b) $X$ rotations result in $X$ rotations in the encoded system, plus $Z$ corrections before and after the rotations in case the $s$ qubit below is $\mid -{⟩}_s$ rather than $\mid +{⟩}_s$. (c) Similarly, the coupling circuit Fig. 3d results in a coupling of the encoded logical qubits, up to the same $Z$ correction on the first logical qubit which depends on the $s$ qubit below in exactly the same way. Thus the $Z$ corrections on each qubit cancel out except for the first and the last, which have no effect due to the initialization and measurement in the computational basis.

Figure 5

(Color online) Weighted graph state as a universal resource. Solid lines correspond to edges that have been entangled using phase gates with phase $\varphi =\pi$, dotted lines correspond to edges entangled with phase gates with $\varphi =\pi ∕2$. This shows that one can replace some edges with weakly entangled bonds.

Figure 6

(Color online) Weighted graph state where the gate is achieved by appropriately bringing two wires together in a “rerouting process.”

Figure 7

(Color online) Cubic lattice of a graph state corresponding to the situation where some edges are missing in a cluster state. If the probability $p$ of having an edge is sufficiently high the processes are independent, then a renormalized perfect sublattice can be found almost certainly, giving rise to a cluster state of smaller size. If $p>p_2=1∕2$, where $p_2$ is the percolation threshold for edge percolation in 2D cubic lattices, then a renormalized lattice can be found almost certainly. Interestingly, even if $1∕2>p>p_3$, $p_3=0.249$ denoting the percolation threshold in 3D, one can almost certainly construct a perfect sublattice using an overhead that is arbitrarily close to being quadratic.