Topological quantum computing with a very noisy network and local error rates approaching one percent

Abstract

A scalable quantum computer could be built by networking together many simple processor cells, thus avoiding the need to create a single complex structure. The difficulty is that realistic quantum links are very error prone. A solution is for cells to repeatedly communicate with each other and so purify any imperfections; however prior studies suggest that the cells themselves must then have prohibitively low internal error rates. Here we describe a method by which even error-prone cells can perform purification: groups of cells generate shared resource states, which then enable stabilization of topologically encoded data. Given a realistically noisy network (≥10% error rate) we find that our protocol can succeed provided that intra-cell error rates for initialisation, state manipulation and measurement are below 0.82%. This level of fidelity is already achievable in several laboratory systems.

Figure 1. Quantum architectures.

Left: a monolithic grid of qubits with neighbours directly connected to enable high fidelity two-qubit operations. (layout from ref. .) Such a structure is a plausible goal for some systems, for example, specific superconducting devices. Right: For other nascent quantum technologies the network paradigm is appropriate. A single nitrogen-vacancy (NV) centre in diamond, with its electron spin and associated nuclear spin(s), would constitute a cell. A small ion trap holding a modest number of ions is another example. Noisy network links with error rates ≥ 10% are acceptable. For photonic links this goal is realistic given imperfections like photon loss and instabilities in path lengths or interaction strengths. Similarly, with solid state ‘wires' formed by spin chains, noisy entanglement distribution of this kind is a reasonable goal.

Figure 2. Protocols for stabilizer measurement.

The three phases for generating and consuming a 4-cell GHZ state among the ancilla qubits (blue-to-green) in order to perform a stabilizer measurement on the data qubits (purple). Phase (1): use purification to create a high quality Bell pair shared between cell A and cell B, while in parallel doing the same thing with cells C and D. (2) Fusion operations, A to C and B to D, create a high fidelity GHZ state. (3) Finally use the GHZ state to perform a one-step stabilizer operation. The parity of the four measured classical bits is also the parity of the stabilizer operation we have performed on the data qubits. Two dashed regions indicate operations that are part of the STRINGENT protocol; omitting them yields the EXPEDIENT alternative.

Figure 3. Scheduling stabilizer operations.

The right side graphic shows the standard arrangement of one complete stabilizer cycle, involving Z and X projectors (square symbols indicate that the four surrounding data qubits are to be stabilized). Because a given cell can only be involved in generating one GHZ resource at a given time, each of these two stabilizer types must be broken into two subsets; see main figure. Fortunately in our stabilizer superoperator we can commute projectors and errors so as to expel errors from the intervening time between subsets, so allowing them to merge.

Figure 4. Performance of the EXPEDIENT and STRINGENT protocols.

We employ a toric code with n rows × n columns of data qubits (2n² in total). A given numerical experiment is a simulation of 100 complete stabilizer measurement cycles on an initially perfect array, after which we attempt to decode a Z measurement of the stored qubit. The result is either a success or a failure; for each data point we perform at least 10,000 experiments to determine the fail probability, and reciprocate this to infer an expected time to failure. Network error rates are 10% in all cases; we set intra-cell gate and measurement error rates equal, p_m=p_g, and plot this on the horizontal axes. For low error rates the system's performance improves with increasing array size. As the error rates pass the threshold this property fails. Insets: typical final states of the toroidal array after error correction. Yellow squares are flipped qubits, green squares indicate the pattern of Z-stabilizers. Closed loops are successful error corrections; while both arrays are therefore successfully corrected, it is visually apparent that the above-threshold case is liable to long paths.