Topologically fault-tolerant measurement-based quantum computing with non-Clifford gates using local measurements in a Reed-Muller quantum code

A scheme for Fault-tolerant measurement-based quantum computing based on topological error-correction.

”In addition to the topological method, we make use of a Reed-Muller quantum code so that only local measurements are needed to perform non-Clifford operations fault-tolerantly.”

“Why are the above two codes chosen? We require a quantum code with the following three properties: 1) The code is a Quantum code from two binary linear codes, 2) The code satisfies Eq. (5), and 3) The code fits with Graph states. For some arrangement of qubits on a translation-invariant two-dimensional lattice, the code has a translation-invariant set of stabilizer generators and these generators each have a small support on the lattice. The Reed-Muller code has properties 1 and 2 but not 3. The Surface codes have properties 1 and 3 but not 2. Thus, neither of the codes alone suffices. But their combination does.”

“We make use of topological error correction capabilities that the Graph states naturally provide and which can be linked to Surface codes. Any 2D CSS code evolving in time can be mapped to a 3D graph state. The main design tool upon which we base our construction are engineered lattice defects which are topologically entangled.”

“We do not use the Surface codes to encode logical qubits. We use them to appropriately “wire” a subset of the Graph qubits, the S-qubits.”

“The picture is the following: quantum computation is performed on a three-dimensional Graph state via a temporal sequence of one-qubit measurements. The Graph lattice is subdivided into three regions, V , D and S. The set S comprises the ‘singular’ qubits which are measured in an adaptive basis. The quantum computation happens essentially there. The sets V and D are to distribute the correct quantum correlations among the qubits of S. V stands for ‘vacuum’, the quantum correlations can propagate and spread freely in V . D stands for ‘defect’. Quantum correlations cannot penetrate the defect regions. They either end in them or wrap around them. In both cases, the defects guide the quantum correlations. V and D are distinguished by the bases in which the respective Graph qubits are measured. The region V fills most of the Graph. Embedded in V are the defects (D) most of which take the shape of loops. These loops are topologically entangled with another. Further, there are defects in the shape of ear clips which each hold an S-qubit in their opening. Such a defect and the belonging S-qubit form again a loop. These are the only locations where the S-qubits occur.”

“The 3D Graph state is structured by two kinds of lattice defects: primal and dual. The defects wind around another. Some of them hold singular qubits (red) which realize the non-Clifford part of the quantum computation.”

“Now the measurement pattern (7) becomes understandable: the edge qubits in the defects are measured in the Z-basis which effectively removes them from the Graph. In this way, the quantum state on the exterior of the defect becomes disentangled from the state with support on interior of the defect. Thereby, a defect is created in the Graph lattice.”

“The purpose of the measurements in V and D is to create on S a Reed-Muller-encoded algorithm-specific resource |Ψalgo〉S. We specify the location of S-qubits with respect to the lattice defects and we give a construction for a topologically protected circuit providing |Ψalgo〉S .”

“The construction of a topologically protected circuit providing |Ψ\baralgo〉S proceeds in three steps. First we show that |Ψ\baralgo〉S is local unitarily equivalent to a bi-colorable Graph state, by local Hadamard-transformations. Second, we show how to create a bi-colorable Graph state. Third, we take care of the Hadamard-transformations.”

Why do we want |Ψalgo〉S?

“The measurements of X and Z on the qubits C2\Q implement the Clifford part. They are performed simultaneously in the first round of measurements.|Ψalgo〉Q is the state of the unmeasured qubits after the first measurement round. It is an algorithm-specific Stabilizer state. Since it is a Stabilizer state, it is easy to create and one may start with this state as an algorithm-specific resource instead of the universal Graph state. Quantum computation with this state proceeds by measuring local observables (X±Y)/√2 .”

“Topologically protected quantum gates are performed by measuring some regions of qubits in the Z-basis, which effectively removes the qubits from the state. The remaining cluster, whose qubits are measured in the X and X±Y -basis, thereby attains a non-trivial topology in which fault-tolerant quantum gates can be encoded.”