Background
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Motivation
Steane’s 7-qubit quantum error correcting code (QEC) admits a set of fault-tolerant generators for the Clifford group, but it does not admit fault-tolerant T gate. On the other hand, the 15-qubit Reed-Muller QEC admits fault-tolerant T and CZ gates, but not a universal fault-tolerant gate set.
By Eastin-Knill Theorem, no quantum error correcting codes can have a continuous symmetry which acts transversely on physical qubits. In other words, no QEC can transversally implement a universal gate set. For example, the Clifford+T circuit generated by {H, S, CX, T} cannot be implemented transversally.
Hence we would like to find a scheme to realize a universal set of fault-tolerant logical gates.
Main deliverable
Anderson et al proposed a scheme that combines the above features of the 7-qubit Steane’s QEC and the 15-qubit Reed-Muller QEC, with a fault-tolerant in-place conversion between these two codes. Thus, the new scheme possess a universal set of transversal gates. Moreover, this set is overcomplete (i.e., with the addition of CZ), so it may help reduce the compilation overhead.
Intuition
Different quantum Reed-Muller codes all correspond to the same subsystem code with different gauge fixing (i.e., fixing some gauge qubits to the logical |0> or |+> states). In fact, the proposed conversion scheme generalizes the method by Paetznick and Reichardt. It could be rephrased using the subsystem code formalism by Kribs et al and Poulin.
Methods
The family of Quantum Reed-Muller codes QRM(m) is derived from the classical Reed-Muller codes RM(m), whose definition is recursive. The conversion scheme relies on this recursiveness to derive stabilizer generators for the extended QRM.
At the bottom of the proposed scheme sits Steane’s 7-qubit code.
Climbing up the conversion hierarchy gives an extension to an infinite family of quantum Reed-Muller codes.
Application
Advantages
The proposed scheme fault-tolerantly and directly convert between QRM(m) and QRM(m+1). By combining the transversal gate sets of these codes, we obtain an over-complete universal gate set that is transversal.
This scheme may reduce the overhead of qubits required to encode a logical state.
This scheme could potentially improve the magic-state distillation procedure.
This scheme could be generalized to high-order QRM (i.e., when r is more than 1).
Technical definitions
For example, in a CNOT circuit, an X error is propagated from the control to the target.
For example, in a CNOT circuit, a Z error is propagated from the target to the control.
Transversality: In an architecture where each logical qubit is encoded in a code block which can protect against up to t errors, a gate is transversal if it does not couple qubits inside a given code block. As a result, the number of errors or faults in a block cannot increase under the application of a gate.
The number of errors after the application of a gate is at most the number of initial errors on the data plus the number of faults in the exercution of the gate. Single-qubit errors can be propogated to other blocks, introducing a single-qubit error to that block. As shown on the right, the two qubits in each example are from distinct code block.
Magic State Distillation: A procedure that uses Clifford operations to increase the fidelity of non-stabilizer states, which can be injected in the computation to realize non-Clifford transformations (e.g., T gate).