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Register a new userEvery Action in Every Code is Fault Tolerant: Fault Tolerance Quantum Computation in Quotient Algebra Partition
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In the framework quotient algebra partition, a general methodology is introduced to construct fault tolerant encodes for an arbitrary action in an error-correcting code.
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We explain how to combine holonomic quantum computation (HQC) with fault tolerant quantum error correction. This establishes the scalability of HQC, putting it on equal footing with other models of computation, while retaining the inherent robustness the method derives from its geometric nature.
We review an approach to fault-tolerant holonomic quantum computation on stabilizer codes. We explain its workings as based on adiabatic dragging of the subsystem containing the logical information around suitable loops along which the information remains protected.
We describe a fault-tolerant version of the one-way quantum computer using a cluster state in three spatial dimensions. Topologically protected quantum gates are realized by choosing appropriate boundary conditions on the cluster. We provide equivalence transformations for these boundary conditions that can be used to simplify fault-tolerant circuits and to derive circuit identities in a topological manner. The spatial dimensionality of the scheme can be reduced to two by converting one spatial axis of the cluster into time. The error threshold is 0.75% for each source in an error model with preparation, gate, storage and measurement errors. The operational overhead is poly-logarithmic in the circuit size.
Certain physical systems that one might consider for fault-tolerant quantum computing where qubits do not readily interact, for instance photons, are better suited for measurement-based quantum-computational protocols. Here we propose a measurement-based model for universal quantum computation that simulates the braiding and fusion of Majorana modes. To derive our model we develop a general framework that maps any scheme of fault-tolerant quantum computation with stabilizer codes into the measurement-based picture. As such, our framework gives an explicit way of producing fault-tolerant models of universal quantum computation with linear optics using protocols developed using the stabilizer formalism. Given the remarkable fault-tolerant properties that Majorana modes promise, the main example we present offers a robust and resource efficient proposal for photonic quantum computation.
Reliable qubits are difficult to engineer, but standard fault-tolerance schemes use seven or more physical qubits to encode each logical qubit, with still more qubits required for error correction. The large overhead makes it hard to experiment with fault-tolerance schemes with multiple encoded qubits. The 15-qubit Hamming code protects seven encoded qubits to distance three. We give fault-tolerant procedures for applying arbitrary Clifford operations on these encoded qubits, using only two extra qubits, 17 total. In particular, individual encoded qubits within the code block can be targeted. Fault-tolerant universal computation is possible with four extra qubits, 19 total. The procedures could enable testing more sophisticated protected circuits in small-scale quantum devices. Our main technique is to use gadgets to protect gates against correlated faults. We also take advantage of special code symmetries, and use pieceable fault tolerance.
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