Universal adapters between quantum LDPC codes: Fault-tolerant Logical gates and measurements
ORAL
Abstract
We propose the repetition code adapter as a novel method to perform joint logical Pauli measurements within a quantum low-density parity check (LDPC) codeblock or between separate such codeblocks. This adapter is universal in the sense that it works regardless of the LDPC codes involved and the Paulis being measured. Since logical Pauli measurements offer
a route to universal quantum computation provided magic states are available, our repetition code adapter is a flexible tool to compute fault-tolerantly with arbitrary LDPC codes. Our construction achieves joint logical Pauli measurement of t weight O(d) operators using O(t d log^2 (d)) additional qubits and checks and O(d) time. To obtain these results, we develop a novel weaker form of graph edge expansion. As a special case, for some geometrically-local codes in fixed D>=2 dimensions, only O(t d) additional qubits and checks are required instead.
By extending the adapter in the case t = 2, we also construct a toric code adapter that uses O(d^2) additional qubits and checks to perform targeted logical CNOT gates on arbitrary LDPC codes via Dehn twists. The toric code adapter is a space-efficient alternative to teleportation in order to mediate between an LDPC code and a toric code. Further, this is an explicit example of how our adapters can be used as a tool during code deformation to connect between codes with different properties, such as those with large rate for space-efficient memory and those with additional symmetries enabling fault-tolerant logical computation.
a route to universal quantum computation provided magic states are available, our repetition code adapter is a flexible tool to compute fault-tolerantly with arbitrary LDPC codes. Our construction achieves joint logical Pauli measurement of t weight O(d) operators using O(t d log^2 (d)) additional qubits and checks and O(d) time. To obtain these results, we develop a novel weaker form of graph edge expansion. As a special case, for some geometrically-local codes in fixed D>=2 dimensions, only O(t d) additional qubits and checks are required instead.
By extending the adapter in the case t = 2, we also construct a toric code adapter that uses O(d^2) additional qubits and checks to perform targeted logical CNOT gates on arbitrary LDPC codes via Dehn twists. The toric code adapter is a space-efficient alternative to teleportation in order to mediate between an LDPC code and a toric code. Further, this is an explicit example of how our adapters can be used as a tool during code deformation to connect between codes with different properties, such as those with large rate for space-efficient memory and those with additional symmetries enabling fault-tolerant logical computation.
*This work was done in part while ES was visiting the Simons Institute for the Theory of Computing, Berkeley, supported by DOE QSA grant #FP00010905. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Colleges and Universities.
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Publication: arXiv preprint, arXiv:2410.03628
Presenters
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Esha Swaroop
- University of Waterloo and Perimeter Institute