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International Conference on Quantum, Nano and Micro Technologies (ICQNM)

Encoding Subsystem Codes with and without Noisy Gauge Qubits

Pradeep Kiran Sarvepalli, Andreas Klappenecker

Published Feb 1, 2009• 3 citations
Publisher

Abstract

In this paper, we investigate a rather neglected aspect of subsystem codes viz. that of encoding. We show that encoding of subsystem codes can be reduced to encoding of a related stabilizer code making it possible to use the known results on encoding of stabilizer codes. Along the way we also show how Clifford codes can be encoded. These encoding schemes can tolerate initialization errors on the gauge qubits. This tolerance can be traded for reduced encoding complexity.

DOI
10.1109/icqnm.2009.10

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In this paper, we investigate a rather neglected aspect of subsystem codes viz. that of encoding. We show that encoding of subsystem codes can be reduced to encoding of a related stabilizer code making it possible to use the known results on encoding of stabilizer codes. Along the way we also show how Clifford codes can be encoded. These encoding schemes can tolerate initialization errors on the gauge qubits. This tolerance can be traded for reduced encoding complexity.
We derive new tradeoffs for reliable quantum information storage in a 2D local architecture based on subsystem quantum codes. Our results apply to stabilizer subsystem codes, that is, stabilizer codes in which part of the logical qubits does not encode any information. A stabilizer subsystem code can be specified by its gauge group - a subgroup of the Pauli group that includes the stabilizers and the logical operators on the unused logical qubits. We assume that the physical qubits are arranged on a two-dimensional grid and the gauge group has spatially local generators such that each generator acts only on a few qubits located close to each other. Our main result is an upper bound kd = O(n), where k is the number of encoded qubits, d is the minimal distance, and n is the number of physical qubits. In the special case when both gauge group and the stabilizer group have spatially local generators, we derive a stronger bound kd2= O(n) which is tight up to a constant factor.
Codeword-stabilized quantum codes provide a unified approach to constructing quantum error-correcting codes, including both additive and nonadditive quantum codes. Standard codeword-stabilized quantum codes encode quantum information into subspaces. The more general notion of encoding quantum information into a subsystem is known as an operator (or subsystem) quantum error-correcting code. Most operator codes studied to date are based in the usual stabilizer formalism. We introduce operator quantum codes based on the codeword-stabilized quantum code framework. Based on the necessary and sufficient conditions for operator quantum error correction, we derive an error-correction condition for operator codeword-stabilized quantum codes. Based on this condition, the word operators of a operator codeword-stabilized quantum code are constructed from a set of classical binary errors induced by generators of the gauge group. We use this scheme to construct examples of both additive and nonadditive codes that encode quantum information into a subsystem.
Operator quantum error correction is a recently developed theory that provides a generalized and unified framework for active error correction and passive error avoiding schemes. In this Letter, we describe these codes using the stabilizer formalism. This is achieved by adding a gauge group to stabilizer codes that defines an equivalence class between encoded states. Gauge transformations leave the encoded information unchanged; their effect is absorbed by virtual gauge qubits that do not carry useful information. We illustrate the construction by identifying a gauge symmetry in Shor's 9-qubit code that allows us to remove 3 of its 8 stabilizer generators, leading to a simpler decoding procedure and a wider class of logical operations without affecting its essential properties. This opens the path to possible improvements of the error threshold of fault-tolerant quantum computing.
Subsystem codes are the most versatile class of quantum error-correcting codes known to date that combine the best features of all known passive and active error-control schemes. The subsystem code is a subspace of the quantum state space that is decomposed into a tensor product of two vector spaces: the subsystem and the co-subsystem. A generic method to derive subsystem codes from existing subsystem codes is given that allows one to trade the dimensions of subsystem and co-subsystem while maintaining or improving the minimum distance. As a consequence, it is shown that all pure MDS subsystem codes are derived from MDS stabilizer codes. The existence of numerous families of MDS subsystem codes is established. Propagation rules are derived that allow one to obtain longer and shorter subsystem codes from given subsystem codes. Furthermore, propagation rules are derived that allow one to construct a new subsystem code by combining two given subsystem codes.
Quantum error-correcting codes provide a powerful framework for analyzing and designing fault-tolerant Clifford circuits. In this work, we construct the subsystem spacetime code and use it to characterize the distances of Clifford circuits. We show that its gauge group can be explicitly constructed and classified using the masking properties of the instantaneous stabilizer groups and consists precisely of faults that are harmless to the circuit. We use this framework to identify where a dynamical code loses distance and to analyze how the fault distance changes when circuits are composed from smaller components. Our results provide a versatile tool for analyzing and designing fault-tolerant circuits, with potential applications to the construction of new dynamical codes.