Bound on quantum computation time: Quantum error correction in a critical environment

TL;DR

This paper derives an upper bound on quantum computation time considering quantum error correction and environmental decoherence, showing how different bath types affect logical qubit fidelity over time.

Contribution

It provides an explicit evolution operator for logical qubits and analyzes the impact of various environmental spectral densities on quantum computation duration.

Findings

Trace distance saturates for super-Ohmic baths

Finite time limit for Ohmic/sub-Ohmic baths

Quantum error correction reduces effective environmental coupling

Abstract

We obtain an upper bound on the time available for quantum computation for a given quantum computer and decohering environment with quantum error correction implemented. First, we derive an explicit quantum evolution operator for the logical qubits and show that it has the same form as that for the physical qubits but with a reduced coupling strength to the environment. Using this evolution operator, we find the trace distance between the real and ideal states of the logical qubits in two cases. For a super-Ohmic bath, the trace distance saturates, while for Ohmic or sub-Ohmic baths, there is a finite time before the trace distance exceeds a value set by the user.