Simulation of fault-tolerant quantum circuits on quantum computational tensor network

TL;DR

This paper explores the simulation of fault-tolerant quantum circuits within the quantum computational tensor network framework, highlighting challenges due to certain physical errors not translating straightforwardly into the correlation space.

Contribution

It analyzes the limitations of simulating fault-tolerant quantum circuits in the tensor network framework, especially regarding error representation in the correlation space.

Findings

Not all physical errors appear as linear CPTP errors in correlation space

Simulation of fault-tolerant circuits is complex due to error translation issues

Challenges in applying existing fault-tolerance theories to tensor network models

Abstract

In the framework of quantum computational tensor network [D. Gross and J. Eisert, Phys. Rev. Lett. {\bf98}, 220503 (2007)], which is a general framework of measurement-based quantum computation, the resource many-body state is represented in a tensor-network form, and universal quantum computation is performed in a virtual linear space, which is called a correlation space, where tensors live. Since any unitary operation, state preparation, and the projection measurement in the computational basis can be simulated in a correlation space, it is natural to expect that fault-tolerant quantum circuits can also be simulated in a correlation space. However, we point out that not all physical errors on physical qudits appear as linear completely-positive trace-preserving errors in a correlation space. Since the theories of fault-tolerant quantum circuits known so far assume such noises, this means that the simulation of fault-tolerant quantum circuits in a correlation space is not so straightforward for general resource states.