[Paper Review] Quantum universality by distilling certain one- and two-qubit states with stabilizer operations

Ben W. Reichardt|arXiv (Cornell University)|Aug 9, 2006

Quantum Computing Algorithms and ArchitectureComputer Science26 references13 citations

TL;DR

This paper demonstrates that quantum universality can be achieved using only stabilizer operations and specific one- and two-qubit ancilla states, even when those states are not pure or stabilizer-like. By introducing a parity-checking operation, the authors extend the set of single-qubit mixed states that enable universality and identify a two-qubit mixed state—non-stabilizer in nature—whose postselected reductions yield only stabilizer states, thereby enabling fault-tolerant quantum computation with tight threshold bounds.

ABSTRACT

Quantum universality can be achieved using stabilizer operations and repeated preparation of certain ancilla states. Which ancilla states suffice for universality? We extend the range of single-qubit mixed states which are known to give universality, by using a simple parity-checking operation. Additionally, we display a two-qubit mixed state which is not a mixture of stabilizer states, but for which every postselected stabilizer reduction from two qubits to one outputs a mixture of stabilizer states. The main application of these techniques is to quantum fault tolerance. Our results imply that recent fault-tolerance threshold upper bounds based on the Gottesman-Knill theorem are tight.

Motivation & Objective

• To identify which mixed one- and two-qubit states enable quantum universality when combined with stabilizer operations.
• To extend the known set of single-qubit mixed states that suffice for universality using a novel parity-checking operation.
• To demonstrate a two-qubit mixed state that is not a mixture of stabilizer states but yields only stabilizer states upon postselected reduction.
• To establish tighter bounds on fault-tolerance thresholds in quantum computing using the Gottesman-Knill theorem.

Proposed method

• Employing a parity-checking operation to extract logical information from multiple qubits, enabling effective distillation of non-stabilizer states.
• Analyzing postselected reductions of two-qubit states to one qubit under stabilizer operations, showing that certain mixed states yield only stabilizer outputs.
• Using stabilizer formalism to characterize the output states after postselection, proving that all outcomes are mixtures of stabilizer states.
• Applying the Gottesman-Knill theorem to show that the resulting computation remains simulatable classically, which constrains fault-tolerance thresholds.
• Constructing a framework where universality is achieved through repeated preparation of specific ancilla states and stabilizer operations alone.

Experimental results

Research questions

• RQ1 Which single-qubit mixed states can enable quantum universality when combined with stabilizer operations?
• RQ2 Can a two-qubit mixed state that is not a mixture of stabilizer states still yield only stabilizer states upon postselected reduction?
• RQ3 What is the impact of such state distillation protocols on the fault-tolerance threshold in quantum computing?
• RQ4 How does the use of parity-checking operations extend the range of distillable states for universality?

Key findings

• The paper identifies a broader class of single-qubit mixed states that achieve quantum universality when used with stabilizer operations, extending prior results.
• A novel parity-checking operation enables effective distillation of non-stabilizer states, increasing the set of usable ancilla states.
• A specific two-qubit mixed state is found that is not a mixture of stabilizer states, yet every postselected one-qubit reduction produces only stabilizer states.
• The results imply that recent upper bounds on the fault-tolerance threshold based on the Gottesman-Knill theorem are tight, as they cannot be improved without new non-stabilizer resources.