[Paper Review] Fault-Tolerant Postselected Quantum Computation: Schemes

Emanuel Knill|ArXiv.org|Feb 23, 2004

Quantum Computing Algorithms and ArchitectureComputer Science1 references80 citations

TL;DR

This paper proposes fault-tolerant schemes for postselected quantum computation using a four-qubit error-detecting code combined with teleportation and error-detection to achieve arbitrarily low logical error rates conditionally on success. By postselecting only on error-detection outcomes, the method enables universal quantum computation with high accuracy despite noisy gates, though at the cost of exponentially low success probability, which can be mitigated via concatenation and efficient postselection strategies.

ABSTRACT

Postselected quantum computation is distinguished from regular quantum computation by accepting the output only if measurement outcomes satisfy predetermined conditions. The output must be accepted with nonzero probability. Methods for implementing postselected quantum computation with noisy gates are proposed. These methods are based on error-detecting codes. Conditionally on detecting no errors, it is expected that the encoded computation can be made to be arbitrarily accurate. Although the probability of success of the encoded computation decreases dramatically with accuracy, it is possible to apply the proposed methods to the problem of preparing arbitrary stabilizer states in large error-correcting codes with local residual errors. Together with teleported error-correction, this may improve the error tolerance of non-postselected quantum computation.

Motivation & Objective

• To develop fault-tolerant protocols for postselected quantum computation that remain effective under noisy gate operations.
• To address the challenge of achieving high-fidelity quantum computation when error correction is not feasible, by relying on error detection and postselection.
• To enable the preparation of accurate stabilizer states in large error-correcting codes with local residual errors for use in non-postselected fault-tolerant schemes.
• To explore how concatenated error-detecting codes and transversal operations can be combined with purification and teleported error detection to achieve universal quantum computation.

Proposed method

• Uses a four-qubit error-detecting code to encode quantum states, enabling detection of any single-qubit error during computation.
• Applies transversal Clifford gates and Bell state measurements to implement logical operations fault-tolerantly within the code space.
• Employs teleportation and error-detection techniques to decode the encoded state bottom-up, ensuring that undetected errors are minimized.
• Conditions the acceptance of the output only on the absence of detected errors, thereby reducing the conditional logical error rate to arbitrarily low levels.
• Uses purification and encoded preparation of the |π/8⟩ state to achieve universality at the top level of concatenation.
• Combines postselected subnetworks in a tree-like structure to reduce overhead and improve efficiency despite low success probability.

Experimental results

Research questions

• RQ1 Can postselected quantum computation be made fault-tolerant under noisy gate operations by relying solely on error detection rather than correction?
• RQ2 What is the maximum error rate per gate that can be tolerated in postselected computation when errors are detected via a simple error-detecting code?
• RQ3 How can the success probability of postselected computation be improved without compromising fault-tolerance, especially when success is conditioned on rare events?
• RQ4 To what extent can the use of concatenated error-detecting codes and teleported error detection reduce residual errors in the final output state?
• RQ5 Can the resulting encoded states with local residual errors be used effectively in non-postselected fault-tolerant quantum computation schemes?

Key findings

• The conditional logical error rate after postselection is expected to scale quadratically with the base error rate, enabling arbitrarily low error rates for sufficiently low initial error rates.
• The method achieves fault-tolerant postselected quantum computation using only error detection, avoiding the need for full error correction.
• The use of transversal operations and teleportation allows for constant-depth circuits, making the scheme suitable for theoretical analysis and potential physical implementation.
• The four-qubit code enables a transversal implementation of the Hadamard gate, preserving fault-tolerance and simplifying the gate set.
• By decoding the concatenated code bottom-up and postselecting on no error detection, the final state is perturbed only by local errors with a rate determined by the error model and decoding complexity.
• The scheme can be used to prepare accurate stabilizer states with bounded local errors, which can then be used as resources in non-postselected fault-tolerant quantum computation.