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arXiv:0801.2608

An upper bound on quantum fault tolerant thresholds

Jesse Fern

Published Jan 17, 2008• 2 citations
arXivHTMLPDF

Abstract

In this paper we calculate upper bounds on fault tolerance, without restrictions on the overhead involved. Optimally adaptive recovery operators are used, and the Shannon entropy is used to estimate the thresholds. By allowing for unrealistically high levels of overhead, we find a quantum fault tolerant threshold of 6.88% for the depolarizing noise used by Knill, which compares well to "above 3%" evidenced by Knill. We conjecture that the optimal threshold is 6.90%, based upon the hashing rate. We also perform threshold calculations for types of noise other than that discussed by Knill.

DOI
10.48550/arxiv.0801.2608

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In this paper we calculate upper bounds on fault tolerance, without restrictions on the overhead involved. Optimally adaptive recovery operators are used, and the Shannon entropy is used to estimate the thresholds. By allowing for unrealistically high levels of overhead, we find a quantum fault tolerant threshold of 6.88% for the depolarizing noise used by Knill, which compares well to "above 3%" evidenced by Knill. We conjecture that the optimal threshold is 6.90%, based upon the hashing rate. We also perform threshold calculations for types of noise other than that discussed by Knill.
We consider the possibility of adding noise to a quantum circuit to make it efficiently simulatable classically. In previous works, this approach has been used to derive upper bounds to fault tolerance thresholds—usually by identifying a privileged resource, such as an entangling gate or a non-Clifford operation, and then deriving the noise levels required to make it 'unprivileged'. In this work, we consider extensions of this approach where noise is added to Clifford gates too and then 'commuted' around until it concentrates on attacking the non-Clifford resource. While commuting noise around is not always straightforward, we find that easy instances can be identified in popular fault tolerance proposals, thereby enabling sharper upper bounds to be derived in these cases. For instance we find that if we take Knill's (2005 Nature 434 39) fault tolerance proposal together with the ability to prepare any possible state in the XY plane of the Bloch sphere, then not more than 3.69% error-per-gate noise is sufficient to make it classical, and 13.71% of Knill's γ noise model is sufficient. These bounds have been derived without noise being added to the decoding parts of the circuits. Introducing such noise in a toy example suggests that the present approach can be optimized further to yield tighter bounds.
We consider the possibility of adding noise to a quantum circuit to make it efficiently simulatable classically. In previous works, this approach has been used to derive upper bounds to fault tolerance thresholds—usually by identifying a privileged resource, such as an entangling gate or a non-Clifford operation, and then deriving the noise levels required to make it ‘unprivileged’. In this work, we consider extensions of this approach where noise is added to Clifford gates too and then ‘commuted’ around until it concentrates on attacking the non-Clifford resource. While commuting noise around is not always straightforward, we find that easy instances can be identified in popular fault tolerance proposals, thereby enabling sharper upper bounds to be derived in these cases. For instance we find that if we take Knill's (2005 Nature 434 39) fault tolerance proposal together with the ability to prepare any possible state in the XY plane of the Bloch sphere, then not more than 3.69% error-per-gate noise is sufficient to make it classical, and 13.71% of Knill's γ noise model is sufficient. These bounds have been derived without noise being added to the decoding parts of the circuits. Introducing such noise in a toy example suggests that the present approach can be optimized further to yield tighter bounds.
We study a comprehensive list of quantum codes as candidates for codes used at the physical level in a fault-tolerant code architecture. Using the Aliferis-Gottesman-Preskill (AGP) ex-Rec method we calculate the pseudo-threshold for these codes against depolarizing noise at various levels of overhead. We estimate the logical noise rate as a function of overhead at a physical error rate of p0=1×10−4p_0=1 \times 10^{-4}p0​=1×10−4. The Bacon-Shor codes and the Golay code are the best performers in our study.
We rigorously analyze Knill's Fibonacci scheme for fault-tolerant quantum computation, which is based on the recursive preparation of Bell states protected by a concatenated error-detecting code. We prove lower bounds on the threshold fault rate of $0.67\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ for adversarial local stochastic noise, and $1.25\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ for independent depolarizing noise. In contrast to other schemes with comparable proved accuracy thresholds, the Fibonacci scheme has a significantly reduced overhead cost because it uses postselection far more sparingly.
We study a comprehensive list of quantum codes as candidates of codes to be used at the bottom, physical, level in a fault-tolerant code architecture. Using the Aliferis-Gottesman-Preskill (AGP) ex-Rec method we calculate the pseudo-threshold for these codes against depolarizing noise at various levels of overhead. We estimate the logical noise rate as a function of overhead at a physical error rate of p0=1×10−4p_0=1\times 10^{-4}p0​=1×10−4. The Bacon-Shor codes and the Golay code are the best performers in our study.
We show that quantum circuits cannot be made fault-tolerant against a depolarizing noise level of thetas = (6 - 2radic2)/7 ap 45%, thereby improving on a previous bound of 50% (due to Razborov, 2004). More precisely, the circuit model for which we prove this bound contains perfect gates from the Clifford group (CNOT, Hadamard, S, X, Y, Z) and arbitrary additional one-qubit gates that are subject to depolarizing noise thetas. We prove that this set of gates cannot be universal for arbitrary (even classical) computation, from which the upper bound on the noise threshold for fault-tolerant quantum computation follows