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Advances in intelligent systems and computing

Clifford+T-based Fault-Tolerant Quantum Implementation of Code Converter Circuit

Laxmidhar Biswal, Chandan Bandyopadhyay, Hafizur Rahaman

Published Jan 1, 2020• 2 citations
Publisher
DOI
10.1007/978-981-15-4032-5_58

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Universal fault-tolerant quantum computers will require the use of efficient protocols to implement encoded operations necessary in the execution of algorithms. In this work, we show how SMT solvers can be used to automate the construction of Clifford circuits with certain fault-tolerance properties and we apply our techniques to a fault-tolerant magic-state-preparation protocol. Part of the protocol requires converting magic states encoded in the color code to magic states encoded in the surface code. Since the teleportation step involves decoding a color code merged with a surface code, we develop a decoding algorithm that is applicable to such codes.
Flagging the Clifford hierarchy:~Fault-tolerant logical π2l\fracπ{2^l}2lπ​ rotations via measuring circuit gauge operators of non-Cliffords
We provide a recursively defined sequence of flag circuits which will detect logical errors induced by non-fault-tolerant RZ‾(π2l)R_{\overline{Z}}(\fracπ{2^l})RZ​(2lπ​) gates on CSS codes with a fault distance of two. As applications, we give a family of circuits with O(l)O(l)O(l) gates and ancillae which implement fault-tolerant logical RZ(π2l)R_{Z}(\fracπ{2^l})RZ​(2lπ​) or RZZ(π2l)R_{ZZ}(\fracπ{2^l})RZZ​(2lπ​) gates on any [[k+2,k,2]][[k + 2, k, 2]][[k+2,k,2]] iceberg code and fault-tolerant circuits of size O(l)O(l)O(l) for preparing ∣π2l⟩|\fracπ{2^l}\rangle∣2lπ​⟩ resource states in the [[7,1,3]][[7,1,3]][[7,1,3]] code, which can be used to perform fault-tolerant RZ‾(π2l)R_{\overline{Z}}(\fracπ{2^l})RZ​(2lπ​) rotations via gate teleportation, allowing for implementations of these gates that bypass the high overheads of gate synthesis when lll is small relative to the precision required. We show how the circuits above can be generalized to π(x0.x1x2…xl)=∑jlπxj2jπ( x_0.x_{1}x_{2}\ldots x_{l}) = \sum_{j}^{l} π\frac{x_j}{2^j}π(x0​.x1​x2​…xl​)=∑jl​π2jxj​​ rotations with identical overheads in lll, which could be useful in quantum simulations where time is digitized in binary. Finally, we illustrate two approaches to increase the fault-distance of our construction. We show how to increase the fault distance of a Cliffordized version of the T gate circuit to 333 in the Steane code and how to increase the fault-distance of the π2\fracπ{2}2π​ iceberg circuit to 444 through concatenation in two-level iceberg codes. This yields a targeted logical RZ‾(π2)R_{\overline{Z}}(\fracπ{2})RZ​(2π​) gate with fault distance 444 on any row of logical qubits in an [[(k2+2)(k1+2),k1k2,4]][[(k_2+2)(k_1+2), k_1k_2, 4]][[(k2​+2)(k1​+2),k1​k2​,4]] code.