Abstract
Magic state distillation is one of the leading candidates for implementing universal fault-tolerant logical gates. However, the distillation circuits themselves are not fault-tolerant, so there is additional cost to first implement encoded Clifford gates with negligible error. In this paper we present a scheme to fault-tolerantly and directly prepare magic states using flag qubits. One of these schemes requires only three ancilla qubits, even with noisy Clifford gates. We compare the physical qubit and gate cost of our scheme to the magic state distillation protocol of Meier, Eastin, and Knill (MEK), which is efficient and uses a small stabilizer circuit. For low enough noise rates, we show that in some regimes the overhead can be improved by several orders of magnitude compared to the MEK scheme which uses Clifford operations encoded in the codes considered in this work.
Highlights
Certain algorithms can be implemented efficiently on quantum computers, whereas the best known classical algorithms require superpolynomial resources [1, 2]
In low noise rate regimes (10−5 ≤ p ≤ 10−4), we show that the qubit and gate overhead cost of our scheme is lower compared to distance-three surface code implementations of magic state distillation
Since the encoded version of the gates and states are implemented in a fault-tolerant way, the failure probability for each logical fault E of a gate G at a physical error rate p resulting from the malignant event mal(E1) can be upper bounded as
Summary
Certain algorithms can be implemented efficiently on quantum computers, whereas the best known classical algorithms require superpolynomial resources [1, 2]. We consider a full circuit level noise model (which includes noisy Clifford operations) where gates can be applied between any pair of qubits In this model, we compare the overhead cost of our scheme to a magic state distillation scheme introduced by Meier, Eastin and Knill (MEK), due to its efficiency and small circuit size [26]. In low noise rate regimes (10−5 ≤ p ≤ 10−4), we show that the qubit and gate overhead cost of our scheme is lower compared to distance-three surface code implementations of magic state distillation. A more thorough analysis of surface code implementations of magic state distillation for error rates mentioned above is required to determine if our scheme has a smaller overhead. Details of the overhead and numerical analysis are provided in Appendices C to E
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