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1. (WO2019050609) CODES AND PROTOCOLS FOR DISTILLING T, CONTROLLED-S, AND TOFFOLI GATES FOR USE IN QUANTUM CIRCUITS
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CLAIMS

1. A method for distilling magic states in a quantum computing device, comprising: generating a magic state distillation protocol for creating magic states in the quantum computing device, wherein the magic state distillation protocol includes (a) Reed-Muller codes, or (b) punctured Reed-Muller codes; and

configuring the quantum computing device to implement the magic state distillation protocol.

2. The method of claim 1, wherein the magic state distillation protocol is for Toffoli gates or controlled-controlled-Z (CCZ) gates.

3. The method of claim 1, wherein the magic state distillation protocol includes punctured Reed-Muller codes, and wherein the punctured Reed-Muller codes are selected based on Hamming distances.

4. The method of claim 1 wherein the magic state distillation protocol includes punctured Reed-Muller codes, and wherein the punctured Reed-Muller codes are selected by random puncturing and unpuncturing.

5. The method of claim 1, wherein the method further comprises:

measuring a controlled-Z operator using a transversal T gate to measure stabilizers of a CCZ magic state.

6. The method of claim 5, wherein the stabilizers of the CCZ magic state achieve a second order error reduction or a fourth order error reduction.

7. The method of claim 1, further comprising simultaneously measuring stabilizers of CCZ magic states using one or more transversal CCZ gates.

8. A method for distilling magic states in a quantum computing device, comprising: generating a magic state distillation protocol for T gates, controlled-S gates, or CCZ gates using (a) a randomized construction process, or (b) level-lifted triorthogonal codes for reducing circuit depth: and

configuring the quantum computing device to implement the magic state distillation protocol.

9. The method of claim 8, wherein the magic state distillation protocol has an asymptotic distillation efficiency -- 1

10. The method of claim 8. wherein the method further comprises measuring a controlled-Z operator using a transversal T gate to measure stabilizers of a CGZ magic state.

11. The method of claim 10, wherein the stabilizers of the CCZ magic state achieve a second order error reduction or a fourt h order error reduction.

12. The method of claim 8, further comprising simultaneously measuring stabilizers of CCZ magic states using one or more transversal CCZ gates.

13. A method for distilling magic states in a quantum computing device, comprising: generating a magic state distillation protocol using only k -I- ηχ total qubits; and configuring the quantum computing device to implement the magic state distillation protocol.

14. The method of claim 13, wherein the magic state distillation protocol is based on a triorthogonal code.

15. A quantum computer system configured to perform the magic state distillation protocol in accordance with claims 1- 14.