# Improved upper bounds on the stabilizer rank of magic states

Hammam Qassim1,2,3, Hakop Pashayan1,4,5, and David Gosset1,4,5

1Institute for Quantum Computing, University of Waterloo, Ontario
2Department of Physics and Astronomy, University of Waterloo, Ontario
4Department of Combinatorics and Optimization, University of Waterloo, Ontario
5Perimeter Institute for Theoretical Physics, Waterloo, Ontario

### Abstract

In this work we improve the runtime of recent classical algorithms for strong simulation of quantum circuits composed of Clifford and T gates. The improvement is obtained by establishing a new upper bound on the stabilizer rank of $m$ copies of the magic state $|T\rangle=\sqrt{2}^{-1}(|0\rangle+e^{i\pi/4}|1\rangle)$ in the limit of large $m$. In particular, we show that $|T\rangle^{\otimes m}$ can be exactly expressed as a superposition of at most $O(2^{\alpha m})$ stabilizer states, where $\alpha\leq 0.3963$, improving on the best previously known bound $\alpha \leq 0.463$. This furnishes, via known techniques, a classical algorithm which approximates output probabilities of an $n$-qubit Clifford + T circuit $U$ with $m$ uses of the T gate to within a given inverse polynomial relative error using a runtime $\mathrm{poly}(n,m)2^{\alpha m}$. We also provide improved upper bounds on the stabilizer rank of symmetric product states $|\psi\rangle^{\otimes m}$ more generally; as a consequence we obtain a strong simulation algorithm for circuits consisting of Clifford gates and $m$ instances of any (fixed) single-qubit $Z$-rotation gate with runtime $\text{poly}(n,m) 2^{m/2}$. We suggest a method to further improve the upper bounds by constructing linear codes with certain properties.

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### Cited by

[1] Sergey Bravyi, David Gosset, and Yinchen Liu, "How to Simulate Quantum Measurement without Computing Marginals", Physical Review Letters 128 22, 220503 (2022).

[2] Salvatore F. E. Oliviero, Lorenzo Leone, Alioscia Hamma, and Seth Lloyd, "Measuring magic on a quantum processor", npj Quantum Information 8 1, 148 (2022).

[3] Nikolaos Koukoulekidis, Hyukjoon Kwon, Hyejung H. Jee, David Jennings, and M. S. Kim, "Faster Born probability estimation via gate merging and frame optimisation", Quantum 6, 838 (2022).

[4] Salvatore F. E. Oliviero, Lorenzo Leone, and Alioscia Hamma, "Magic-state resource theory for the ground state of the transverse-field Ising model", Physical Review A 106 4, 042426 (2022).

[5] Aleks Kissinger and John van de Wetering, "Simulating quantum circuits with ZX-calculus reduced stabiliser decompositions", Quantum Science and Technology 7 4, 044001 (2022).

[6] Shir Peleg, Amir Shpilka, and Ben Lee Volk, "Lower Bounds on Stabilizer Rank", Quantum 6, 652 (2022).

[7] Benjamin Lovitz and Vincent Steffan, "New techniques for bounding stabilizer rank", Quantum 6, 692 (2022).

[8] Sahar Atallah, Michael Garn, Sania Jevtic, Yukuan Tao, and Shashank Virmani, "Efficient classical simulation of cluster state quantum circuits with alternative inputs", arXiv:2201.07655, (2022).

[9] Lucas Kocia and Genele Tulloch, "More Optimal Simulation of Universal Quantum Computers", arXiv:2202.01233, (2022).

[10] Srinivasan Arunachalam, Sergey Bravyi, Chinmay Nirkhe, and Bryan O'Gorman, "The Parameterized Complexity of Quantum Verification", arXiv:2202.08119, (2022).

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