Symmetry Protected Quantum Computation

Michael H. Freedman1,2, Matthew B. Hastings1,2, and Modjtaba Shokrian Zini3,4

1Station Q, Microsoft Research, Santa Barbara, CA 93106-6105, USA
2Microsoft Quantum, Redmond, WA 98052, USA
3Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada
4Research Consultant, Microsoft

Abstract

We consider a model of quantum computation using qubits where it is possible to measure whether a given pair are in a singlet (total spin $0$) or triplet (total spin $1$) state. The physical motivation is that we can do these measurements in a way that is protected against revealing other information so long as all terms in the Hamiltonian are $SU(2)$-invariant. We conjecture that this model is equivalent to BQP. Towards this goal, we show: (1) this model is capable of universal quantum computation with polylogarithmic overhead if it is supplemented by single qubit $X$ and $Z$ gates. (2) Without any additional gates, it is at least as powerful as the weak model of "permutational quantum computation" of Jordan [14, 18]. (3) With postselection, the model is equivalent to PostBQP.

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Cited by

[1] Constantin Schrade, Charles M. Marcus, and András Gyenis, "Protected Hybrid Superconducting Qubit in an Array of Gate-Tunable Josephson Interferometers", PRX Quantum 3 3, 030303 (2022).

[2] Josiah Couch, Yale Fan, and Sanjit Shashi, "Circuit Complexity in Topological Quantum Field Theory", Fortschritte der Physik 70 9-10, 2200102 (2022).

[3] Josiah Couch, Yale Fan, and Sanjit Shashi, "Circuit Complexity in Topological Quantum Field Theory", arXiv:2108.13427, (2021).

[4] Terry Rudolph and Shashank Soyuz Virmani, "Relational quantum computing using only maximally mixed initial qubit states", arXiv:2107.03239, (2021).

[5] Matthew Brooks and Charles Tahan, "Hybrid Exchange-Measurement-Based Qubit Operations in Semiconductor Double-Quantum-Dot Qubits", Physical Review Applied 16 6, 064019 (2021).

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