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

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

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

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