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We consider a computational model composed of ideal Gottesman-Kitaev-Preskill stabilizer states, Gaussian operations - including all rational symplectic operations and all real displacements -, and homodyne measurement.
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Note that in the main text, we simplify the class of simulatable operations to those which have a rational symplectic matrix. However, the class of simulatable operations also includes those given in the multimode case of Ref. Calcluth et al. 2022 . We provide the broader requirements of the class of simulatable symplectic matrices in Appendix C
2022
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C. Calcluth, A. Ferraro, and G. Ferrini, arXiv:2203.11182 (2022)
2022
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Integer eigenvectors of a rational matrix (2022a), mathematics Stack Exchange. Available at https://math.stackexchange.com/questions/4391454/integer-eigenvectors-of-a-rational-matrix/4391951 (accessed: 2022-04-26)
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Gaussian operations parameterized by irrational symplectic operations cannot in general be simulated with our method. We refer to our previous work Calcluth et al. 2022 which demonstrates that when the symplectic matrix is irrational, the wavefunction of the transformed state corresponds to a periodic distribution which cannot be analytically reduced. Measuring in the position basis of a state which has been transformed by a general irrational symplectic matrix will have a PDF which will give random integer combinations of irrational numbers. Except for specific choices of irrational symplectic matrices, the measurement values will be randomly selected from a set dense on the real number line
2022
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Is the symplectic group over the rationals dense on the symplectic group over the reals? (2022b), Mathematics Stack Exchange. Available at https://math.stackexchange.com/q/4510323/ (accessed: 2022-08-18)
2022
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