Abstract
Encoding logical quantum information in harmonic oscillator modes is a promising and hardware-efficient approach to the realization of a quantum computer. In this work, we propose to encode logical qubits in grid states of an ensemble of harmonic oscillator modes. We first discuss general results about these multimode bosonic codes; how to design them, how to practically implement them in different experimental platforms and how lattice symmetries can be leveraged to perform logical non-Clifford operations. We then introduce in detail two two-mode grid codes based on the hypercubic and D4 lattices, respectively, showing how to perform a universal set of logical operations. We demonstrate numerically that multimode grid codes have, compared to their single-mode counterpart, increased robustness against propagation of errors from ancillas used for error correction. Finally, we highlight some interesting links between multidimensional lattices and single-mode grid codes concatenated with qubit codes.
Highlights
Encoding logical quantum information in harmonic oscillator modes is a promising and hardwareefficient approach to the realization of a quantum computer
One promising approach to quantum error correction (QEC) is to encode logical qubits in harmonic oscillator modes, a strategy named bosonic codes from the statistics obeyed by the oscillator excitations
Amongst the attractive features of bosonic codes is their large Hilbert space, which allows a high degree of redundancy and their relatively simple error model compared to multiqubit QEC codes
Summary
We consider m harmonic oscillator modes, and we aim to establish correspondences between translations in a (symplectic) vector space R2m and the. √ By defining translations in units of l = 2π , the translation operators associated with two vectors u and v commute if and only if their symplectic form is an integer, [T(u), T(v)] = 0 ⇔ uT v ∈ Z. Quantum unitaries generated by quadratic Hamiltonians are represented by 2m × 2m real symplectic matrices M ∈ Sp(2m, R) respecting M T M =. We define the unitary-valued function Qthat takes as input a symplectic matrix M and outputs the corresponding quantum unitary. We define a beamsplitter operation between two modes j , k, Bj →k = exp{−iπ(qj pk − pj qk)/4}, which has a symplectic representation.
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