Abstract
The Gottesman–Knill theorem established that stabilizer states and Clifford operations can be efficiently simulated classically. For qudits with odd dimension three and greater, stabilizer states and Clifford operations have been found to correspond to positive discrete Wigner functions and dynamics. We present a discrete Wigner function-based simulation algorithm for odd-d qudits that has the same time and space complexity as the Aaronson–Gottesman algorithm for qubits. We show that the efficiency of both algorithms is due to harmonic evolution in the symplectic structure of discrete phase space. The differences between the Wigner function algorithm for odd-d and the Aaronson–Gottesman algorithm for qubits are likely due only to the fact that the Weyl–Heisenberg group is not in S U ( d ) for d = 2 and that qubits exhibit state-independent contextuality. This may provide a guide for extending the discrete Wigner function approach to qubits.
Highlights
The cost of brute-force classical simulation of the time evolution of n-qubit states grows exponentially with n
Recent progress has been the result of work by Wootters [6], Gross [7], Veitch et al [8,9], Mari et al [4], and Howard et al [5], who have formulated a new perspective based on the discrete phase spaces of states and operators in finite Hilbert spaces using discrete Wigner functions
We show that the original Aaronson–Gottesman tableau algorithm for qubit stabilizer states is equivalent to such a discrete Wigner function propagation and that the tableau matrix coincides with the discrete Wigner function of a stabilizer state
Summary
The cost of brute-force classical simulation of the time evolution of n-qubit states grows exponentially with n. This set of states plays an important role in quantum error correction [1] and is closed under action by Clifford gates Efficient simulation of such systems was demonstrated with the tableau algorithm of Aaronson and Gottesman [1,2] for qubits (d = 2). In odd-dimensional systems, stabilizer states have been shown to be the discrete analogue to Gaussian states in continuous systems [7] and Clifford group gates have been shown to have underlying harmonic Hamiltonians that preserve the discrete Weyl phase space points [10] This means Clifford circuits are expressible by path integrals truncated at order h0 and are manifestly classical [10,11]. By instead working in the Schrödinger picture we are able to more reveal the purely classical basis of both algorithms and the physically intuitive phase space structures and symplectic properties on which they rely
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