Bases for optimising stabiliser decompositions of quantum states

Stabiliser states play a central role in the theory of quantum computation. For example, they are used to encode data in quantum error correction schemes. Arbitrary quantum states admit many stabiliser decompositions: ways of being expressed as a superposition of stabiliser states. Understanding the structure of stabiliser decompositions has applications in verifying and simulating near-term quantum computers. We introduce and study the vector space of linear dependencies of $n$-qubit stabiliser states. These spaces have canonical bases containing vectors whose size grows exponentially in $n$. We construct elegant bases of linear dependencies of constant size three. We apply our methods to computing the stabiliser extent of large states and suggest potential future applications to improving bounds on the stabiliser rank of magic states

Stationary Phase Method in Discrete Wigner Functions and Classical Simulation of Quantum Circuits

One of the lowest-order corrections to Gaussian quantum mechanics in infinite-dimensional Hilbert spaces are Airy functions: a uniformization of the stationary phase method applied in the path integral perspective. We introduce a ``periodized stationary phase method'' to discrete Wigner functions of systems with odd prime dimension and show that the $\frac{\pi}{8}$ gate is the discrete analog of the Airy function. We then establish a relationship between the stabilizer rank of states and the number of quadratic Gauss sums necessary in the periodized stationary phase method. This allows us to develop a classical strong simulation of a single qutrit marginal on $t$ qutrit $\frac{\pi}{8}$ gates that are followed by Clifford evolution, and show that this only requires $3^{\frac{t}{2}+1}$ quadratic Gauss sums. This outperforms the best alternative qutrit algorithm (based on Wigner negativity and scaling as $\sim\hspace{-3pt} 3^{0.8 t}$ for $10^{-2}$ precision) for any number of $\frac{\pi}{8}$ gates to full precision

Entropy Cones and Entanglement Evolution for Dicke States

The $N$-qubit Dicke states $|D^N_k\rangle$, of Hamming-weight $k$, are a class of entangled states which play an important role in quantum algorithm optimization. We present a general calculation of entanglement entropy in Dicke states, which we use to describe the $|D^N_k\rangle$ entropy cone. We demonstrate that all $|D^N_k\rangle$ entropy vectors emerge symmetrized, and use this to define a min-cut protocol on star graphs which realizes $|D^N_k\rangle$ entropy vectors. We identify the stabilizer group for all $|D^N_k\rangle$, under the action of the $N$-qubit Pauli group and two-qubit Clifford group, which we use to construct $|D^N_k\rangle$ reachability graphs. We use these reachability graphs to analyze and bound the evolution of $|D^N_k\rangle$ entropy vectors in Clifford circuits.Comment: 31 pages, 13 figures, 1 Mathematica packag