Papers › A polynomial time and space heuristic algorithm for T-count
A polynomial time and space heuristic algorithm for T-count
Michele Mosca, Priyanka Mukhopadhyay
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This work focuses on reducing the physical cost of implementing quantum algorithms when using the state-of-the-art fault-tolerant quantum error correcting codes, in particular, those for which implementing the T gate consumes vastly more resources than the other gates in the gate set. More specifically, we consider the group of unitaries that can be exactly implemented by a quantum circuit consisting of the Clifford+T gate set, a universal gate set. Our primary interest is to compute a circuit for a given n-qubit unitary U, using the minimum possible number of T gates (called the T-count of unitary U). We consider the problem COUNT-T, the optimization version of which aims to find the T-count of U. In its decision version the goal is to decide if the T-count is at most some positive integer m. Given an oracle for COUNT-T, we can compute a T-count-optimal circuit in time polynomial in the T-count and dimension of U. We give a provable classical algorithm that solves COUNT-T (decision) in time O(N^(2(c-1)⌈m/c⌉)poly(m,N)) and space O(N^(2⌈m/c⌉)poly(m,N)), where N=2ⁿ and c≥2. This gives a space-time trade-off for solving this problem with variants of meet-in-the-middle techniques. We also introduce an asymptotically faster multiplication method that shaves a factor of N^(0.7457) off of the overall complexity. Lastly, beyond our improvements to the rigorous algorithm, we give a heuristic algorithm that outputs a T-count-optimal circuit and has space and time complexity poly(m,N), under some assumptions. While our heuristic method still scales exponentially with the number of qubits (though with a lower exponent, there is a large improvement by going from exponential to polynomial scaling with m.
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