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Efficient synthesis of arbitrary quantum states and unitaries from auniversal fault-tolerant gate-set e.g. Clifford+T is a key subroutine inquantum computation. As large quantum algorithms feature many qubits thatencode coherent quantum information but remain idle for parts of thecomputation, these should be used if it minimizes overall gate counts,especially that of the expensive T-gates. We present a quantum algorithm forpreparing any dimension-$N$ pure quantum state specified by a list of $N$classical numbers, that realizes a trade-off between space and T-gates. Ourscheme uses $\mathcal{O}(\log{(N/\epsilon)})$ clean qubits and a tunable numberof $\sim(\lambda\log{(\frac{\log{N}}{\epsilon})})$ dirty qubits, to reduce theT-gate cost to$\mathcal{O}(\frac{N}{\lambda}+\lambda\log^2{\frac{N}{\epsilon}})$. Thistrade-off is optimal up to logarithmic factors, proven through an unconditionalgate counting lower bound, and is, in the best case, a quadratic improvement inT-count over prior ancillary-free approaches. We prove similar statements forunitary synthesis by reduction to state preparation.

Trading T-gates for dirty qubits in state preparation and unitary synthesis