. The Solovay-Kitaev theorem provides an elegant solution which is based on the construction of successively tighter ``nets'' around unity comprised of successively longer sequences of gates. The procedure for the construction of the nets, according to this theorem, requires accessibility to the inverse of the gates as well. In this work, we propose a method for constructing nets around unity without this requirement. The algorithmic procedure is applicable to sets of gates which are diffusive enough, in the sense that sequences of moderate length cover the space of unitary matrices in a uniform way. We prove that the number of gates sufficient for reaching a precision

\ensuremath{\varepsilon}

scales as

log{(1/\ensuremath{\varepsilon})}^{log3/log2}

while the precompilation time is increased as compared to that of the Solovay-Kitaev algorithm by the exponential factor

3/2

DOI: 10.1103/physreva.98.012325

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\ensuremath{\varepsilon}

\ensuremath{\varepsilon}

scales as

log{(1/\ensuremath{\varepsilon})}^{log3/log2}

while the precompilation time is increased as compared to that of the Solovay-Kitaev algorithm by the exponential factor

3/2

\varepsilon

. Solovay-Kitaev theorem provides an elegant solution which is based on the construction of successively tighter `nets' around the unity comprised by successively longer sequences of gates. The procedure for the construction of the nets, according to this theorem, requires accessibility to the inverse of the gates as well. In this work, we propose a method for constructing nets around unity without this requirement. The algorithmic procedure is applicable to sets of gates which are diffusive enough, in the sense that sequences of moderate length cover the space of unitary matrices in a uniform way. We prove that the number of gates sufficient for reaching a precision

\varepsilon

scales as $ \log (1/\varepsilon )^{\log 3 / log 2} $ while the pre-compilation time is increased as compared to thatof the Solovay-Kitaev algorithm by the exponential factor 3/2.

\text{SU}(d)

and

\text{SL}(d, \mathbb{C})

. The algorithm works by showing that approximate gate implementations of the generalized Pauli group can self-correct their errors.