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Constructing All Qutrit Controlled Clifford+T gates in Clifford+T

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Part of the Lecture Notes in Computer Science book series (LNCS,volume 13354)


For a number of useful quantum circuits, qudit constructions have been found which reduce resource requirements compared to the best known or best possible qubit construction. However, many of the necessary qutrit gates in these constructions have never been explicitly and efficiently constructed in a fault-tolerant manner. We show how to exactly and unitarily construct any qutrit multiple-controlled Clifford+T unitary using just Clifford+T gates and without using ancillae. The T-count to do so is polynomial in the number of controls k, scaling as \(O(k^{3.585})\). With our results we can construct ancilla-free Clifford+T implementations of multiple-controlled T gates as well as all versions of the qutrit multiple-controlled Toffoli, while the analogous results for qubits are impossible. As an application of our results, we provide a procedure to implement any ternary classical reversible function on n trits as an ancilla-free qutrit unitary using \(O(3^n n^{3.585})\) T gates.


  • Qutrits
  • Gate Synthesis
  • Clifford+T

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The authors wish to thank Andrew Glaudell and Neil J. Ross for discussions regarding the consequences of our results and Andrew Glaudell specifically for pointing out Eq. (23). We additionally wish to thank Shuxiang Cao and Razin Shaikh for assistance in preparing the figures in an early draft of this paper. JvdW is supported by an NWO Rubicon personal fellowship. LY is supported by an Oxford - Basil Reeve Graduate Scholarship at Oriel College with the Clarendon Fund.

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Yeh, L., van de Wetering, J. (2022). Constructing All Qutrit Controlled Clifford+T gates in Clifford+T. In: Mezzina, C.A., Podlaski, K. (eds) Reversible Computation. RC 2022. Lecture Notes in Computer Science, vol 13354. Springer, Cham.

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