Linear-in-dimension T-count synthesis or stronger lower bound

Determine whether arbitrary effectively specified unitaries on a d-dimensional Hilbert space can be synthesized over Clifford+T with a T-count of \widetilde{O}(d), or establish a stronger worst-case lower bound than the known \widetilde{\Omega}(d) bound at constant accuracy.

Background

The paper develops a Clifford+T synthesis method for arbitrary effectively specified d-dimensional unitaries, achieving a T-count of O(d{5/4}L{5/8}\log d) in the regime L\le d, where L=n+\log(1/\epsilon). This improves the preceding exponent but leaves a multiplicative gap between the upper bound and the known \widetilde{\Omega}(d) lower bound at constant accuracy.

The authors identify closing this gap in either direction as the central unresolved issue: one possibility is an essentially linear T-count construction, while the other is a stronger lower bound demonstrating that such a construction is impossible.

References

The known lower bound at constant accuracy remains \Omega(d), leaving a multiplicative gap of \widetilde O(d{1/4}). Determining whether arbitrary unitaries can be synthesized with \widetilde O(d) T gates, or establishing a stronger lower bound, remains the central open problem.

— Quantum Circuit for General Unitary: Improved T-count via Block Flattening and Dilation  (2608.17846 - Yuan et al., 18 Aug 2026) in Section 5, Conclusion

At $n$ qubits the resource landscape is less settled, and the contrast matters for what follows, since the pipelines used in practice are multi-qubit ones. General lower bounds on non-Clifford resources are due to Beverland, Campbell, Howard, and Kliuchnikov and to Low, Kliuchnikov, and Schaeffer, recent upper-bound improvements to Tan, and the gap between the two remains open.

— An Exactness Barrier for ZX-Calculus Optimization of Synthesized Clifford+T Circuits  (2608.22801 - Kam et al., 24 Aug 2026) in Section 1, Introduction