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Algorithmic Assertions - Craig Gidney's Computer Science Blog Why haven't quantum computers factored 21 yet? 30 Aug 2025 In 2001, quantum computers factored the number 15. It’s now 2025, and quantum computers haven’t yet factored the number 21. It’s sometimes claimed this is proof there’s been no progress in quantum computers. But there’s actually a much more surprising reason 21 hasn’t been factored yet, which jumps out at you when contrasting the operations used to factor 15 and to factor 21. The circuit (the series of quantum logic gates) that was run to factor 15 can be seen in Figure 1b of “Experimental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance”: The important cost here is the number of entangling gates. This factoring-15 circuit has 6 two-qubit entangling gates (a mix of CNOT and CPHASE gates). It also has 2 Toffoli gates, which each decompose into 6 two-qubit entangling gates. So there’s a total of 21 entangling gates in this circuit

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