Extended Data Fig. 2: Example of mapping a quantum circuit to a signature tensor.

(Left) A circuit formed by CNOT and T gates, along with the diagonal unitary matrix that it implements. The output of the circuit is the quantum state \({e}^{i\phi (x,y)}\left\vert x,y\right\rangle\), where x, y ∈ {0, 1} and the phase rotation is captured by the so-called phase polynomial ϕ(x, y). (Middle) The phase polynomial ϕ(x, y) can be directly mapped to an alternative implementation of the circuit. Here, implementing the phase rotation ϕ(x, y) requires one T gate on the first qubit (term \(\frac{\pi }{4}x\)), two T gates on the second qubit (term \(\frac{\pi }{4}2y\)), and three CS gates (term \(-\frac{2\pi }{4}xy\)). Since two consecutive T gates form a (Clifford) S gate, and two consecutive CS gates form a (Clifford) CZ gate, the only non-Clifford components that are needed are a T gate acting on the first qubit and a CS gate. (Right) The 2 × 2 × 2 signature tensor \({\mathcal{T}}\) describing the non-Clifford components. The diagonal entries of the tensor describe single-qubit interactions and correspond to the T gates. The off-diagonal entries describe two-qubit interactions and correspond to the CS gate. The signature tensor indicates a T gate acting on the first qubit and a CS gate acting on both qubits. If the circuit had more than two qubits, then entries of the tensor \({{\mathcal{T}}}_{ijk}\) with i ≠ j, i ≠ k, j ≠ k would describe three-qubit interactions and correspond to CCZ gates.