4 · Digital quantum circuits

Quantum gates are small unitaries on one or two qubits, which makes a quantum circuit a special case of the engine of Chapter 3. The first notebook introduces rotations as exponentials of Pauli matrices, the standard gate set, circuits as data that can be compiled with jit, circuit identities, universality and the cost of approximating a rotation with the finite Clifford+\(T\) gate set, a Trotter step written as a circuit, and circuits with noise after every gate on the density tensor.

The second notebook turns to randomness: Haar-random states and unitaries, the Porter–Thomas distribution of output probabilities, random brick-wall circuits and the growth of their entanglement, the single-qubit Clifford group as a unitary 3-design, and cross-entropy benchmarking of random circuit sampling. Read the two notebooks in the listed order; the second uses the gates, the circuit lists and the Clifford gates of the first.

Prerequisites are notebook 01 (JAX) of Chapter 1, the Trotterization of notebook 04 in Chapter 2, and Chapter 3: apply_gate of notebook 05 above all, with entanglement entropy (06), Kraus channels and trajectories (07) and sampled bit strings (08). The Trotter circuits continue as TEBD in Chapter 5, and the parametrised circuits as variational algorithms in Chapter 13.

Notebooks in this chapter

Quantum gates and circuits — the digital viewpoint

Qubits are spins and gates are small unitaries: rotations as exponentials of Pauli matrices, circuits as lists of einsum contractions, jit-compiled circuits, circuit identities, a Trotter step as a circuit, and noisy circuits on the density tensor.

Random unitaries and random circuits

Haar-random states and unitaries (Mezzadri’s QR recipe and why its phase fix matters), the Porter-Thomas distribution, random brick-wall circuits and the growth of entanglement to the Page value, the single-qubit Clifford group, random circuit sampling and cross-entropy benchmarking.