Classical simulations of Abelian-group normalizer circuits with intermediate measurements

Quantum normalizer circuits were recently introduced as generalizations of Clifford circuits [arXiv:1201.4867]: a normalizer circuit over a finite Abelian group $G$ is composed of the quantum Fourier transform (QFT) over G, together with gates which compute quadratic functions and automorphisms. In [arXiv:1201.4867] it was shown that every normalizer circuit can be simulated efficiently classically. This result provides a nontrivial example of a family of quantum circuits that cannot yield exponential speed-ups in spite of usage of the QFT, the latter being a central quantum algorithmic primitive. Here we extend the aforementioned result in several ways. Most importantly, we show that normalizer circuits supplemented with intermediate measurements can also be simulated efficiently classically, even when the computation proceeds adaptively. This yields a generalization of the Gottesman-Knill theorem (valid for n-qubit Clifford operations [quant-ph/9705052, quant-ph/9807006] to quantum circuits described by arbitrary finite Abelian groups. Moreover, our simulations are twofold: we present efficient classical algorithms to sample the measurement probability distribution of any adaptive-normalizer computation, as well as to compute the amplitudes of the state vector in every step of it. Finally we develop a generalization of the stabilizer formalism [quant-ph/9705052, quant-ph/9807006] relative to arbitrary finite Abelian groups: for example we characterize how to update stabilizers under generalized Pauli measurements and provide a normal form of the amplitudes of generalized stabilizer states using quadratic functions and subgroup cosets.

💡 Research Summary

The paper studies a broad class of quantum circuits called Abelian‑group normalizer circuits and shows that even when these circuits are equipped with intermediate measurements and adaptive control, they can still be simulated efficiently on a classical computer. A normalizer circuit over a finite Abelian group (G) is built from three elementary primitives: (i) the quantum Fourier transform (QFT) over (G), (ii) gates that implement group automorphisms (linear changes of basis in the group), and (iii) phase gates that multiply basis states (|x\rangle) by a complex phase (e^{2\pi i q(x)}) where (q) is a quadratic function on (G). Earlier work (arXiv:1201.4867) proved that circuits consisting solely of these primitives admit a polynomial‑time classical simulation, thereby providing a non‑trivial example of quantum circuits that cannot yield exponential speed‑ups despite containing the QFT.

The present work extends that result in three major directions. First, it allows generalized Pauli measurements to be performed at arbitrary points during the computation. In the context of an arbitrary Abelian group, Pauli operators are defined as the natural shift‑and‑multiply operators acting on the group basis; measuring such an operator yields a classical outcome that determines a coset of a subgroup of (G). Second, the authors permit adaptive choice of subsequent gates based on previous measurement outcomes, which is the quantum analogue of classical branching. Third, they provide two complementary simulation algorithms: (a) a sampling algorithm that efficiently draws outcomes from the exact measurement probability distribution, and (b) an amplitude‑tracking algorithm that computes the exact complex amplitudes of the quantum state after each step.

Both algorithms rely on a generalized stabilizer formalism for arbitrary finite Abelian groups. In the traditional Clifford‑stabilizer picture (for (G=\mathbb{Z}_2^n)), a stabilizer group is generated by commuting Pauli operators, and its evolution under Clifford gates can be tracked by updating the generator matrix. Here the stabilizer group is described by a pair ((H,a,q)) where (H\le G) is a subgroup, (a\in G) a coset representative, and (q:G\to\mathbb{Q}/\mathbb{Z}) a quadratic function. Any normalizer state can be written in the standard form

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