Fidelity decay and error accumulation in random quantum circuits

Dear SciPost Editor,

We are very grateful for your consideration.
We sincerely appreciate the time and effort both Referees have dedicated to reviewing our work, in particular for the level of commitment required for thoughtful analysis of the technical results. Their detailed comments and critical assessments have been invaluable in refining our manuscript. We firmly believe that we adequately addressed their feedback. Primarily, we tried to clarify the argumentation for the solvable model used for analytical calculations, and extended the motivation and background for our work, the problems raised by both referees.
With these revisions, we believe that the new version of our paper now meets the standards for publication in SciPost.

With our best wishes,

The Authors

1.- Added the following paragraph in the introduction that emphasizes the motivation and usefulness of our work.

``In the context of quantum chaos and dynamic phase transitions, quantum fidelity is termed Loschmidt echo[25,31], measuring the extent to which a complex system is recovered after applying an imperfect (perturbed) time-reversal. In the framework of time-independent Hamiltonians, the behaviour of the Loschmidt echo is well understood for single-particle quantum systems whose dynamics are fully chaotic in the classical limit: it typically exhibits an initial parabolic decay, followed by an exponential one, and eventually saturates [31]. This pattern has also been observed in many-body systems [32], and similar behaviour is expected in systems governed by time-dependent Hamiltonians, such as quantum circuits [33]. While a quantitative understanding is valuable in its own right, it becomes particularly pertinent in light of the technological relevance of quantum circuits.’'

2- Expanded the paragraph in the introduction, where we discuss the circuit with random permutations emphasizing the versatility of RQCs:

``For example, the brick-wall circuit—consisting of a sequential alternation of leftward and rightward single-qubit shifts—is a paradigmatic model of local random quantum circuits (RQCs)[4]. It has been studied extensively in the context of information spreading [5], thermalization[6], and measurement-induced phase transitions [1]. In contrast, random permutations have been used to model black hole dynamics with non-local interactions[7,8], to establish bounds on entanglement generation[9,10], to study pseudo-randomness and unitary $k$-designs—ensembles that reproduce Haar-random statistics up to the $k$-th moment [11,12]—and to investigate quantum complexity [13], among other applications.''

3- To better motivate the explicit noise model chosen for our calculations, we extended the last paragraph in Section 2.1:

``Since the two-qubit gates are already random, noise must be modelled as a random deviation from the uniform sampling defined by the Haar measure. To this end, to preserve the symmetry with which the gates were sampled we consider that each random unitary $u_{r,r'}$ is independently perturbed by unstructured noise: $\tilde{u}{r,r'} = e^{i\alpha h$}} u_{r,r', where $h_{r,r'}$ is drawn from the Gaussian Unitary Ensemble (GUE), and $\alpha \geq 0$ controls the noise strength. Notably, the ability to model noise independently in both the permutations and the two-qubit gates provides significant flexibility and control in our framework.’'

4- Added a sentence to the final paragraph of the conclusions highlighting a possible research direction beyond the quantum-computing-inspired topics already discussed.

``In addition, it could be interesting to explore the connection between average fidelity in the generic models considered and out-of-time order correlators (OTOCs), inspired by the known relation between the Loschmidt echo and OTOCs in systems governed by time-independent Hamiltonians [48].’’

5- Modified discussion around Eq, (B13):



``It is convenient to exploit the fact that the GUE measure is invariant under unitary transformations. In particular, this implies that the eigenvectors of $H$ do not favour any specific direction in Hilbert space. Therefore, the unitary matrix $U$ that diagonalizes $e^{i\alpha H}$, namely

must be distributed according to the Haar measure. As a result, the average in Eq.~(B.12) can be decomposed into an average over two copies of the Haar-distributed unitaries, as discussed above, together with an average over the eigenvalues encoded in $D_\lambda \otimes D^*_\lambda$, which must be performed with respect to the GUE joint probability distribution. ‘'

6.- Corrected typos throughout the text