Local random quantum circuits are approximate polynomial-designs: numerical results

By P. Ćwikliński, Michal Horodecki, Marek Mozrzymas, Łukasz Pankowski, Michal Studzinski. 2012

158 citations recorded by the source corpus.

We numerically investigate the statement that local random quantum circuits acting on n qubits composed of polynomially many nearest-neighbor two-qubit gates form an approximate unitary poly(n)-design (Brandão et al 2012 arXiv:1208.0692). Using a group theory formalism, spectral gaps that give a ratio of convergence to a given t-design are evaluated for a different number of qubits n (up to 20) and degrees t (t = 2, 3, 4 and 5), improving previously known results for n = 2 in the case of t = 2 and 3. Their values lead to the conclusion that the previously used lower bound that bounds spectral gaps values may give very little information about the real situation and in most cases, only tells us that a gap is closed. We compare our results to another lower bounding technique, again showing that its results may not be tight.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/238323558a3ea5c85bffce82599b60e83e1b1664

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Citations in the corpus

Unitary designs from statistical mechanics in random quantum circuits. (2019, background): “Lastly, we note that [35] numerically investigated the gap of the moment operator and found evidence for a linear growth in design from the ﬁrst few moments.”
The role of quantum information in thermodynamics—a topical review (2015, background): “There has been a strong effort to generalize the concept of typicality for different sets of states [43–45].”
Quantum Entanglement Growth Under Random Unitary Dynamics (2016, background): “Simple quantum circuits — quantum evolutions in discrete time — serve as useful toy models for entanglement growth and scrambling [28].”
Dynamical Purification Phase Transition Induced by Quantum Measurements (2019, background): “It turns out some basic features of the mixed phase already appear when the measurements occur only at a single given site at an arbitrarily slow rate (technically, a sufficient condition for large L is p 1/L(3) [60]).”
Random Quantum Circuits (2022, background): “Random circuits are also important in quantum information as ingredients of proofs or algorithms (121), and for benchmarking (non-random) quantum circuits (122, 123).”
Many-Body Quantum Magic (2020, background): “Combining with the well-known result that t-designs can be approximated by O(poly(t)) random gates [73], we expect that all divergence-based measures of entanglement and magic as probes of complexity become nearly maximal with O(poly(n)) gates, or O(poly(n)) depth and time.”
Chaos and complexity by design (2016, background): “(33)It is believed that this is actually saturated by the local random circuit of [41].”
Variational quantum algorithms (2020, background): “The proof of this result relies on the fact that these unstructured ansatze become 2-designs when their depth grows polynomially with the number of qubits n [166, 167].”
Onset of random matrix behavior in scrambling systems (2018, methodology): “erators [28]. An important advance was the use of Markov chain techniques to estimate convergence times [29]. In particular these ideas have been applied to the study of approximate unitary k-designs [30,31,32,33,34,35,36,37,38,39,40]. These techniques were used to give the rst logarithmic estimate ”
Entanglement Membrane in Chaotic Many-Body Systems (2019, background): “[5] Fernando GSL Brandão, Aram W Harrow, and Micha l Horodecki.”