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1. Approximate $t$ -Designs in Generic Circuit Architectures

   https://doi.org/10.1103/PRXQuantum.5.040344  

   Belkin, Daniel; Allen, James; Ghosh, Soumik; Kang, Christopher; Lin, Sophia; Sud, James; Chong, Frederic T; Fefferman, Bill; Clark, Bryan K (December 2024, PRX Quantum)

   Unitary $t$-designs are distributions on the unitary group whose first $t$moments appear maximally random. Previous work has established several upper bounds on the depths at which certain specific random quantum circuit ensembles approximate $t$-designs. Here we show that these bounds can be extended to any fixed architecture of Haar-random two-site gates. This is accomplished by relating the spectral gaps of such architectures to those of one-dimensional brickwork architectures. Our bound depends on the details of the architecture only via the typical number of layers needed for a block of the circuit to form a connected graph over the sites. When this quantity is bounded, the circuit forms an approximate $t$-design in at most linear depth. We give numerical evidence for a stronger bound that depends only on the number of connected blocks into which the architecture can be divided. We also give an implicit bound for nondeterministic architectures in terms of properties of the corresponding distribution over fixed architectures. Published by the American Physical Society2024 

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2. Quantum chaos and circuit parameter optimization

   https://doi.org/10.1088/1742-5468/acb52d  

   Kim, Joonho; Oz, Yaron; Rosa, Dario (February 2023, Journal of Statistical Mechanics: Theory and Experiment)

   Abstract We consider quantum chaos diagnostics of the variational circuit states at random parameters and explore their connection to the circuit expressibility and optimizability. By measuring the operator spreading coefficient and the eigenvalue spectrum of the modular Hamiltonian of the reduced density matrix, we identify the emergence of universal random matrix ensembles in high-depth circuit states. The diagnostics that use the eigenvalue spectrum, e.g. operator spreading and entanglement entropy, turn out to be more accurate measures of the variational quantum algorithm optimization efficiency than those that use the level spacing distribution of the entanglement spectrum, such as r -statistics or spectral form factors. 

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3. q-analog qudit Dicke states

   https://doi.org/10.1088/1751-8121/ad1ea4  

   Raveh, David; Nepomechie, Rafael_I (February 2024, Journal of Physics A: Mathematical and Theoretical)

   Abstract Dicke states are completely symmetric states of multiple qubits (2-level systems), and qudit Dicke states are theird-level generalization. We define hereq-deformed qudit Dicke states using the quantum algebra $s{u}_q\left(d\right)$. We show that these states can be compactly expressed as a weighted sum over permutations withq-factors involving the so-called inversion number, an important permutation statistic in Combinatorics. We use this result to compute the bipartite entanglement entropy of these states. We also discuss the preparation of these states on a quantum computer, and show that introducing aq-dependence does not change the circuit gate count. 

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4. Approximate Unitary t-Designs by Short Random Quantum Circuits Using Nearest-Neighbor and Long-Range Gates

   https://doi.org/10.1007/s00220-023-04675-z  

   Harrow, Aram W.; Mehraban, Saeed (May 2023, Communications in Mathematical Physics)

   Abstract We prove that$${{\,\textrm{poly}\,}}(t) \cdot n^{1/D}$$ $\text{poly}\left(t\right)·n^{1/D}$-depth local random quantum circuits with two qudit nearest-neighbor gates on aD-dimensional lattice withnqudits are approximatet-designs in various measures. These include the “monomial” measure, meaning that the monomials of a random circuit from this family have expectation close to the value that would result from the Haar measure. Previously, the best bound was$${{\,\textrm{poly}\,}}(t)\cdot n$$ $\text{poly}\left(t\right)·n$due to Brandão–Harrow–Horodecki (Commun Math Phys 346(2):397–434, 2016) for$$D=1$$ $D=1$. We also improve the “scrambling” and “decoupling” bounds for spatially local random circuits due to Brown and Fawzi (Scrambling speed of random quantum circuits, 2012). One consequence of our result is that assuming the polynomial hierarchy ($${{\,\mathrm{\textsf{PH}}\,}}$$ $\mathrm{PH}$) is infinite and that certain counting problems are$$\#{\textsf{P}}$$ $\#P$-hard “on average”, sampling within total variation distance from these circuits is hard for classical computers. Previously, exact sampling from the outputs of even constant-depth quantum circuits was known to be hard for classical computers under these assumptions. However the standard strategy for extending this hardness result to approximate sampling requires the quantum circuits to have a property called “anti-concentration”, meaning roughly that the output has near-maximal entropy. Unitary 2-designs have the desired anti-concentration property. Our result improves the required depth for this level of anti-concentration from linear depth to a sub-linear value, depending on the geometry of the interactions. This is relevant to a recent experiment by the Google Quantum AI group to perform such a sampling task with 53 qubits on a two-dimensional lattice (Arute in Nature 574(7779):505–510, 2019; Boixo et al. in Nate Phys 14(6):595–600, 2018) (and related experiments by USTC), and confirms their conjecture that$$O(\sqrt{n})$$ $O\left(\sqrt{n}\right)$depth suffices for anti-concentration. The proof is based on a previous construction oft-designs by Brandão et al. (2016), an analysis of how approximate designs behave under composition, and an extension of the quasi-orthogonality of permutation operators developed by Brandão et al. (2016). Different versions of the approximate design condition correspond to different norms, and part of our contribution is to introduce the norm corresponding to anti-concentration and to establish equivalence between these various norms for low-depth circuits. For random circuits with long-range gates, we use different methods to show that anti-concentration happens at circuit size$$O(n\ln ^2 n)$$ $O\left(n{ln}^{2}n\right)$corresponding to depth$$O(\ln ^3 n)$$ $O\left({ln}^{3}n\right)$. We also show a lower bound of$$\Omega (n \ln n)$$ $\Omega (nlnn)$for the size of such circuit in this case. We also prove that anti-concentration is possible in depth$$O(\ln n \ln \ln n)$$ $O(lnnlnlnn)$(size$$O(n \ln n \ln \ln n)$$ $O(nlnnlnlnn)$) using a different model. 

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5. Connecting physics to systems with modular spin-circuits

   https://doi.org/10.1038/s44306-024-00059-8  

   Selcuk, Kemal; Bunaiyan, Saleh; Singh, Nihal Sanjay; Sayed, Shehrin; Ganguly, Samiran; Finocchio, Giovanni; Datta, Supriyo; Camsari, Kerem Y (December 2024, npj Spintronics)

   Abstract An emerging paradigm in modern electronics is that of CMOS+$${\mathsf{X}}$$ $X$requiring the integration of standard CMOS technology with novel materials and technologies denoted by$${\mathsf{X}}$$ $X$. In this context, a crucial challenge is to develop accurate circuit models for$${\mathsf{X}}$$ $X$that are compatible with standard models for CMOS-based circuits and systems. In this perspective, we present physics-based, experimentally benchmarked modular circuit models that can be used to evaluate a class of CMOS+$${\mathsf{X}}$$ $X$systems, where$${\mathsf{X}}$$ $X$denotes magnetic and spintronic materials and phenomena. This class of materials is particularly challenging because they go beyond conventional charge-based phenomena and involve the spin degree of freedom which involves non-trivial quantum effects. Starting from density matrices—the central quantity in quantum transport—using well-defined approximations, it is possible to obtain spin-circuits that generalize ordinary circuit theory to 4-component currents and voltages (1 for charge and 3 for spin). With step-by-step examples that progressively become more complex, we illustrate how the spin-circuit approach can be used to start from the physics of magnetism and spintronics to enable accurate system-level evaluations. We believe the core approach can be extended to include other quantum degrees of freedom like valley and pseudospins starting from corresponding density matrices. 

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