Random unitary operations are a fundamental resource for quantum information processing, yet provably generating them with experimentally available controls remains challenging. Existing approaches often require calculating higher-order operator expectation values, relying on statistical assumptions about the experimental Hamiltonian's spectrum to make these calculations tractable. We show how to implement random unitaries by a sequence of evolutions generated by a fixed Hamiltonian conjugated by a global control operator. We give a geometric argument that establishes exponential convergence to the uniform random (Haar) measure on the accessible unitary group under uniform parameter sampling, beyond a finite sequence-length threshold that we bound. We show this holds when the connected Lie subgroup generated by the conjugated Hamiltonians is closed. Native dynamics and global control thus provide a constructive route to uniform randomness, with applications to many experimental platforms including Rydberg atoms, bosons and fermions in optical lattices, transmons, and trapped ions.
Random unitary operations are a fundamental resource for quantum information processing, yet provably generating them with experimentally available controls remains challenging. Existing approaches often require calculating higher-order operator expectation values, relying on statistical assumptions about the experimental Hamiltonian's spectrum to make these calculations tractable. We show how to implement random unitaries by a sequence of evolutions generated by a fixed Hamiltonian conjugated by a global control operator. We give a geometric argument that establishes exponential convergence to the uniform random (Haar) measure on the accessible unitary group under uniform parameter sampling, beyond a finite sequence-length threshold that we bound. We show this holds when the connected Lie subgroup generated by the conjugated Hamiltonians is closed. Native dynamics and global control thus provide a constructive route to uniform randomness, with applications to many experimental platforms including Rydberg atoms, bosons and fermions in optical lattices, transmons, and trapped ions.
According to one of the fundamental axioms of quantum mechanics, unitary operators rule the evolution of any quantum system; it is thus of prominent importance to investigate the corresponding group and discover its properties. Random processes over the unitary group have indeed a wide range of applications in the context of quantum information; in particular, they are involved in the construction of peculiar distributions called unitary designs which mimic the uniform Haar measure by matching its moments and are thus deeply connected to the description of phenomena such as quantum tomography, equilibration, thermalization, encryption and scrambling. Previous approaches have shown that unitary designs are efficiently approximated by random quantum circuits with local unitary operators. Moreover, these circuits are used to characterize the precision of experimental implementations of unitary gates via randomized benchmarking protocols and to rapidly decouple a system from the environment. The main novel contribution of this work is to extend these mixing properties to a continuous-time framework, namely, Brownian motion over the unitary group induced by stochastic local Hamiltonians. In order to achieve these new results, on the one hand we move to a representation theoretic formulation and make use of its tools to establish the gap of local generators linked to the moments of the distribution induced by the diffusion process; from this we then derive an expression for the length of time it takes to ensure convergence toward the moments of the Haar measure. On the other hand we project the stochastic evolution onto a random walk on Pauli matrices and tie this description to the analogous one for random quantum circuits to achieve decoupling in a run time scaling almost linearly with respect to the system size. In addition to providing a unifying framework for random processes over the unitary group, we hence aim at presenting new mathematical results and techniques for quantum information. We furthermore discuss applications to black holes dynamics in the perspective of the information paradox. As an additional novel result not related to Brownian motion, we propose a randomized benchmarking protocol exploiting the symmetry of the gate whose accuracy has to be estimated, in order to overcome current shortcomings afflicting the known schemes. We again rely on representation theory to reduce the computational effort and the amount of employed quantum resources.
Unitary designs provide efficient substitutes for Haar-random unitaries in quantum information processing, randomized measurements, and many-body quantum dynamics. We propose a protocol based on time evolutions of a single chaotic Hamiltonian interleaved with unitary perturbations to generate unitary designs. We derive the frame potential of the resulting ensemble in terms of those of the intermediate ensemble. The intermediate ensemble need not itself be Haar random or exhibit Haar-level frame potentials, provided that its trace moments and frame-potential growth are sufficiently suppressed. The nontrivial Pauli set and the Clifford ensemble are simple examples, whereas ensembles whose cardinality does not grow with the Hilbert-space dimension generally fail. We further establish a frame-potential correspondence between multi-Hamiltonian temporal protocols and one-Hamiltonian protocols interleaved with repeated fixed trace-suppressed perturbations, showing that two perturbations suffice to recover Haar-level frame potentials. For local Hamiltonians, we derive a lower bound on the space--depth control resource, while more generally we establish a universal finite-window bound that constrains the required temporal sampling range. Together, these results identify complementary spatial and temporal resource requirements for perturbation-assisted randomness generation.
As random operations for quantum systems are intensively used in various quantum information tasks, a trustworthy measure of the randomness in quantum operations is highly demanded. The Haar measure of randomness is a useful tool with wide applications, such as boson sampling. Recently, a theoretical protocol was proposed to combine quantum control theory and driven stochastic quantum walks to generate Haar-uniform random operations. This opens up a promising route to converting classical randomness to quantum randomness. Here, we implement a two-dimensional stochastic quantum walk on the integrated photonic chip and demonstrate that the average of all distribution profiles converges to the even distribution when the evolution length increases, suggesting the 1-pad Haar-uniform randomness. We further show that our two-dimensional array outperforms the one-dimensional array of the same number of waveguide for the speed of convergence. Our Letter demonstrates a scalable and robust way to generate Haar-uniform randomness that can provide useful building blocks to boost future quantum information techniques.
Unitary designs provide finite-moment approximations to Haar-random unitaries, with wide-ranging applications across physics and quantum information, from scrambling and black-hole dynamics to foundational primitives in quantum algorithms. Strong unitary designs capture a more demanding operational notion of approximation, requiring indistinguishability from Haar randomness even for quantum algorithms that may access a unitary not only in the forward direction, but also through its inverse, transpose, and complex conjugate. Motivated by the physical requirement that scrambling arise within the system itself, Schuster, Ma, Lombardi, Brandão, and Huang (arXiv:2509.26310) left open whether strong unitary designs can be generated in logarithmic depth using only the system qubits. For every fixed design order k and measurable-error tolerance, we construct strong approximate unitary k-designs in optimal Θ(logn) all-to-all circuit depth using only the n original system qubits. Our new ingredient is a logarithmic-depth Pauli-mixing bound for the perfect-matching ensemble, whose layers pair the qubits uniformly at random and apply independent random two-qubit gates. This bound controls the mixed forward-reverse two-query case, which we combine with existing design and gluing results to obtain strong unitary designs of arbitrary fixed order.
We investigate the generation of uniform quantum state ensembles via continuous measurement. Using the SU(d) Bloch representation, we derive the associated Langevin and Fokker-Planck equations and identify geometric conditions under which homogeneous monitoring causes global convergence to the uniform pure-state ensemble. We then extend the analysis to mixed states, showing that homogeneous purity-dependent decoherence rates generate uniform Hilbert-Schmidt and Bures ensembles of qubit states through an effective nonlinear stochastic evolution. Additionally, we introduce a post-mixing protocol for qubits: target mixed-state ensembles are assembled by classically sampling trajectories generated with different fixed efficiencies (or decoherence rates). This provides an experimentally feasible route to reconstructing Hilbert-Schmidt and Bures-random mixed-state ensembles, demonstrating that continuous monitoring provides both an exact dynamical generator of Haar-random pure states and a practical route to constructing mixed-state ensembles.
Unitary randomness underpins both fundamental tasks in quantum information and the modern theory of quantum chaos. On one side, a central concept is that of approximate unitary designs: circuits that look random according to small moments and for forward-in-time protocols. In a distinct setting, out-of-time-ordered correlators (OTOCs), intensely studied as a measure of information scrambling, have recently been shown to probe freeness between Heisenberg operators, the noncommutative generalization of statistical independence. Bridging these two concepts, we study the emergence of freeness in a random matrix product unitary ensemble. We prove that, with only polynomial bond dimension, these unitaries reproduce Haar values of higher-order OTOCs for local, finite-trace observables, while traceless observables instead require exponential resources. Indeed, local observables are precisely those predicted to thermalize in chaotic many-body systems according to the eigenstate thermalization hypothesis. Moreover, adding to previous literature, we show how random matrix product unitaries constitute approximate designs: we exactly compute the frame potential of the ensemble, showing convergence to the Haar value with polynomial deviations and so indicating that global observables are freely independent on-average. Our results highlight the need to refine previous notions of unitary design in the context of operator dynamics, guiding us towards protocols for quantum advantage and shedding light on the emergent complexity of chaotic many-body systems.