The entangling layer of a quantum circuit Born machine can supply both entanglement and nonstabilizerness. In this work we show that the entangling layer can be a Clifford circuit, which supplies no nonstabilizerness. We use an architecture in which a fixed, non-trainable entangler is reused in every layer and only single-qubit rotations are optimized. There, a Clifford entangler reaches the same Kullback--Leibler divergence as a near-Haar one at sufficient circuit depth, and entanglers spanning zero to Haar-typical nonstabilizerness reach the same loss, up to a small offset for a free-fermion matchgate entangler. A doping control that raises the entangler's nonstabilizerness while leaving the entangling power of every gate unchanged leaves the loss flat. With a Clifford entangler the nonstabilizerness of the trained state comes entirely from the rotations, which take the trained circuit outside the stabilizer formalism. We also show that, among all pure states whose Born distribution is a given target, the one whose amplitudes are the positive square roots of the probabilities has the smallest stabilizer Rényi entropy, so this minimum can be computed without training.
The entangling layer of a quantum circuit Born machine can supply both entanglement and nonstabilizerness. In this work we show that the entangling layer can be a Clifford circuit, which supplies no nonstabilizerness. We use an architecture in which a fixed, non-trainable entangler is reused in every layer and only single-qubit rotations are optimized. There, a Clifford entangler reaches the same Kullback--Leibler divergence as a near-Haar one at sufficient circuit depth, and entanglers spanning zero to Haar-typical nonstabilizerness reach the same loss, up to a small offset for a free-fermion matchgate entangler. A doping control that raises the entangler's nonstabilizerness while leaving the entangling power of every gate unchanged leaves the loss flat. With a Clifford entangler the nonstabilizerness of the trained state comes entirely from the rotations, which take the trained circuit outside the stabilizer formalism. We also show that, among all pure states whose Born distribution is a given target, the one whose amplitudes are the positive square roots of the probabilities has the smallest stabilizer Rényi entropy, so this minimum can be computed without training.
Quantum generative modeling, where the Born rule naturally defines probability distributions through measurement of parameterized quantum states, is a promising near-term application of quantum computing. We propose a Quantum Scrambling Born Machine in which a fixed entangling unitary -- acting as a scrambling reservoir -- provides multi-qubit entanglement, while only single-qubit rotations are optimized. We consider three entangling unitaries -- a Haar random unitary and two physically realizable approximations, a finite-depth brickwork random circuit and analog time evolution under nearest-neighbor spin-chain Hamiltonians -- and show that, for the benchmark distributions and system sizes considered, once the entangler produces near-Haar-typical entanglement the model learns the target distribution with weak sensitivity to the scrambler's microscopic origin. Finally, promoting the Hamiltonian couplings to trainable parameters casts the generative task as a variational Hamiltonian problem, with performance competitive with representative classical generative models at matched parameter count.
Nonstabilizerness describes the distance of a quantum state to its closest stabilizer state. It is—like entanglement—a necessary resource for a quantum advantage over classical computing. We study nonstabilizerness, quantified by stabilizer entropy, in a hybrid quantum circuit with projective measurements and a controlled injection of non-Clifford resources. We discover a phase transition between a power law and constant scaling of nonstabilizerness with system size controlled by the rate of measurements. The same circuit also exhibits a phase transition in entanglement that appears, however, at a different critical measurement rate. This mechanism shows how, from the viewpoint of a quantum advantage, hybrid circuits can host multiple distinct transitions where not only entanglement, but also other nonlinear properties of the density matrix come into play. Published by the American Physical Society 2024
Quantum generative models use the intrinsic probabilistic nature of quantum mechanics to learn and reproduce complex probability distributions. In this paper, we present an implementation of a 3-qubit quantum circuit Born machine trained to model a 3-bit Gaussian distribution using a Kullback-Leibler (KL) divergence loss and parameter-shift gradient optimization. The variational quantum circuit consists of layers of parameterized rotations and entangling gates, and is optimized such that the Born rule output distribution closely matches the target distribution. We detail the mathematical formulation of the model distribution, the KL divergence cost function, and the parameter-shift rule for gradient evaluation. Training results on a statevector simulator show that the KL divergence is minimized to near zero, and the final generated distribution aligns quantitatively with the target probabilities. We analyze the convergence behavior and discuss the implications for scalability and quantum advantage. Our results demonstrate the feasibility of small-scale quantum generative learning and provide insight into the training dynamics of quantum circuit models.
We investigate the dynamics of nonstabilizerness—also known as “magic”—in monitored quantum circuits composed of random Clifford unitaries and local measurements. For measurements in the computational basis, we derive an analytical model for dynamics of the stabilizer nullity, showing that it decays in quantized steps and requires exponentially many measurements to vanish, which reveals the strong protection through Clifford scrambling. On the other hand, for measurements performed in rotated non-Clifford bases, measurements can both create and destroy nonstabilizerness. Here, the dynamics leads to a steady state with nontrivial nonstabilizerness, independent of the initial state. We find that Haar-random states equilibrate in constant time, whereas stabilizer states exhibit linear-in-size relaxation time. While the stabilizer nullity is insensitive to the rotation angle, stabilizer Rényi entropies expose a richer structure in their dynamics. Our results uncover sharp distinctions between coarse and fine-grained nonstabilizerness diagnostics and demonstrate how measurements can both suppress and sustain quantum computational resources.
Determining the quantum-classical boundary between quantum circuits that can be efficiently simulated classically and those that cannot remains a fundamental question. One approach to classical simulation is to represent the output of a quantum circuit as a Clifford-augmented matrix product state (CAMPS), which, via a disentangling algorithm, decomposes the wave function into Clifford and MPS components, and from which Pauli expectation values can be computed in time polynomial in the MPS bond dimension. In this work, we develop an optimization-free disentangling (OFD) algorithm for Clifford circuits either doped with T gates or, equivalently, preceded by multiqubit gates of the form α I + β P . We give a simple and easily computed algebraic criterion that characterizes the individual quantum circuits for which OFD generates an efficient CAMPS—the bond dimension is exponential in the null space of a Galois binary field GF(2) matrix induced by a tableau of the twisted Pauli strings P . This significantly increases the number of circuits with rigorous polynomial-time classical simulations. We also give evidence that the typical N qubit random Clifford circuit doped with N uniformly distributed T gates of polylogarithmic depth or greater has a CAMPS with polynomial bond dimension. In addition, we compare OFD against disentangling by optimization. We further explore the representability of CAMPS for random Clifford circuits doped with more than N T gates. We also propose algorithms for sampling, probability, and amplitude estimation of bitstrings, and evaluation of entanglement Rényi entropy from CAMPS, which, though still having exponential complexity, are more efficient than standard MPS simulations. This work establishes a versatile framework for understanding classical simulatability of Clifford + T circuits and explores the interplay between quantum entanglement and quantum magic in quantum systems.
Encoding quantum information with low circuit overhead is a fundamental challenge in fault-tolerant quantum computation. Random circuits provide a natural mechanism for rapidly spreading logical information through simple gates applied in parallel. Brown and Fawzi showed that random Clifford circuits on two-qubit Clifford gates provide such encoders that achieve the quantum Gilbert-Varshamov rate-distance tradeoff with depth O(log3n). We show that the same asymptotic tradeoff is attained in optimal O(logn) depth under a gate distribution with a more restricted support. For every fixed δ>0 and sufficiently large n, if nk<1−H(nd)−ndlog23−δ, we can construct random circuits of depth O(logn) which define, with high probability, an [n,k] stabilizer code of distance at least d+1, which matches the Ω(logn) light-cone lower bound for linear distance encoders. Our ensemble employs a random matching circuit architecture consisting of T independent permutation-invariant layers. In each layer, the qubits are paired up by a uniformly random perfect matching, and a random independent two-qubit Clifford gate is applied to each pair. The gate distribution need not be uniform over, or even have full support on, the two-qubit Clifford group; rather, we allow for very general distributions on Clifford gates satisfying three regularity conditions. In particular, the construction can be implemented using n/2 CNOT gates on randomly matched pairs in each layer, with parallel one-qubit Clifford twirls. These regularity conditions allow us to reduce the second-moment dynamics of our random circuits to a reversible Markov chain on binary support strings. We establish logarithmic hitting-time bounds for this Markov chain and comparisons of its stationary distribution to prove the coding properties of the circuits.