Local random quantum circuits can occasionally produce unusually large output probabilities. However, the final output distribution containing such a peak does not by itself reveal how the peak formed during circuit evolution. We investigate whether states formed during circuit evolution carry structural signatures that predict subsequent peak formation. In numerical experiments, circuits selected for large terminal peaks exhibit stronger pairwise correlations and probability redistribution, together with lower bipartite entanglement and output entropy, than controls generated by the same random-circuit model. Their intermediate states retain higher fresh-peak event probability when the remaining gates are independently regenerated. Within an unconditioned ensemble, a fixed structural score computed from the initial circuit segment predicts new-peak probability. Averaged equally across n=8-16, a replay with equal numbers of remaining-circuit evaluations shows that score-based allocation yields 1.80 times as many peaks as uniform allocation and about 18% more than a fixed baseline based on probability concentration and redistribution. Randomizing relative phases while preserving computational-basis probabilities strongly attenuates the selected-state advantage, demonstrating a functional contribution from relative-phase organization. Controlled gate-selection tests further show that the identified structure can guide circuit construction to promote peak formation. Together, these results connect rare output events to intermediate-state structure with predictive and constructive value under finite-depth local dynamics.
Local random quantum circuits can occasionally produce unusually large output probabilities. However, the final output distribution containing such a peak does not by itself reveal how the peak formed during circuit evolution. We investigate whether states formed during circuit evolution carry structural signatures that predict subsequent peak formation. In numerical experiments, circuits selected for large terminal peaks exhibit stronger pairwise correlations and probability redistribution, together with lower bipartite entanglement and output entropy, than controls generated by the same random-circuit model. Their intermediate states retain higher fresh-peak event probability when the remaining gates are independently regenerated. Within an unconditioned ensemble, a fixed structural score computed from the initial circuit segment predicts new-peak probability. Averaged equally across n=8-16, a replay with equal numbers of remaining-circuit evaluations shows that score-based allocation yields 1.80 times as many peaks as uniform allocation and about 18% more than a fixed baseline based on probability concentration and redistribution. Randomizing relative phases while preserving computational-basis probabilities strongly attenuates the selected-state advantage, demonstrating a functional contribution from relative-phase organization. Controlled gate-selection tests further show that the identified structure can guide circuit construction to promote peak formation. Together, these results connect rare output events to intermediate-state structure with predictive and constructive value under finite-depth local dynamics.
An algorithm is proposed for constructing quasi-random "peaked" quantum circuits, i.e., circuits whose final qubit state exhibits a high probability concentration on a specific computational basis state. These circuits consist of random gates arranged in a brick-wall architecture. While the multiqubit state in the middle of the circuit can exhibit significant entanglement, the final state is, with high probability, a predetermined pure bitstring. A technique is introduced to obscure the final bitstring in the structure of the quantum circuit. The algorithm allows precise control over the probability of the final peaked state. A modified version of the algorithm enables the construction of double- or multi-peaked quantum circuits. The matrix product state (MPS) method is evaluated for simulating such circuits; it performs effectively for shallow peaked circuits but offers no significant advantage for deeper ones.
Over a decade after its proposal, the idea of using quantum computers to sample hard distributions has remained a key path to demonstrating quantum advantage. Yet a severe drawback remains: verification seems to require classical computation exponential in the system size, n. As an attempt to overcome this difficulty, we propose a new candidate for quantum advantage experiments with otherwise random "peaked circuits", i.e., quantum circuits whose outputs have high concentrations on a computational basis state. Naturally, the heavy output string can be used for classical verification. In this work, we analytically and numerically study an explicit model of peaked circuits, in which τr layers of uniformly random gates are augmented by τp layers of gates that are optimized to maximize peakedness. We show that getting 1/poly(n) peakedness from such circuits requires τp=Ω((τr/n)0.19) with overwhelming probability. However, we also give numerical evidence that nontrivial peakedness is possible in this model -- decaying exponentially with the number of qubits, but more than can be explained by any approximation where the output of a random quantum circuit is treated as a Haar-random state. This suggests that these peaked circuits have the potential for future verifiable quantum advantage experiments. Our work raises numerous open questions about random peaked circuits, including how to generate them efficiently, and whether they can be distinguished from fully random circuits in classical polynomial time.
Random circuit models often describe local dynamics using generic two-qubit gates, which have proven successful in capturing entanglement growth and operator spreading in many contexts. This approach naturally leads to the expectation that detailed gate structure plays only a limited role in coarse-grained entanglement and scrambling diagnostics. We show that the internal structure of multipartite circuit primitives can significantly influence these dynamical rates, even within a fixed random-circuit architecture. To investigate this, we study an exactly simulable family of Clifford quantum circuits built from fixed n-qubit graph-state preparation unitaries, which we treat as elementary building blocks. Specifically, we consider a one-dimensional chain of N qubits initialized in a product state and evolved by layers in which nonoverlapping length-n blocks are placed at uniformly random positions with sparsity α. We find that different choices of graph-state building blocks lead to strongly varying dynamical rates. Graph states that are inequivalent under local Clifford (LC) transformations generate sharply different entanglement velocities vE and butterfly velocities vB, even though the circuits are drawn from the same ensemble with identical architecture and randomness parameters. We further show that this hierarchy is captured by two complementary block-level characteristics: the distribution of entanglement across internal bipartitions of the graph state, which correlates with vE, and a graph-theoretic connectivity profile across bipartitions, which correlates with vB. Neither descriptor alone fully determines the dynamics; rather, entanglement growth and operator spreading are controlled by distinct structural features of the local circuit blocks. Notably, AME states appear among the fastest scrambling building blocks within the ensembles studied here.
Random quantum circuits have been extensively explored for quantum supremacy demonstrations [1]–[4]. However, verifying their output distributions remains challenging [2], [5], [6]. Here, we propose the infinite-temperature correlation function (ITCF) as a physically meaningful observable for noisy intermediate-scale quantum (NISQ) devices one that can be extracted using engineered circuits rather than relying on fully random constructions. This is realized by leveraging peaked quantum states whose probability distributions are sharply peaked at specific outcomes due to constructive interference thus offering more efficient verifiability and stronger signal observability. Rather than using Haar-random states, which often yield vanishing signals through destructive interference, we construct purposefully biased quantum states using either Grover-based amplitude amplification or shallow structured circuits. These engineered states amplify contributions from relevant operator subspaces, enabling robust detection of non-zero ITCF values that would otherwise be suppressed under random-state sampling. Our results highlight a problem-specific state preparation framework that mitigates signal loss from random averaging and facilitates the detection of physically meaningful observables in NISQ devices. We also discuss future extensions to multiqubit observables, scrambling diagnostics, and variational circuit optimization, underscoring the broader potential of Peaked States for quantum simulation and verification.
Random quantum circuits aim to efficiently reproduce the statistical properties of ideal random quantum evolution. One approach is to construct approximate unitary designs, which match the moments of Haar-random unitaries up to a prescribed order with controlled error. We establish quantitative guarantees for how accurately these designs reproduce the distribution of each output probability. We show that, for every strong ε-approximate unitary k-design on n qubits, the distribution of each output probability is within O(k−1/2(1+k/2n)−1/2+ε) in Kolmogorov distance of the Porter-Thomas distribution, a beta distribution with parameters 1 and 2n−1. This bound is optimal up to constant factors, implying that a substantially better uniform bound requires additional structure. We then show that a mild form of local invariance suffices. We prove that any such design that is invariant under unitary translations acting on a single qubit approximates Porter-Thomas to total variation distance O(k−1/2+ε). Every strong ε-approximate unitary k-design invariant on logk+O(1) qubits achieves error 2−Ω(k)+O(ε) in Kolmogorov distance and 2−Ω(k)+O(εlog(2/ε)) in total variation distance. These results identify the distributional accuracy guaranteed by moment matching and the structure that improves it. As an application, we show that the Shannon entropy of the output distribution of a strong ε-approximate k-design is concentrated around its Haar value n−Θ(1): its mean differs from the Haar mean by O(k−1/4+ε1/2) and its standard deviation is O(2−n/2+k−1/8+ε1/4). Consequently, a trusted device sampling such circuits generates n−O(1) bits of entropy per sample, which makes them useful for random number generation.
With the accelerating development of quantum technologies and their growing computational potential, quantum systems are being adapted for simulations and other critical tasks across diverse domains, making the reliability of the corresponding quantum software an essential concern. Although recent efforts have started to incorporate quantum-specific properties such as magnitude, phase, and entanglement under the form of input-coverage criteria into software testing, the unique structure of the quantum state space demands for more comprehensive testing. In particular, the notion of complete state-space exploration has so far received little attention. To address this gap, we propose a framework for evaluating test circuit generators with respect to their coverage of the quantum state space. Our contribution is threefold: we develop a set of diversity scores that capture both local and global indicators of the extent to which the state space is explored; we propose a test circuit generator that produces test input states via a Brick-Circuit (BC) construction designed to approximate ideal random states using hardware-compatible gates; we compare the proposed construction with existing generators based on their ability to generate uniformly distributed random test input states. Our extended diversity scores quantify the local correlations and global spread of magnitude, phase and entanglement. Using these scores, we evaluate the expressibility, defined as the capability to span the quantum state space uniformly, and entangling capabilities of existing generators relative to the BC generator. Our results show that the hardware-compatible BC generator achieves higher expressibility and entanglement performance at shallower depths than existing circuit generators.