[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81895-en":3,"doc-seo-81895-105":31,"detail-sidebar-cat-0-en-105":93},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},81895,8796095462418,"Noah","https://ap-avatar.wpscdn.com/avatar/80000253c1241d02b47?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778826106357471780",8,"Research & Report","Frozen-Tree Sampling Refutes Quantum Advantage of Random Circuit Sampling","Random circuit sampling of bitstrings from a Haar-random quantum state is commonly used to argue classical intractability and demonstrate quantum advantage. This work proposes an efficient classical frozen-tree sampling algorithm that leverages the conditional scale invariance of Haar-random quantum states. The sampler generates n-qubit bitstrings with O(n) time per sample, and its output distribution matches that of random quantum circuits because both correspond to independent realizations of the same Dirichlet distribution. Therefore, sample-only statistical tests cannot distinguish them, undermining the claimed advantage.","arXiv :2607 .04054v1 [ quant-ph] 4 Jul 2026  \nFrozen-Tree Sampling Refutes Quantum Advantage of Random Circuit Sampling  \nSangchul Oh∗  \nSchool of Physics and Applied Physics, Southern Illinois University Carbondale, IL 62906, USA  \n(Dated: July 7, 2026)  \nRandom circuit sampling of bitstrings from a Haar-random quantum state is widely believed tobe classically intractable, and has therefore been implemented as a primary benchmark for demonstrating quantum advantage. Here, we challenge this premise by proposing an efficient classical frozen-tree sampling algorithm that exploits the conditional scale invariance of Haar-random quantum states [Oh, arXiv:2602.19448] . The frozen-tree sampler draws bitstrings of n qubits in O (n) time per sample. Moreover, its output probability pF (x) is statistically identical to the probability pC (x) of a random quantum circuit, since both are independent instances of the same Dirichlet distribution. Consequently, no statistical test acting on samples alone can distinguish the classical frozen-tree sampler from a quantum random circuit. The claimed quantum advantage of random circuit sampling therefore does not withstand scrutiny: its hardness lies not in sampling from the Dirichlet distribution, which is classically efficient, but in identifying a specific circuit realization.  \nIntroduction — Quantum advantage, the outperformance of quantum computers over classical digital computers on certain tasks, is considered one of the most important milestones in quantum computation. Random circuit sampling (RCS) is regarded as a primary benchmark for demonstrating quantum advantage on current noisy intermediate-scale quantum computers. Claims of quantum advantage in RCS have recently been reported using superconducting qubits [1–5] and ion-trap qubits [6–8] . Operationally, RCS is the task of sampling bitstrings from a Haar-random quantum state, hereafter referred to as a random quantum state, generated by a random quantum circuit [9] . RCS is believed to be classically intractable because a random quantum state is highly entangled [10] and appears too chaotic for a classical algorithm to exploit any pattern or structure [11– 14] . Statistical properties of output bitstrings such as the exponential distribution, the linear cross-entropy benchmark [1, 9 , 15 , 16], heavy output generation [17] and anticoncentration [18–20] have been proposed as evidence for the quantum advantage of RCS.  \nIn this paper, we challenge the premise of quantum advantage in RCS by introducing frozen-tree sampling, which exploits the exact conditional scale invariance of a random quantum state [21, 22], as shown in Fig. 1. We prove that the probability of finding bitstrings of a random quantum state can be represented by a binary tree with a precise recursive structure: each bit is drawn from a Beta-distributed conditional probability determined by the preceding bits. We present a frozen-tree sampler that classically samples bitstrings from an n-qubit random quantum state in O (n) time per sample. Both the probability pC (x) of a random quantum circuit and the probability pF (x) of the frozen-tree sampler are independent realizations of the Dirichlet vector characterizing a random quantum state, and are therefore statistically identical. It follows that no statistical verification method can serve as evidence of quantum advantage in RCS.  \nn  \npF (x) = |⟨x|ψ⟩| 2 = Y R1xxkk 􀀀1 − Rx \u003Ck 􀀁 x k  \nk=1  \npF (x) ∼ Beta(1, N) , N = 2n − 1 Ru ∼ Beta(K, K) , K = 2 n−|u| − 1  \nFigure 1 . Binary-tree representation of the probability p(x) =|⟨x|ψ⟩|2 = Qk R1xxkk (1 − Rx\u003Ck)xk of finding a bitstring x = x1 ··· xn for a random quantum state |ψ⟩ of n qubits. The leaf probability p (x) is uniquely determined by the product of branch ratios Ru at depth d = |u| = k − 1 with prefix u = x \u003Ck = x1 ··· xk−1, where Ru ∼ Beta(K, K) with K = 2n−|u|−1 . Each subtree is conditionally scale invariant and statistically identical to the whole system [21] .  \n","cbCaicJE02j2Y8k3","https://ap.wps.com/l/cbCaicJE02j2Y8k3","pdf",894880,6,1,7,"English","en",105,"# Introduction\n## Random circuit sampling as a quantum-advantage benchmark\n## Frozen-tree sampling and conditional scale invariance\n# Dirichlet distribution of a Haar-random quantum state\n## Probability and marginal distributions\n# Binary-tree representation of output probabilities","[{\"question\":\"What does the document claim about classical difficulty in random circuit sampling?\",\"answer\":\"It claims that the classical intractability premise does not hold when sample-only statistics are used, because a frozen-tree classical sampler produces bitstrings with the same distribution as a random quantum circuit.\"},{\"question\":\"How does the proposed frozen-tree sampler generate n-qubit samples?\",\"answer\":\"It exploits the conditional scale invariance of Haar-random quantum states to build a binary-tree sampling procedure, drawing each bit from Beta-distributed conditional probabilities, with O(n) time per sample.\"},{\"question\":\"Why can’t statistical tests on samples alone distinguish the classical and quantum samplers?\",\"answer\":\"Because the probability distributions produced by the frozen-tree sampler and by random quantum circuits are statistically identical: both are independent realizations of the same Dirichlet distribution, making sample-only tests unable to tell them apart.\"}]","Frozen-Tree Sampling Refutes Quantum Advantage of Random Circuit Sampling | 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does the document claim about classical difficulty in random circuit sampling?","Question",{"text":77,"@type":78},"It claims that the classical intractability premise does not hold when sample-only statistics are used, because a frozen-tree classical sampler produces bitstrings with the same distribution as a random quantum circuit.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"How does the proposed frozen-tree sampler generate n-qubit samples?",{"text":82,"@type":78},"It exploits the conditional scale invariance of Haar-random quantum states to build a binary-tree sampling procedure, drawing each bit from Beta-distributed conditional probabilities, with O(n) time per sample.",{"name":84,"@type":75,"acceptedAnswer":85},"Why can’t statistical tests on samples alone distinguish the classical and quantum samplers?",{"text":86,"@type":78},"Because the probability distributions produced by the frozen-tree sampler and by random quantum circuits are statistically identical: both are independent realizations of the same Dirichlet distribution, making sample-only tests unable to tell them apart.","https://schema.org",{"og:url":53,"og:type":89,"og:title":13,"og:site_name":60,"og:description":14},"article",{"robots":91,"canonical":53},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":94},[95,99,103,107,112,116,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":96,"show_sort_weight":97,"slug":98},"Story & 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