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arXiv:2510.00132

Complexity and hardness of random peaked circuits

Yuxuan Zhang

Published Sep 30, 2025• 0 citations
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Abstract

Near-term feasibility, classical hardness, and verifiability are the three requirements for demonstrating quantum advantage; most existing quantum advantage proposals achieve at most two. A promising candidate recently proposed is through randomly generated peaked circuits. In this work, we study an explicit construction for random peaked circuits: first selecting a random circuit $C$ of polynomial size, which forms a $k$-design. Subsequently, a second random circuit $C'$ is chosen from the same architecture, subject to a postselection criterion: $C'$ must exhibit a high overlap with $C$ in one of their rows. Utilizing unitary design properties, we demonstrate that the circuits generated by this method are non-trivial; specifically, $C'$ is provably far from $C^\dagger$. Indeed, with overwhelmingly high probability, a random peaked circuit generated this way is non-compressible and is of circuit complexity $\tilde Ω(nk)$. This resolves an open problem posed by Aaronson in 2022. Secondly, we analytically establish that estimating the peakedness of a random peaked circuit to within a $2^{-\text{poly}(n)}$ additive error, is average-case \#P-hard. When the additive error is relaxed to $1/\text{poly}(n)$, we note that the worst-case scenario for this problem is BQP-complete. Under widely accepted assumptions on random quantum circuits, we identify a regime where no classical polynomial-time sequential simulator attains inverse-polynomial additive accuracy on the peak on a non-negligible fraction of instances. Thirdly, we study using peaked circuits as a practical attempt for a verifiable quantum advantage protocol. While the postselection method for generating peaked circuits could be costly, we demonstrate that numerical search for $C'$ with randomized initialization successfully returns a random peaked circuit, achieving the properties as theoretically predicted.

DOI
10.48550/arxiv.2510.00132

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Near-term feasibility, classical hardness, and verifiability are the three requirements for demonstrating quantum advantage; most existing quantum advantage proposals achieve at most two. A promising candidate recently proposed is through randomly generated peaked circuits. In this work, we study an explicit construction for random peaked circuits: first selecting a random circuit CCC of polynomial size, which forms a kkk-design. Subsequently, a second random circuit C′C'C′ is chosen from the same architecture, subject to a postselection criterion: C′C'C′ must exhibit a high overlap with CCC in one of their rows. Utilizing unitary design properties, we demonstrate that the circuits generated by this method are non-trivial; specifically, C′C'C′ is provably far from C†C^\daggerC†. Indeed, with overwhelmingly high probability, a random peaked circuit generated this way is non-compressible and is of circuit complexity Ω~(nk)\tilde Ω(nk)Ω~(nk). This resolves an open problem posed by Aaronson in 2022. Secondly, we analytically establish that estimating the peakedness of a random peaked circuit to within a 2−poly(n)2^{-\text{poly}(n)}2−poly(n) additive error, is average-case #P-hard. When the additive error is relaxed to 1/poly(n)1/\text{poly}(n)1/poly(n), we note that the worst-case scenario for this problem is BQP-complete. Under widely accepted assumptions on random quantum circuits, we identify a regime where no classical polynomial-time sequential simulator attains inverse-polynomial additive accuracy on the peak on a non-negligible fraction of instances. Thirdly, we study using peaked circuits as a practical attempt for a verifiable quantum advantage protocol. While the postselection method for generating peaked circuits could be costly, we demonstrate that numerical search for C′C'C′ with randomized initialization successfully returns a random peaked circuit, achieving the properties as theoretically predicted.
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, nnn. 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 τrτ_rτr​ layers of uniformly random gates are augmented by τpτ_pτp​ layers of gates that are optimized to maximize peakedness. We show that getting 1/poly(n)1/\text{poly}(n)1/poly(n) peakedness from such circuits requires τp=Ω((τr/n)0.19)τ_{p} = Ω((τ_r/n)^{0.19})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.
Peaked circuits are random quantum circuits whose measurement returns one bitstring far more often than chance. The probability of that string is the peakedness, and a verifier who knows the string checks the device in a few shots. Peaked circuits are a candidate route to verifiable quantum advantage, and classical generation is the bottleneck. Aaronson and Zhang train a variational circuit by gradient descent, and the peakedness they reach decays exponentially with the number of qubits. They state that either their optimizer stalls or no efficient generation method exists. Here we show that no bare fixed base fits the decay we measure and that neither the barren plateau nor fragmentation of the solution set explains it. We measure the optimization landscape on eighteen instances per size, from eight to sixteen qubits, under a protocol whose falsifiers were fixed in advance. The decay steepens as the qubit number grows, leaving extrapolations to larger devices unsupported. A better optimizer exists, but it gains a few percent and the peakedness it reaches decays at nearly the same rate. Aaronson and Zhang attribute the difficulty to a barren plateau. The plateau is present, but the second-order amplitude data are depth-independent while the attained peakedness is not. Independent runs land on uncorrelated solutions, yet paths connecting them stay well above the scale of a random state, so fragmentation fails at that scale. In the deep limit we take the scrambled state Haar-random. There we prove that no search over a polynomial-parameter Lipschitz family, exhaustive search included, beats the scale of a random state by more than a poly(n) factor on average. Both branches of the Aaronson-Zhang alternative therefore stay open. A generation method must beat a preregistered baseline, and a hardness argument must accommodate a solution set connected at the scale of a random state.
Random circuit sampling (RCS) is a leading candidate for demonstrating quantum advantage, supported by strong complexity-theoretic evidence of hardness in the ideal setting and by rapid experimental progress to date. In practice, however, noise is unavoidable, and a central problem is to identify the noise-strength boundary between classically simulable and classically hard regimes. In this work, we establish an architecture-general hardness bound for this boundary for the standard local depolarizing noise of strength γγγ. Assuming the standard average-case #P-hardness conjecture for ideal RCS, we show that, for any circuit architecture satisfying this conjecture, noisy RCS on the same architecture remains hard to simulate classically within any inverse-polynomial total variation distance whenever γ=O(log⁡n/(nd))γ=O(\log n/(nd))γ=O(logn/(nd)) for nnn-qubit circuits of depth ddd, unless the polynomial hierarchy collapses. Crucially, noisy-RCS hardness follows without any additional conjectural or architecture-specific assumption beyond those already entering the ideal-RCS hardness framework. Our proof combines a low-degree polynomial extrapolation with a monotonicity reduction showing that efficient classical simulation at one depolarizing noise strength implies efficient simulation at every larger strength. Together, these ingredients transfer the standard ideal-RCS hardness conjecture to sampling hardness at a prespecified noise strength. Finally, combining the convergence-to-uniformity result of Dalzell et al. [Commun. Math. Phys. 405, 78 (2024)] with our monotonicity reduction yields efficient classical simulation for γ=ω(log⁡n/(nd))γ=ω(\log n/(nd))γ=ω(logn/(nd)) on layered, regularly connected architectures. Thus, wherever the two architectural settings overlap, this identifies γ=Θ(log⁡n/(nd))γ=Θ(\log n/(nd))γ=Θ(logn/(nd)) as the asymptotic complexity-transition scale.
Motivated by the recent experimental demonstrations of quantum supremacy, proving the hardness of the output of random quantum circuits is an imperative near term goal. We prove under the complexity theoretical assumption of the non-collapse of the polynomial hierarchy that approximating the output probabilities of random quantum circuits to withinexp⁡(−Ω(mlog⁡m))\exp(-\Omega(m\log m))exp(−Ω(mlogm))additive error is hard for any classical computer, wheremmmis the number of gates in the quantum computation. More precisely, we show that the above problem is #P-hard under BPPNPreduction. In the recent experiments, the quantum circuit has n-qubits and the architecture is a two-dimensional grid of sizen×n\sqrt{n}\times\sqrt{n}n​×n​[1]. Indeed for constant depth circuits approximating the output probabilities to within2−Ω(nlog⁡n)2^{-\Omega(n\log n)}2−Ω(nlogn)is hard. For circuits of depthlog⁡n\log nlognorn\sqrt{n}n​for which the anti-concentration property holds, approximating the output probabilities to within2−Ω(nlog⁡2n)2^{-\Omega(n\log^{2}n)}2−Ω(nlog2n)and2−Ω(n3/2log⁡n)2^{-\Omega(n^{3/2}\log n)}2−Ω(n3/2logn)is hard respectively. We then show that the hardness results extend to any open neighborhood of an arbitrary (fixed) circuit including the trivial circuit with identity gates. We made an effort to find the best proofs and proved these results from first principles, which do not use the standard techniques such as the Berlekamp–Welch algorithm, the usual Paturi's lemma, and Rakhmanov's result.
We construct protocols for classically verifiable quantum advantage and classical verification of BQP\mathsf{BQP}BQP computations using \emph{quantum indistinguishability obfuscation} (qiO). Specifically, given qiO and assuming a slightly stronger version of BQP≠BPP\mathsf{BQP}\neq\mathsf{BPP}BQP=BPP, we construct a two-message quantum-advantage protocol that is efficiently and publicly verifiable. Our result can be viewed as a rigorous cryptographic foundation for the heuristic quantum advantage proposals based on \emph{peaked random circuit sampling} of Aaronson and Zhang (arXiv:2404.14493). We also construct two simple protocols for classically verifying arbitrary BQP\mathsf{BQP}BQP computations. The first protocol is privately verifiable and assumes only the existence of qiO. This gives a rare example of a nontrivial cryptographic application of (quantum) iO that does not make additional computational hardness assumptions. The second protocol additionally assumes post-quantum one-way functions and is \emph{publicly verifiable}. To our knowledge, this is the first publicly verifiable protocol for classical verification of BQP\mathsf{BQP}BQP computations under computational assumptions in the standard model. As qiO is the basis for our results, we propose an approach to constructing qiO by adapting the local-mixing framework of Canetti--Chamon--Mucciolo--Ruckenstein.
Motivated by the recent experimental demonstrations of quantum supremacy, proving the hardness of the output of random quantum circuits is an imperative near term goal. We prove under the complexity theoretical assumption of the non-collapse of the polynomial hierarchy that approximating the output probabilities of random quantum circuits to within exp⁡(−Ω(mlog⁡m))\exp(-\Omega(m\log m))exp(−Ω(mlogm)) additive error is hard for any classical computer, where mmm is the number of gates in the quantum computation. More precisely, we show that the above problem is #P\#\mathsf{P}#P-hard under BPPNP\mathsf{BPP}^{\mathsf{NP}}BPPNP reduction. In the recent experiments, the quantum circuit has nnn-qubits and the architecture is a two-dimensional grid of size n×n\sqrt{n}\times\sqrt{n}n​×n​. Indeed for constant depth circuits approximating the output probabilities to within 2−Ω(nlog⁡n)2^{-\Omega(n\log{n})}2−Ω(nlogn) is hard. For circuits of depth log⁡n\log{n}logn or n\sqrt{n}n​ for which the anti-concentration property holds, approximating the output probabilities to within 2−Ω(nlog⁡2n)2^{-\Omega(n\log^2{n})}2−Ω(nlog2n) and 2−Ω(n3/2log⁡n)2^{-\Omega(n^{3/2}\log n)}2−Ω(n3/2logn) is hard respectively. We made an effort to find the best proofs and proved these results from first principles, which do not use the standard techniques such as the Berlekamp--Welch algorithm, the usual Paturi's lemma, and Rakhmanov's result.