A key issue with existing quantum advantage experiments is that their verification requires exponential classical time. In this work, we address this challenge by designing a new proposal---Hidden Code Sampling---with efficient classical verification. We use properties of quantum error correction to build an experiment that is "conditionally peaked": conditioned on a subset of qubits, the distribution on another subset is peaked. We give evidence for the classical intractability of this protocol by showing complexity-theoretic hardness of classical simulation, putting our scheme on par with other quantum advantage schemes. A major hurdle in instantiating the scheme concerns distinguishing between two noise channels, one involving local coherent noise and the other involving local Pauli noise. We identify algebraic properties of the underlying codes that enable an efficient distinguisher and construct an explicit code family satisfying these properties while preserving the hardness guarantees. We provide further evidence for soundness of our verification tests by proving an exponential query lower bound for classical algorithms that pass our verification tests in a black-box model.
A central question in the theory of quantum advantage is whether there are quantum advantage protocols with similar resource requirements as random circuit sampling that are also verifiable just from the classical outputs of the quantum computation. Here, we develop the idea of simulation secrets for verifiable advantage. A verifier can use a simulation secret to evaluate a cross-entropy test faster than it would take a classical adversary to pass the test. We instantiate this idea using IQP circuits described by cubic polynomials with planted independent spaces. These correspond to the largest independent set in the orbit of a polynomial under the general linear group and yield a low-rank stabilizer decomposition of the corresponding state. We conjecture that large independent spaces are invisible to a computationally bounded adversary, and therefore they cannot exploit them to pass the protocol. A second conjecture regards the fine-grained complexity of producing samples that pass the cross-entropy test for uniformly random polynomials. Under these conjectures, our scheme results in a polynomial gap between the verification time and the time a classical adversary would need to pass the protocol---both are exponential. It has a potential application to generating classically certifiable randomness, since the output distributions have high min-entropy. We estimate that the planted polynomial scheme is implementable using 100 logical qubits at logical error rates around
10−6.
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.
Peaked quantum circuits, whose output distribution is sharply concentrated on a single bitstring, have emerged as a promising candidate for verifiable quantum advantage, as the correctness of the quantum output can be checked by simply comparing against the known peak. Recent work by Gharibyan et al. arXiv:2510.25838 claimed heuristic quantum advantage using peaked circuits executed on Quantinuum's 56-qubit H2 processor. These peaked circuits concentrate their output on a single hidden bitstring by training a shallow simulable circuit variationally and inserting an obfuscated permutation to increase the depth to a level that makes classical simulation intractable, with estimated runtimes of years for the largest instances. We show that these circuits can be efficiently simulated classically. We describe a method that efficiently performs a full tensor network contraction, allowing near-exact sampling and extraction of the peaked bitstring. The method exploits the mirrored structure of the circuit and iteratively cancels both halves into a Matrix Product Operator (MPO), and avoids the obfuscated permutation by greedily reducing the MPO bond dimension through a process we call unswapping. The method can fully contract and extract the peak of the largest circuit in approximately one hour on a single GPU, around half the time it took to run on the quantum hardware.
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†. Indeed, with overwhelmingly high probability, a random peaked circuit generated this way is non-compressible and is of circuit complexity
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) additive error, is average-case \#P-hard. When the additive error is relaxed to
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′ with randomized initialization successfully returns a random peaked circuit, achieving the properties as theoretically predicted.
Demonstrating quantum advantage in a manner that can be independently verified by classical means remains one of the most pressing open problems in quantum computing. Here we propose a new heuristic approach for constructing quantum circuits that are hard to classically simulate yet whose output distributions can be efficiently verified classically. Our approach is based on Clifford circuit obfuscation. This obfuscation scheme hides the Clifford structure and injects a controlled and rapidly growing non-stabilizer resources that resist known classical attack strategies including reverse engineering and direct simulation. Our protocol is heuristic but is supported by both numerical and theoretical evidence. This work provides a new avenue toward classically verifiable quantum advantage that avoids the stringent implementation requirements of known approaches.