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.

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