Abstract

We introduce a simple sub-universal quantum computing model, which we call the Hadamard-classical circuit with one-qubit (HC1Q) model. It consists of a classical reversible circuit sandwiched by two layers of Hadamard gates, and therefore it is in the second level of the Fourier hierarchy. We show that output probability distributions of the HC1Q model cannot be classically efficiently sampled within a multiplicative error unless the polynomial-time hierarchy collapses to the second level. The proof technique is different from those used for previous sub-universal models, such as IQP, Boson Sampling, and DQC1, and therefore the technique itself might be useful for finding other sub-universal models that are hard to classically simulate. We also study the classical verification of quantum computing in the second level of the Fourier hierarchy. To this end, we define a promise problem, which we call the probability distribution distinguishability with maximum norm (PDD-Max). It is a promise problem to decide whether output probability distributions of two quantum circuits are far apart or close. We show that PDD-Max is BQP-complete, but if the two circuits are restricted to some types in the second level of the Fourier hierarchy, such as the HC1Q model or the IQP model, PDD-Max has a Merlin-Arthur system with quantum polynomial-time Merlin and classical probabilistic polynomial-time Arthur.

Highlights

  • It consists of a classical reversible circuit sandwiched by two layers of Hadamard gates, and it is in the second level of the Fourier hierarchy [1]

  • We show that output probability distributions of the HC1Q model cannot be classically efficiently sampled within a multiplicative error unless the polynomial-time hierarchy collapses to the second level

  • We show that Probability Distribution Distinguishability with Maximum Norm (PDD-Max) is BQP-complete, but if the two circuits are restricted to some types in the second level of the Fourier hierarchy, such as the HC1Q model or the instantaneous quantum polynomialtime (IQP) model, PDD-Max has a Merlin-Arthur system with quantum polynomial-time Merlin and classical probabilistic polynomial-time Arthur

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Summary

Quantum supremacy of the HC1Q model

The kth level of the Fourier hierarchy, FHk, is the class of quantum circuits with k layers of Hadamard gates and all other gates preserving the computational basis. 2. The unitary H⊗N DH⊗N is applied. The IQP model is a well-known example of sub-universal quantum computing models whose output probability distributions cannot be classically efficiently sampled unless the polynomial-time hierarchy collapses. Other sub-universal models that exhibit similar quantum supremacy are known, such as the depth four model [6], the Boson sampling model [7], the DQC1 model [8,9,10,11,12], the Fourier sampling model [13], the conjugated Clifford model [14], and the random circuit model [15].

A polynomial-time uniformly-generated classical reversible circuit
Preliminary
PROOF OF THEOREM 3
H XWH 1 n
PROOF OF THEOREM 5
PROOF OF THEOREM 6
BQP-hardness
In BQP
Calculate pz
PROOF OF THEOREM 13
Generalizations to other circuits

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