Hardness of approximation for quantum problems

The polynomial hierarchy plays a central role in classical complexity theory. Here, we define a quantum generalization of the polynomial hierarchy, and initiate its study. We show that not only are there natural complete problems for the second level of this quantum hierarchy, but that these problems are in fact hard to approximate. Using these techniques, we also obtain hardness of approximation for the class QCMA. Our approach is based on the use of dispersers, and is inspired by the classical results of Umans regarding hardness of approximation for the second level of the classical polynomial hierarchy [Umans, FOCS 1999]. The problems for which we prove hardness of approximation for include, among others, a quantum version of the Succinct Set Cover problem, and a variant of the local Hamiltonian problem with hybrid classical-quantum ground states.

💡 Research Summary

The paper introduces a quantum analogue of the classical polynomial hierarchy (PH), called the Quantum Polynomial Hierarchy (QPH), and investigates its structural and hardness properties. QPH is defined by alternating existential and universal quantifiers over quantum witnesses, yielding classes QΣ_i and QΠ_i for each level i. The first level coincides with QMA (QΣ_1) and QCMA (QΠ_1), while the second level (QΣ_2 and QΠ_2) captures more intricate interactions between two quantum provers.

The authors present two natural QΣ_2‑complete and QΠ_2‑complete problems. The first, Quantum Succinct Set Cover (QSSC), is a quantum version of the classic Succinct Set Cover: an input circuit encodes a family of subsets, and the question is whether there exists a collection of at most k quantum states (each a hybrid of classical bits and qubits) that “covers” every element. The second, Hybrid Local Hamiltonian (HLH), modifies the k‑local Hamiltonian problem by requiring the ground state to be a classical‑quantum hybrid and asks whether its energy falls below a given threshold. Both problems are shown to be complete for their respective second‑level quantum classes.

To prove hardness of approximation, the paper adapts Umans’ disperser‑based technique from the classical Σ₂^P setting. A disperser is a bipartite graph that spreads any small subset of left vertices across a large fraction of right vertices. The authors construct a gap‑preserving reduction from a classical Σ₂^P‑complete instance (such as Succinct Set Cover) to a QΣ_2 instance by first encoding the instance with a disperser and then translating it into a quantum verification circuit. In the “YES” case, there exists a quantum witness that makes the verifier accept with probability at least 2/3; in the “NO” case, every quantum witness leads to acceptance probability at most 1/3. Because the reduction preserves the acceptance gap, any algorithm that approximates the optimum within a constant factor (e.g., 1/2) would solve the underlying Σ₂^P problem, implying NP‑hardness of approximation for QSSC and HLH.

The paper further extends these results to QCMA, the class of problems verifiable by a quantum computer given a classical witness. By restricting the existential quantum witness in the QΣ_2 reduction to a classical string while keeping the verifier quantum, the authors obtain QCMA‑complete problems that inherit the same approximation gap. Consequently, even for QCMA, achieving any non‑trivial approximation ratio for the derived problems is NP‑hard, strengthening known QCMA‑completeness results.

Key technical contributions include:

1. A formal definition of QPH and an analysis of its basic properties.
2. Identification of natural, expressive complete problems for the second level of QPH.
3. A novel application of disperser‑based reductions to quantum verification circuits, establishing constant‑factor hardness of approximation.
4. An extension of the hardness results to QCMA, demonstrating that approximation remains difficult even when the witness is classical.

The authors discuss several avenues for future work. One direction is to explore higher levels of QPH (QΣ_3, QΠ_3, etc.) and to seek complete problems with similar approximation gaps. Another is to investigate alternative combinatorial constructions—such as expanders or randomness extractors—to build more efficient or stronger reductions. Finally, the paper hints at a possible connection between quantum hardness of approximation and a quantum PCP theorem, suggesting that the techniques introduced could serve as building blocks toward such a profound result.