The quantum PCP conjecture

Prove or refute that estimating a local Hamiltonian's ground energy to within a constant fraction of its number of terms is QMA-hard.

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1	# The quantum PCP conjecture
2
3	The PCP theorem says that for constraint satisfaction problems it is NP-hard even to approximate the largest number of simultaneously satisfiable constraints, up to an error of a constant fraction of all constraints. The quantum PCP conjecture asks the same of local Hamiltonians, the quantum analogue of constraint systems: is estimating their ground-state energy, up to an error proportional to the number of terms, still as hard as any problem with efficiently checkable quantum proofs?
4
5	## Problem
6
7	A $k$-local Hamiltonian on $n$ qudits of fixed dimension $q$ is a sum
8
9	$$H=\sum_{i=1}^{m}H_i,\qquad 0\preceq H_i\preceq I,$$
10
11	of $m=\mathrm{poly}(n)$ terms, each acting on at most $k$ qudits and specified by $\mathrm{poly}(n)$ bits. For thresholds $a<b$, the local Hamiltonian problem asks whether the ground-state energy $\lambda_{\min}(H)$ is at most $a$ or at least $b$, promised that one of the two holds.
12
13	$\mathsf{QMA}$ is the class of promise problems where a polynomial-time quantum verifier, given a quantum proof of polynomially many qubits, accepts some proof with probability at least $2/3$ on yes-instances and every proof with probability at most $1/3$ on no-instances.
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15	Conjecture. For some constants $k$, $q$ and $0\le\alpha<\beta\le1$, the local Hamiltonian problem with $a=\alpha m$ and $b=\beta m$ is $\mathsf{QMA}$-hard under quantum polynomial-time reductions.
16
17	## Background
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19	Kitaev proved in 1999 that the local Hamiltonian problem is $\mathsf{QMA}$-complete when $b-a\ge 1/\mathrm{poly}(n)$, a quantum Cook–Levin theorem [6]. The quantum PCP question was first raised by Aharonov and Naveh in 2002 and formulated precisely by Aharonov, Arad, Landau and Vazirani in 2009 [1, 6]. The form above is Conjecture 1.3 of the survey by Aharonov, Arad and Vidick, which also gives an equivalent form: a verifier that reads only $O(1)$ qubits of the proof [6].
20
21	## What is known
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23	- A quantum analogue of gap amplification, the first step of Dinur's proof of the PCP theorem, holds for Hamiltonians whose interaction graph is an expander [1].
24	- Brandão and Harrow showed that 2-local Hamiltonians of high degree, small expansion or low entanglement have product states of nearly optimal energy, which can be checked in $\mathsf{NP}$; Hamiltonians for a quantum PCP must avoid all three [2].
25	- If the conjecture holds and $\mathsf{QMA}\ne\mathsf{NP}$, the NLTS conjecture of Freedman and Hastings follows: some local Hamiltonians have no state within $cm$ of the ground energy, for a constant $c>0$, that a constant-depth circuit can prepare [6]. Anshu, Breuckmann and Nirkhe proved NLTS in 2023 using good quantum LDPC codes [3].
26	- Buhrman, Helsen and Weggemans gave quantum oracles relative to which some quantum PCP statements fail, so, as for the classical PCP theorem, a proof needs (quantumly) non-relativizing techniques. They also showed that a quantum PCP for $\mathsf{QMA}(k)$, for any $k\le\mathrm{poly}(n)$, would give $\mathsf{QMA}(2)=\mathsf{QMA}$ [4].
27	- Bergamaschi, Metger, Vidick and Zhang gave a derandomized tensor-product gap amplification for Hamiltonians, following Dinur's model with random walks on expander graphs (CCC 2026) [7].
28	- In October 2026, Bafna, Nguyen and Zhang gave the first template for locality-preserving gap amplification of local Hamiltonians and proved that it amplifies their combinatorial gap (to appear at FOCS 2026) [5].
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30	## What would count as progress
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32	- A proof: a quantum polynomial-time reduction from the inverse-polynomial-gap problem to a constant-relative-gap one. Or a refutation, for example placing the constant-gap problem in $\mathsf{NP}$, which contradicts the conjecture unless $\mathsf{QMA}=\mathsf{NP}$.
33	- A locality-preserving procedure that amplifies the energy gap of general local Hamiltonians, not only the combinatorial gap of [5].
34	- Placing the constant-gap problem in $\mathsf{QCMA}$, where proofs are classical; by [4] this would put every quantum PCP with a constant number of queries in $\mathsf{QCMA}$.
35
36	## References
37
38	1. D. Aharonov, I. Arad, Z. Landau and U. Vazirani, "The detectability lemma and quantum gap amplification", in Proceedings of the 41st Annual ACM Symposium on Theory of Computing (STOC), 417–426 (2009). [doi:10.1145/1536414.1536472](https://doi.org/10.1145/1536414.1536472)
39	2. F. G. S. L. Brandão and A. W. Harrow, "Product-state approximations to quantum ground states", in Proceedings of the 45th Annual ACM Symposium on Theory of Computing (STOC), 871–880 (2013). [doi:10.1145/2488608.2488719](https://doi.org/10.1145/2488608.2488719)
40	3. A. Anshu, N. P. Breuckmann and C. Nirkhe, "NLTS Hamiltonians from good quantum codes", in Proceedings of the 55th Annual ACM Symposium on Theory of Computing (STOC), 1090–1096 (2023). [doi:10.1145/3564246.3585114](https://doi.org/10.1145/3564246.3585114)
41	4. H. Buhrman, J. Helsen and J. Weggemans, "Quantum PCPs: on adaptivity, multiple provers and reductions to local Hamiltonians", Quantum 9, 1791 (2025). [doi:10.22331/q-2025-07-11-1791](https://doi.org/10.22331/q-2025-07-11-1791)
42	5. M. Bafna, Q. T. Nguyen and T. Zhang, "Gap amplification for local Hamiltonians with combinatorial soundness", preprint (2026), to appear at FOCS 2026. [arXiv:2610.04952](https://arxiv.org/abs/2610.04952)
43	6. D. Aharonov, I. Arad and T. Vidick, "The quantum PCP conjecture", ACM SIGACT News 44(2), 47–79 (2013). [doi:10.1145/2491533.2491549](https://doi.org/10.1145/2491533.2491549)
44	7. T. Bergamaschi, T. Metger, T. Vidick and T. Zhang, "Derandomised tensor product gap amplification for quantum Hamiltonians", in 41st Computational Complexity Conference (CCC 2026), LIPIcs, 15 (2026). [doi:10.4230/LIPIcs.CCC.2026.15](https://doi.org/10.4230/LIPIcs.CCC.2026.15)
45
46	## Source
47
48	Imported from [QIQCOP Zoo op_bca77ec42ddd1d5c](https://qiqc-op.com/problem/op_bca77ec42ddd1d5c/), curated by Bikun Li, Qicheng Tang, Changhao Li, Chengkai Zhu, Minbo Gao, Zhong-Xia Shang, Bin Cheng and Naixu Guo. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.