Probabilistically Checking Quantum Proofs, with Interaction

Abstract: The model of interactive oracle proofs (IOP) generalizes the notion of probabilistically checkable proof (PCP), in which a static proof is verified probabilistically by querying a small number of bits, to the interactive setting: a polynomial-time verifier interacts with an unbounded prover, but is restricted to only reading a small number of bits, in total, from the messages sent by the prover. IOPs provide a relaxed setting in which to study local probabilistic verification. They have proved instrumental in devising efficient methods for verification through subsequent compilation into non-interactive or succinct protocols. We study a quantum analogue of interactive oracle proofs (qIOP) in which the verifier and communication are both allowed to be quantum; yet the verifier is restricted to perform measurements only on a small number of qubits received from the prover. Our main result is a qIOP for any language in QMA, in which the total communication is polynomial but the verifier only reads a polylogarithmic number of qubits in total. The protocol has completeness parameter exponentially close to $1$ and soundness bounded away from $1$ by a constant. In the absence of a quantum PCP theorem, this provides the first information-theoretically sound local and robust characterization of QMA, albeit interactive. Our protocol combines the use of a quantum locally testable code (LTC) with classical techniques, notably probabilistically checkable proofs of proximity (PCPP). We avoid the necessity for complex multi-qubit tests employed in other settings by leveraging the local indistinguishability property of the quantum LTC.

• The paper establishes that every QMA language has an information-theoretically sound, robust quantum interactive proof with polynomial communication and only polylogarithmic verifier queries, using amplified dual-basis Hamiltonians.
• The protocol encodes witnesses in quantum locally testable CSS codes and combines measurement-extraction tests with PCPPs to verify logical Hamiltonian constraints against a single potentially entangled quantum prover.
• The result shows that interaction overcomes the global-information barrier blocking static quantum PCPs, while constant query complexity remains dependent on improved quantum locally testable code parameters.

Overview and main result

This paper by Sun and Vidick establishes the first information-theoretically sound, local, and robust characterization of QMA via a quantum analogue of interactive oracle proofs. The central theorem states that every language in QMA admits a quantum interactive oracle proof (qIOP) — more precisely, a quantum interactive probabilistically checkable proof (qIPCP) system — in which total communication is polynomial but the verifier reads only a polylogarithmic number of qubits across the entire interaction:

$$\mathrm{QMA} \subseteq \mathrm{QIPCP}(O(\mathrm{poly}),\ O(\mathrm{polylog}),\ O(\mathrm{polylog}))$$

with completeness exponentially close to 1, constant soundness gap after sequential repetition, and an honest prover that runs in quantum polynomial time given a valid witness (2606.09588). In the absence of a quantum PCP theorem — open for over two decades — this is significant: it shows that interaction plus quantum communication suffices to overcome the fundamental obstruction that entanglement encodes information globally, inaccessible to local probes of a static proof.

The model considered is deliberately restrictive. The prover first sends an $M$-qubit register $W$ (the encoded witness). In each subsequent round, the verifier may measure up to $q_i$ qubits from the prover's messages, send back a deterministically pre-chosen subset $M_i$ of the original witness register, and send a classical message; the prover responds with classical bits. Once the verifier relinquishes part of the witness to the prover, that portion is permanently lost — a malicious prover need not return it. The soundness condition must hold even against provers whose initial state is entangled with the witness register.

The starting point: amplified dual-basis Hamiltonians

The protocol reduces from a tailored QMA-complete problem rather than the standard local Hamiltonian problem. The authors use the "dual-basis projector Hamiltonian" problem $\mathcal{DLH}$: instances are Hamiltonians $H = \frac{1}{m}\sum_i H_i$ where each term is a projection diagonal either in the computational ($Z$) or Hadamard ($X$) basis. Building on Ma and Natarajan's proof that two bases suffice for $\mathrm{QMA}_1$-completeness (Ma et al., 29 Sep 2025), they establish that $M$0 is QMA-complete with completeness exponentially close to zero (as minimum eigenvalue) and inverse-polynomial soundness.

A key technical contribution is amplifying this promise gap while preserving the two-bases structure. Standard tensoring would produce mixed terms involving both bases, which the protocol cannot handle. Instead, the authors apply the layer-wise amplification of Bergamaschi–Metger–Vidick–Zhang (Bergamaschi et al., 1 Oct 2025): since all $M$1-terms commute among themselves (and likewise for $M$2), the Hamiltonian splits into two commuting layers, each amplified independently by repeating its tests sequentially $M$3 times within the amplified term. The resulting Hamiltonian $M$4 has minimum eigenvalue at most $M$5 on YES instances and at least a universal constant $M$6 on NO instances. A consequence worth noting: the amplified terms are non-local averages over exponentially many local projections, though each can be efficiently sampled and described — this is why the final classical message in the protocol has polynomial length (reducible to logarithmic using derandomized amplification, a modification the authors defer).

Verifying measurement outcomes via local indistinguishability

The core difficulty is certifying that the prover honestly measures the witness in the requested basis. Existing approaches rely on multi-qubit tests with two entangled provers, computational assumptions (Mahadev-style trapdoor claw-free functions), or compilation via quantum fully homomorphic encryption. Here there is a single prover, no computational assumptions, but quantum communication is available.

The construction exploits local indistinguishability of quantum error-correcting codes: if the witness is encoded in a CSS code of distance $M$7, then any subset of fewer than $M$8 qubits is statistically independent of the logical content — formally equivalent, up to rotations, to halves of EPR pairs. This reduces measurement certification to a single-qubit test the verifier can run with high fidelity: prepare an EPR pair, send one half, ask for a measurement in a random basis, and check agreement.

Concretely, the witness is encoded in the qLTC of Dinur–Lin–Vidick based on cubical complexes [dlv24], which has linear rate, distance $M$9, soundness $W$0, and constant-weight parity checks. The $W$1 physical qubits are partitioned into $W$2 blocks with $W$3. In round $W$4, the verifier sends block $W$5 to the prover, who reports $W$6- or $W$7-basis outcomes. In one randomly chosen round, the verifier runs the qLTC's local tester, substituting the prover's reported bits for qubits already sent out and measuring held qubits directly. Because only bit-flip errors in the complementary basis can corrupt reported outcomes, and the LTC property bounds such errors' weight while the code's linear distance ensures bounded-weight errors do not affect logical operators, any cheating is detected with probability proportional to the deviation.

The formal guarantee (Lemma on global measurement extraction) is that if validity tests pass with per-round rejection probabilities $W$8, then the operator implementing honest measurement of any logical function agrees with the actual protocol dynamics up to squared error

$W$9

where $q_i$0 is the LTC tester soundness and $q_i$1 the decoding radius. This bound drives the parameter choices throughout: setting $q_i$2 yields polylogarithmic tester query complexity.

Constraint checking with PCPPs

Given certified measurement outcomes $q_i$3, the verifier must check a sampled Hamiltonian constraint $q_i$4. Since constraints are non-local functions of the logical string, the verifier delegates evaluation to the prover via a PCP of proximity: the prover returns the claimed value $q_i$5 together with a PCPP proof that $q_i$6 is a codeword of $q_i$7 consistent with $q_i$8. The proximity parameter $q_i$9 must satisfy $M_i$0 so that any string accepted by the PCPP decodes to the correct logical value — a compatibility condition between the PCPP and the code's decoding radius that pins down $M_i$1.

The soundness analysis rests on a workhorse lemma comparing two ways of evaluating a logical observable on a corrupted codeword: measuring the decoded quantum word versus measuring the physical word and classically decoding. These agree on the low-weight error component of the stabilizer decomposition, and the qLTC local test forces the high-weight component to carry negligible amplitude — with rejection probability governed by $M_i$2 over $M_i$3 repetitions.

Assembling the pieces, the verifier samples one of three phases: a global-measurement-extraction consistency check, a PCPP consistency check, or the energy test proper. On YES instances acceptance probability is at least $M_i$4; on NO instances it is at most $M_i$5, strictly below 1, and constant sequential repetition closes the gap to the standard $M_i$6 versus $M_i$7 parameters.

Parameter limitations and their sources

The polylogarithmic query and round complexity are direct consequences of the sub-optimal qLTC parameters currently available. The paper is explicit that a qLTC with positive rate, constant relative distance and soundness, and constant-weight parity checks would collapse both query and round complexity to constants — i.e., the protocol's non-constant locality is an artifact of current code constructions, not of the approach. Similarly, total communication is quasi-linear except for the PCPP proof, which may be super-linear (e.g., quadratic); linear-size PCPPs would bring communication to near-linear. The last classical message is polynomial-length due to the non-derandomized amplification, reducible to logarithmic at the cost of additional machinery the authors chose not to include, noting they do not know whether that message length matters.

Limitations and open questions

Several restrictions are acknowledged plainly. The model forbids adaptive selection of which qubits to return and disallows the verifier sending qubits not drawn from the initial witness; the general qIOP framework permits these, but the protocol does not use them. The protocol is private-coin, and it requires quantum communication from verifier to prover — both obstacles to Fiat-Shamir-style compilation into non-interactive or succinct arguments. The authors identify as open problems: making the protocol public-coin (they note most private choices appear straightforwardly publicizable); removing verifier-to-prover quantum messages; achieving succinctness via quantum Merkle-tree commitments; compiling to a classical verifier under weaker assumptions than quantum FHE; and mapping the boundary of the model — determining which communication or locality restrictions separate QMA from NP or QCMA in this setting. Whether the query complexity can be made constant hinges entirely on progress in quantum locally testable code constructions.

Conclusion

The paper demonstrates that a single unbounded quantum prover, interacting with a polynomial-time quantum verifier under information-theoretic soundness, can be forced to reveal verifiable local measurement data from a quantum witness, provided the witness is protected by a quantum locally testable code and the verification target is an amplified dual-basis Hamiltonian. The result repositions the quantum PCP question: even without static local proofs for QMA, interactive local verification is achievable with polylogarithmic probing, and its further compression to constants is now a concrete question about code parameters rather than about verification techniques.