QMA equals QMA1: proof of perfect completeness in quantum verification

A clean answer to a stubborn quantum-verification question

Sabee Grewal and Dorian Rudolph have posted a 13-page preprint titled “QMA has perfect completeness,” submitted to arXiv on September 11, 2026, that proves the equality QMA = QMA_1 according to the paper’s main theorem [1]. In plain terms, the claim is that any quantum Merlin-Arthur proof system with ordinary bounded error can be converted into a system where every true instance has a quantum proof accepted with probability exactly one [1].

That exact “one” is the point. In a standard QMA protocol, a computationally unbounded prover, Merlin, sends a polynomial-size quantum witness to an efficient quantum verifier, Arthur. Completeness says that for every YES instance, some valid witness makes Arthur accept with probability at least c; soundness says that for every NO instance, no witness makes Arthur accept above s, for s < c [2]. QMA usually permits c below one, such as 2/3. QMA_1 asks for perfect completeness: on YES instances, an honest witness is never rejected [2].

The new work says this demand costs no computational power. More strongly, Grewal and Rudolph show that the transformation can be carried out using a fixed, exact gate set built from elementary operations: X, CNOT, Toffoli, and paired Hadamards H ⊗ H; they further state that Hadamard, Toffoli, and X gates also suffice because CNOT can be implemented using Toffoli with a fixed control, and two Hadamards implement H ⊗ H [2].

Why perfect completeness was not just a technical preference

Perfect completeness is natural in verification. If a statement is true and the prover supplies the right witness, it is satisfying to demand that the verifier accept with certainty. Classical proof systems often allow this kind of cleanup, and the preprint notes that related classes such as MA, QCMA, QIP, and PreciseQMA were already known not to lose power when perfect completeness is required [2].

QMA was different. Quantum witnesses can be arbitrary states, and the best acceptance probability is tied to the maximum eigenvalue of a verifier’s acceptance operator. Even when the verifier’s matrix entries are dyadic rationals, that maximum eigenvalue may have algebraic degree exponential in the witness size, making it unsuitable as a short exact certificate [2]. The obstacle was not merely that researchers lacked a clever amplification trick. The paper’s introduction recalls known oracle and black-box barriers, including results showing that generic amplification cannot simply drive completeness error all the way to zero in the finite-dimensional QMA setting [2].

This is why the result is conceptually sharp. It does not say that the usual probability-boosting machinery was finally pushed hard enough. Instead, it introduces a different certificate that can be represented exactly and verified exactly.

The theorem and its gate-set strengthening

The paper’s Theorem 1.1 states QMA = QMA_1 for the gate set G₂ = {X, CNOT, Toffoli, H ⊗ H}, written in the paper as QMA = QMA_1^G₂ [2]. The choice of gate set matters because approximate synthesis, so useful in bounded-error quantum computation, is dangerous when the target acceptance probability is exactly one. A tiny approximation error can turn probability 1 into 0.999999, which is no longer perfect completeness [2].

That explains the importance of rational or exactly controlled gates. The authors compile QMA verifiers into a gate set whose arithmetic is sufficiently rigid: after compilation, the acceptance operator has entries with a common dyadic denominator, meaning a polynomially bounded power of two clears the denominators [2]. This arithmetic regularity is the backbone of the proof.

The paper then derives a more accessible corollary: Hadamard, Toffoli, and X gates form a universal gate set for QMA_1 as well [2]. For specialists, that is more than a convenience. It says there is a stable exact circuit language for perfectly complete quantum verification, rather than a class whose definition depends sensitively on which exact gates are allowed.

The proof idea: stop chasing the top eigenvalue

The most interesting move in the proof is what Grewal and Rudolph choose not to certify. A first attempt might ask Merlin to send the maximum eigenvalue λ_max(E) of the acceptance operator E, plus an eigenvector achieving it. The verifier would then try to check that the eigenvalue is high enough and that the alleged eigenvector satisfies the right equation [2]. But the paper explains why that route is blocked: λ_max(E) may not have a succinct exact description, even when E itself has nicely structured dyadic entries [2].

The authors replace that difficult object with a family of rational diagonal quantities. After compiling the verifier, they set A = 2E. In YES instances, the largest eigenvalue of A is at least 4/3; in NO instances, the operator norm of A is at most 2/3 [2]. Powers of A therefore grow in some direction for YES instances and contract for NO instances.

For a polynomially chosen exponent ℓ, the paper considers quantities of the form Γ_z = ⟨z|A^ℓ|z⟩ over computational-basis states z [2]. In YES cases, at least one such Γ_z is large. Crucially, because of the dyadic arithmetic of the compiled verifier, Γ_z has a succinct exact rational description: it can be represented by polynomially many bits [2].

Merlin’s new role is to provide z, the exact value of Γ_z, and a quantum history state built from the sequence |z⟩, A|z⟩, A²|z⟩, and so on [2]. This is not ordinary physical time evolution; A is generally nonunitary. The sequence is a mathematical history whose consistency can be checked.

A finite history closed into a cycle

The proof’s distinctive construction closes that finite history into a cycle. If Γ_z is the correct value, then a scaled projector involving Γ_z⁻¹ maps the final term back to the initial basis state, allowing the authors to define a cyclic operator C_z,β with β = Γ_z⁻¹ [2]. The honest history state is an exact fixed point of this operator, so it lies in the kernel of I − C_z,β and can be accepted with certainty [2].

Soundness comes from the NO case. If the input is a NO instance, A is a strict contraction with norm at most 2/3. The verifier rejects claimed Γ values below a threshold, so β is also bounded. As a result, every allowed cyclic operator remains a contraction, and no proposed history state can satisfy the cyclic consistency condition [2].

This is the central elegance of the argument. The YES proof relies on an exact rational value that closes the cycle perfectly; the NO proof relies on contraction, so no dishonest witness can fake a fixed point. The paper then implements the necessary kernel test exactly over the G₂ gate set, completing the conversion to a QMA_1 verifier [2].

Consequences: quantum 3-SAT and oracle separations

The preprint lists several consequences. One headline corollary is that quantum 3-SAT is QMA-complete [3]. Previously, quantum k-SAT was a canonical setting for QMA_1 because satisfiable instances naturally have zero-energy states that pass every local constraint with certainty. By showing that QMA and QMA_1 coincide, the new result upgrades QMA_1-completeness statements into QMA-completeness statements in the relevant gate settings [3].

The same logic applies to other problems discussed in the paper, including XZ-quantum 6-SAT and gapped clique homology, which the authors identify as QMA-complete under the new equality and earlier completeness results [3]. The paper also states that the construction relativizes to classical oracles, so known classical-oracle separations of QMA from QCMA extend to QMA_1 in the described setting [1].

That last point is subtle. The proof does not relativize to arbitrary quantum oracles, and the authors explicitly connect this limitation to earlier oracle barriers [2]. But for classical oracle access represented by permutation-like query gates, the arithmetic structure needed by the proof survives [2].

What remains open

The result, if accepted by the community, resolves the QMA versus QMA_1 problem as framed in the preprint. But the authors also flag nearby questions that remain unsettled. Their methods do not appear to extend directly to BQP versus BQP_1, where there is no Merlin to supply the exact rational value used in the cyclic test [2]. Nor do they appear to extend directly to QMA(2), where the verifier receives two unentangled witnesses and soundness constraints apply only to product states rather than yielding the operator-norm bound used in this proof [2].

That boundary is important. The paper does not collapse all questions about exact acceptance in quantum complexity. It solves a precise one: for single-witness quantum Merlin-Arthur proof systems, perfect completeness can be obtained without changing the recognized class of promise problems [1].

For quantum complexity theory, this is a structural simplification. Researchers can now reason about QMA verifiers as if honest YES witnesses can be accepted with certainty, while retaining the full power of ordinary QMA. The surprise is that the route to that simplification is not more amplification, but a carefully engineered exact certificate built from dyadic arithmetic, powers of the acceptance operator, and a cyclic consistency test [2].

Sources from the last 72 hours

1. [1]QMA has perfect completenessSep 11, 2026, 4:31 PM UTC
2. [2]QMA has perfect completenessSep 11, 2026, 4:31 PM UTC
3. [3]QMA has perfect completenessSep 11, 2026, 4:31 PM UTC

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