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arXiv:2403.02532 · Quantum

Superposition detection and QMA with non-collapsing measurements

Roozbeh Bassirian, Kunal Marwaha

Published Aug 28, 2025• 1 citation
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Abstract

We prove that QMA where the verifier may also make a single non−collapsing measurement \cite{aaronson2014space} is equal to NEXP , resolving an open question of Aaronson \cite{scott_blogpost_pdqp}. We show this is a corollary to a modified proof of QMA+=NEXP \cite{bassirian2023quantum}. At the core of many results inspired by Blier and Tapp \cite{blier2010quantum} is an unphysical property testing problem deciding whether a quantum state is close to an element of a fixed basis.

DOI
10.22331/q-2025-08-28-1839

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We prove that QMA where the verifier may also make a single non−collapsing measurement \cite{aaronson2014space} is equal to NEXP , resolving an open question of Aaronson \cite{scott_blogpost_pdqp}. We show this is a corollary to a modified proof of QMA+=NEXP \cite{bassirian2023quantum}. At the core of many results inspired by Blier and Tapp \cite{blier2010quantum} is an unphysical property testing problem deciding whether a quantum state is close to an element of a fixed basis.
We define and study a variant of QMA (Quantum Merlin Arthur) in which Arthur can make multiple non-collapsing measurements to Merlin’s witness state, in addition to ordinary collapsing measurements. By analogy to the class PDQP defined by Aaronson, Bouland, Fitzsimons, and Lee (2014), we call this class PDQMA. Our main result is that PDQMA = NEXP; this result builds on the PCP theorem and complements the result of Aaronson (2018) that PDQP/qpoly = ALL. While the result has little to do with quantum mechanics, we also show a more "quantum" result: namely, that QMA with the ability to inspect the entire history of a hidden variable is equal to NEXP, under mild assumptions on the hidden-variable theory. We also observe that a quantum computer, augmented with quantum advice and the ability to inspect the history of a hidden variable, can solve any decision problem in polynomial time.
We define and study a variant of QMA (Quantum Merlin Arthur) in which Arthur can make multiple non-collapsing measurements to Merlin's witness state, in addition to ordinary collapsing measurements. By analogy to the class PDQP defined by Aaronson, Bouland, Fitzsimons, and Lee (2014), we call this class PDQMA. Our main result is that PDQMA = NEXP; this result builds on the PCP theorem and complements the result of Aaronson (2018) that PDQP/qpoly = ALL. While the result has little to do with quantum mechanics, we also show a more "quantum" result: namely, that QMA with the ability to inspect the entire history of a hidden variable is equal to NEXP, under mild assumptions on the hidden-variable theory. We also observe that a quantum computer, augmented with quantum advice and the ability to inspect the history of a hidden variable, can solve any decision problem in polynomial time.
We study a variant of QMA where quantum proofs have no relative phase (i.e. non-negative amplitudes, up to a global phase). If only completeness is modified, this class is equal to QMA [arXiv:1410.2882]; but if both completeness and soundness are modified, the class (named QMA+ by Jeronimo and Wu) can be much more powerful. We show that QMA+ with some constant gap is equal to NEXP, yet QMA+ with some *other* constant gap is equal to QMA. One interpretation is that Merlin's ability to "deceive" originates from relative phase at least as much as from entanglement, since QMA(2) ⊆\subseteq⊆ NEXP.
We initiate the systematic study of QMA algorithms in the setting of property testing, to which we refer as QMA proofs of proximity (QMAPs). These are quantum query algorithms that receive explicit access to a sublinear-size untrusted proof and are required to accept inputs having a property Π and reject inputs that are ε -far from Π , while only probing a minuscule portion of their input.We investigate the complexity landscape of this model, showing that QMAPs can be exponentially stronger than both classical proofs of proximity and quantum testers. To this end, we extend the methodology of Blais, Brody, and Matulef (Computational Complexity, 2012) to prove quantum property testing lower bounds via reductions from communication complexity. This also resolves a question raised in 2013 by Montanaro and de Wolf (cf. Theory of Computing, 2016).Our algorithmic results include a purpose an algorithmic framework that enables quantum speedups for testing an expressive class of properties, namely, those that are succinctly decomposable . A consequence of this framework is a QMA algorithm to verify the Parity of an n -bit string with O(n2/3) queries and proof length. We also propose a QMA algorithm for testing graph bipartitneness, a property that lies outside of this family, for which there is a quantum speedup.
We find a modification to QMA where having one quantum proof is strictly less powerful than having two unentangled proofs, assuming EXP ≠\ne= NEXP. This gives a new route to prove QMA(2) = NEXP that overcomes the primary drawback of a recent approach [arXiv:2402.18790 , arXiv:2306.13247] (QIP 2024). Our modification endows each proof with a form of *multipartite* unentanglement: after tracing out one register, a small number of qubits are separable from the rest of the state.
We identify a formal connection between physical problems related to the detection of separable (unentangled) quantum states and complexity classes in theoretical computer science. In particular, we show that to nearly every quantum interactive proof complexity class (including BQP, QMA, QMA(2), and QSZK), there corresponds a natural separability testing problem that is complete for that class. Of particular interest is the fact that the problem of determining whether an isometry can be made to produce a separable state is either QMA-complete or QMA(2)-complete, depending upon whether the distance between quantum states is measured by the one-way LOCC norm or the trace norm. We obtain strong hardness results by employing prior work on entanglement purification protocols to prove that for each n-qubit maximally entangled state there exists a fixed one-way LOCC measurement that distinguishes it from any separable state with error probability that decays exponentially in n.