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arXiv:1510.06789 · Physical Review A

Quantum proofs can be verified using only single-qubit measurements

Tomoyuki Morimae, Daniel Nagaj, Norbert Schuch

Published Feb 22, 2016• 32 citations
arXivHTMLPDF

Abstract

Quantum Merlin Arthur (QMA) is the class of problems which, though potentially hard to solve, have a quantum solution that can be verified efficiently using a quantum computer. It thus forms a natural quantum version of the classical complexity class NP (and its probabilistic variant MA, Merlin-Arthur games), where the verifier has only classical computational resources. In this paper, we study what happens when we restrict the quantum resources of the verifier to the bare minimum: individual measurements on single qubits received as they come, one by one. We find that despite this grave restriction, it is still possible to soundly verify any problem in QMA for the verifier with the minimum quantum resources possible, without using any quantum memory or multiqubit operations. We provide two independent proofs of this fact, based on measurement-based quantum computation and the local Hamiltonian problem. The former construction also applies to ${\mathrm{QMA}}_{1}$, i.e., QMA with one-sided error.

DOI
10.1103/physreva.93.022326

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What happens if in QMA the quantum channel between Merlin and Arthur is noisy? It is not difficult to show that such a modification does not change the computational power as long as the noise is not too strong so that errors are correctable with high probability, since if Merlin encodes the witness state in a quantum error-correction code and sends it to Arthur, Arthur can correct the error caused by the noisy channel. If we further assume that Arthur can do only single-qubit measurements, however, the problem becomes nontrivial, since in this case Arthur cannot do the universal quantum computation by himself. In this paper, we show that such a restricted complexity class is still equivalent to QMA. To show it, we use measurement-based quantum computing: honest Merlin sends the graph state to Arthur, and Arthur does fault-tolerant measurement-based quantum computing on the noisy graph state with only single-qubit measurements. By measuring stabilizer operators, Arthur also checks the correctness of the graph state. Although this idea itself was already used in several previous papers, these results cannot be directly used to the present case, since the test that checks the graph state used in these papers is so strict that even honest Merlin is rejected with high probability if the channel is noisy. We therefore introduce a more relaxed test that can accept not only the ideal graph state but also noisy graph states that are error-correctable.

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This paper investigates the role of interaction and coins in quantum Arthur-Merlin games (also called public-coin quantum interactive proof systems). While the existing model restricts the messages from the verifier to be classical even in the quantum setting, the present work introduces a generalized version of quantum Arthur-Merlin games where the messages from the verifier can be quantum as well: the verifier can send not only random bits, but also halves of EPR pairs. This generalization turns out to provide several novel characterizations of quantum interactive proof systems with a constant number of turns. First, it is proved that the complexity class corresponding to two-turn quantum Arthur-Merlin games where both of the two messages are quantum, denoted qq-QAM in this paper, does not change by adding a constant number of turns of classical interaction prior to the communications of qq-QAM proof systems. This can be viewed as a quantum analogue of the celebrated collapse theorem for AM due to Babai. To prove this collapse theorem, this paper presents a natural complete problem for qq-QAM: deciding whether the output of a given quantum circuit is close to a totally mixed state. This complete problem is on the very line of the previous studies investigating the hardness of checking properties related to quantum circuits, and thus, qq-QAM may provide a good measure in computational complexity theory. It is further proved that the class qq-QAM_1, the perfect-completeness variant of qq-QAM, gives new bounds for standard well-studied classes of two-turn quantum interactive proof systems. Finally, the collapse theorem above is extended to comprehensively classify the role of classical and quantum interactions in quantum Arthur-Merlin games: it is proved that, for any constant m >= 2, the class of problems having $m$-turn quantum Arthur-Merlin proof systems is either equal to PSPACE or equal to the class of problems having two-turn quantum Arthur-Merlin proof systems of a specific type, which provides a complete set of quantum analogues of Babai's collapse theorem.

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Quantum-Merlin-Arthur Problems Have Perfect Completeness with an Infinite Counter

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A longstanding open problem in quantum complexity theory is whether Quantum Merlin-Arthur (QMA), the quantum analog of nondeterministic polynomial time, is equal to QMA_{1}, its one-sided error variant. We show that QMA=QMA^{∞}=QMA_{1}^{∞}, where QMA_{1}^{∞} is like QMA_{1}, but the verifier has an infinite register, as part of their witness system, in which they can efficiently perform a shift (increment) operation. We call this register an "infinite counter," and compare it to a program counter in a Las Vegas algorithm. The result, QMA=QMA^{∞} means such an infinite register does not increase the power of QMA, but does imply perfect completeness. By truncating our construction to finite dimensions, we get a QMA-amplifier that only amplifies completeness, not soundness, but does so in significantly less time than previous QMA amplifiers. Our new construction achieves completeness 1-2^{-q} using O(1) calls to each of the original verifier and its inverse, and O(logq) other gates, proving that QMA has completeness doubly exponentially close to 1, i.e., QMA=QMA(1-2^{-2^{r}},2^{-r}) for any polynomial r.

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A quantum verifier decides membership in QMA using a (single) quantum witness state. This witness may be arbitrary, and no requirement is placed on it beyond its acceptance probability. A protocol claiming to classically verify QMA should offer the same guarantee: any witness that would convince a quantum verifier should also suffice to convince a classical verifier. Unfortunately, existing classical verification of quantum computation (CVQC) protocols behave quite differently from the original QMA verifier: they require many copies of the witness and destroy every one, effectively demanding that the prover solve the hard problem of witness duplication before verification can begin. We build a protocol to classically verify QMA languages that uses a single copy of the prover's QMA witness. Whenever the original QMA verifier accepts that witness with overwhelming probability, our protocol has negligible completeness and soundness errors and returns the witness negligibly disturbed after interaction with an honest verifier. The soundness of our CVQC is based on the post-quantum Learning With Errors (LWE) assumption. As an intermediate step of independent interest, we introduce witness-preserving classical arguments for NP which enable provers that start out with a superposition over NP witnesses to convince a verifier without disturbing their superposition.

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Quantum Merlin Arthur (QMA) is the class of problems which, though potentially hard to solve, have a quantum solution that can be verified efficiently using a quantum computer. It thus forms a natural quantum version of the classical complexity class NP (and its probabilistic variant MA, Merlin-Arthur games), where the verifier has only classical computational resources. In this paper, we study what happens when we restrict the quantum resources of the verifier to the bare minimum: individual measurements on single qubits received as they come, one by one. We find that despite this grave restriction, it is still possible to soundly verify any problem in QMA for the verifier with the minimum quantum resources possible, without using any quantum memory or multiqubit operations. We provide two independent proofs of this fact, based on measurement-based quantum computation and the local Hamiltonian problem. The former construction also applies to QMA1{\mathrm{QMA}}_{1}QMA1​, i.e., QMA with one-sided error.
We show that the class quantum-Arthur-Merlin (QAM) does not change even if the verifier's ability is restricted to only single-qubit measurements. To show the result, we use the idea of measurement-based quantum computing: the verifier, who can do only single-qubit measurements, can test the graph state sent from the prover and use it for his measurement-based quantum computing. Inspired by this construction, we also introduce a problem which we call stabilizer state optimization, and show that it is QMA-complete.
What happens if in QMA the quantum channel between Merlin and Arthur is noisy? It is not difficult to show that such a modification does not change the computational power as long as the noise is not too strong so that errors are correctable with high probability, since if Merlin encodes the witness state in a quantum error-correction code and sends it to Arthur, Arthur can correct the error caused by the noisy channel. If we further assume that Arthur can do only single-qubit measurements, however, the problem becomes nontrivial, since in this case Arthur cannot do the universal quantum computation by himself. In this paper, we show that such a restricted complexity class is still equivalent to QMA. To show it, we use measurement-based quantum computing: honest Merlin sends the graph state to Arthur, and Arthur does fault-tolerant measurement-based quantum computing on the noisy graph state with only single-qubit measurements. By measuring stabilizer operators, Arthur also checks the correctness of the graph state. Although this idea itself was already used in several previous papers, these results cannot be directly used to the present case, since the test that checks the graph state used in these papers is so strict that even honest Merlin is rejected with high probability if the channel is noisy. We therefore introduce a more relaxed test that can accept not only the ideal graph state but also noisy graph states that are error-correctable.
This paper investigates the role of interaction and coins in quantum Arthur-Merlin games (also called public-coin quantum interactive proof systems). While the existing model restricts the messages from the verifier to be classical even in the quantum setting, the present work introduces a generalized version of quantum Arthur-Merlin games where the messages from the verifier can be quantum as well: the verifier can send not only random bits, but also halves of EPR pairs. This generalization turns out to provide several novel characterizations of quantum interactive proof systems with a constant number of turns. First, it is proved that the complexity class corresponding to two-turn quantum Arthur-Merlin games where both of the two messages are quantum, denoted qq-QAM in this paper, does not change by adding a constant number of turns of classical interaction prior to the communications of qq-QAM proof systems. This can be viewed as a quantum analogue of the celebrated collapse theorem for AM due to Babai. To prove this collapse theorem, this paper presents a natural complete problem for qq-QAM: deciding whether the output of a given quantum circuit is close to a totally mixed state. This complete problem is on the very line of the previous studies investigating the hardness of checking properties related to quantum circuits, and thus, qq-QAM may provide a good measure in computational complexity theory. It is further proved that the class qq-QAM_1, the perfect-completeness variant of qq-QAM, gives new bounds for standard well-studied classes of two-turn quantum interactive proof systems. Finally, the collapse theorem above is extended to comprehensively classify the role of classical and quantum interactions in quantum Arthur-Merlin games: it is proved that, for any constant m >= 2, the class of problems having mmm-turn quantum Arthur-Merlin proof systems is either equal to PSPACE or equal to the class of problems having two-turn quantum Arthur-Merlin proof systems of a specific type, which provides a complete set of quantum analogues of Babai's collapse theorem.
This paper investigates the role of interaction and coins in quantum Arthur-Merlin games(also called public-coin quantum interactive proof systems). While the existing model restricts the messages from the verifier to be classical even in the quantum setting, the present work introduces a generalized version of quantum Arthur-Merlin games where the messages from the verifier can be quantum as well: the verifier can send not only random bits, but also halves of EPR pairs. This generalization turns out to provide several novel characterizations of quantum interactive proof systems with a constant number of turns. First, it is proved that the complexity class corresponding to two-turn quantum Arthur-Merlin games where both of the two messages are quantum, denoted qq-QAM in this paper, does not change by adding a constant number of turns of classical interaction prior to the communications of qq-QAM proof systems. This can be viewed as a quantum analogue of the celebrated collapse theorem for AM due to Babai. To prove this collapse theorem, this paper presents a natural complete problem for qq-QAM: deciding whether the output of a given quantum circuit is close to a totally mixed state. This complete problem is on the very line of the previous studies investigating the hardness of checking properties related to quantum circuits, and thus, qq-QAM may provide a good measure in computational complexity theory. It is further proved that the class qq-QAM1, the perfect-completeness variant of qq-QAM, gives new bounds for standard well-studied classes of two-turn quantum interactive proof systems. Finally, the collapse theorem above is extended to comprehensively classify the role of classical and quantum interactions in quantum Arthur-Merlin games: it is proved that, for any constant m ≥ 2, the class of problems having m-turn quantum Arthur-Merlin proof systems is either equal to PSPACE or equal to the class of problems having two-turn quantum Arthur-Merlin proof systems of a specific type, which provides a complete set of quantum analogues of Babai's collapse theorem.
A longstanding open problem in quantum complexity theory is whether Quantum Merlin-Arthur (QMA), the quantum analog of nondeterministic polynomial time, is equal to QMA_{1}, its one-sided error variant. We show that QMA=QMA^{∞}=QMA_{1}^{∞}, where QMA_{1}^{∞} is like QMA_{1}, but the verifier has an infinite register, as part of their witness system, in which they can efficiently perform a shift (increment) operation. We call this register an "infinite counter," and compare it to a program counter in a Las Vegas algorithm. The result, QMA=QMA^{∞} means such an infinite register does not increase the power of QMA, but does imply perfect completeness. By truncating our construction to finite dimensions, we get a QMA-amplifier that only amplifies completeness, not soundness, but does so in significantly less time than previous QMA amplifiers. Our new construction achieves completeness 1-2^{-q} using O(1) calls to each of the original verifier and its inverse, and O(logq) other gates, proving that QMA has completeness doubly exponentially close to 1, i.e., QMA=QMA(1-2^{-2^{r}},2^{-r}) for any polynomial r.
A quantum verifier decides membership in QMA using a (single) quantum witness state. This witness may be arbitrary, and no requirement is placed on it beyond its acceptance probability. A protocol claiming to classically verify QMA should offer the same guarantee: any witness that would convince a quantum verifier should also suffice to convince a classical verifier. Unfortunately, existing classical verification of quantum computation (CVQC) protocols behave quite differently from the original QMA verifier: they require many copies of the witness and destroy every one, effectively demanding that the prover solve the hard problem of witness duplication before verification can begin. We build a protocol to classically verify QMA languages that uses a single copy of the prover's QMA witness. Whenever the original QMA verifier accepts that witness with overwhelming probability, our protocol has negligible completeness and soundness errors and returns the witness negligibly disturbed after interaction with an honest verifier. The soundness of our CVQC is based on the post-quantum Learning With Errors (LWE) assumption. As an intermediate step of independent interest, we introduce witness-preserving classical arguments for NP which enable provers that start out with a superposition over NP witnesses to convince a verifier without disturbing their superposition.