The way entanglement influences the power of quantum and classical multi-prover interactive proof systems is a long-standing open question. We show that the class of languages recognized by quantum multi-prover interactive proof systems, QMIP, is equal to MIP*, the class of languages recognized by classical multi-prover interactive proof systems where the provers share entanglement. After the recent result by Jain, Ji, Upadhyay and Watrous showing that QIP=IP, our work completes the picture from the verifier's perspective by showing that also in the setting of multiple provers with shared entanglement, a quantum verifier is no more powerful than a classical one: QMIP=MIP*. Our techniques are based on the adaptation of universal blind quantum computation (a protocol recently introduced by us) to the context of interactive proof systems. We show that in the multi-prover scenario, shared entanglement has a positive effect in removing the need for a quantum verifier. As a consequence, our results show that the entire power of quantum information in multi-prover interactive proof systems is captured by the shared entanglement and not by the quantum communication.
The way entanglement influences the power of quantum and classical multi-prover interactive proof systems is a long-standing open question. We show that the class of languages recognized by quantum multi-prover interactive proof systems, QMIP, is equal to MIP*, the class of languages recognized by classical multi-prover interactive proof systems where the provers share entanglement. After the recent result by Jain, Ji, Upadhyay and Watrous showing that QIP=IP, our work completes the picture from the verifier's perspective by showing that also in the setting of multiple provers with shared entanglement, a quantum verifier is no more powerful than a classical one: QMIP=MIP*. Our techniques are based on the adaptation of universal blind quantum computation (a protocol recently introduced by us) to the context of interactive proof systems. We show that in the multi-prover scenario, shared entanglement has a positive effect in removing the need for a quantum verifier. As a consequence, our results show that the entire power of quantum information in multi-prover interactive proof systems is captured by the shared entanglement and not by the quantum communication.
Multi Prover Interactive Proof systems (MIPs)were first presented in a cryptographic context, but ever since they were used in various fields. Understanding the power of MIPs in the quantum context raises many open problems, as there are several interesting models to consider. For example, one can study the question when the provers share entanglement or not, and the communication between the verifier and the provers is quantum or classical. While there are several partial results on the subject, so far no one presented an efficient scheme for recognizing NEXP (or NP with logarithmic communication), except for [KM03], in the case there is no entanglement (and of course no communication between the provers). We introduce another variant of Quantum MIP, where the provers do not share entanglement, the communication between the verifier and the provers is quantum, but the provers are unlimited in the classical communication between them. At first, this model may seem very weak, as provers who exchange information seem to be equivalent in power to a simple prover. This in fact is not the case - we show that any language in NEXP can be recognized in this model efficiently, with just two provers and two rounds of communication, with a constant completeness-soundness gap.
We introduce a new variant of quantum multiprover interactive proofs (QMIP) where the provers and the verifier are quantum. The verifier can exchange quantum messages with the provers. The provers cannot communicate quantumly between themselves and do not share entanglement, but are unlimited in the classical communication between them, even after receiving messages from the verifier. We show that any language in nondeterministic exponential time (NEXP) can be recognized in this model efficiently, with just two provers and two rounds of communication, and with a constant completeness/soundness gap. This is in contrast to the result of [R. Jain et al., Comm. ACM, 53 (2010), pp. 102--109], which shows that QIP = PSPACE, or equivalently that the set of languages that can be recognized by a quantum verifier communicating with a single quantum prover is equal to PSPACE. To analyze the cheating power of the provers, we give them more power and allow them to perform any separable operation. We then show a unique two-phase protocol in which the provers first commit to a superposition of correct answers to all possible questions, and then in the second phase the verifier opens up the committed answer and checks for correctness and consistency.
We introduce another variant of quantum MIP, where the provers do not share entanglement, the communication between the verifier and the provers is quantum, but the provers are unlimited in the classical communication between them. At first, this model may seem very weak, as provers who exchange information seem to be equivalent in power to a simple prover. This in fact is not the case-we show that any language in NEXP can be recognized in this model efficiently, with just two provers and two rounds of communication, with a constant completeness-soundness gap. Similar ideas and techniques may help help with other models of quantum MIP, including the dual question, of non communicating provers with unlimited entanglement.
The study of interactive proofs in the quantum setting has yielded profound insights in complexity theory and quantum information. A curious feature of these results is that the advantage, in terms of computational power, of quantum models over their classical counterparts is usually due to entanglement phenomena rather than quantum communication with the verifier. For example, it is known that QIP = IP = PSPACE, and QMIP with unentangled provers is equal to NEXP = MIP; on the other hand, MIP* = RE. In this work we initiate the general study of (quantum) positional multi-prover interactive proofs ((Q)PMIP), in which provers and verifiers positioned in space communicate freely save for the constraints imposed by the speed of light. We investigate how the class of languages decidable by (Q)PMIPs depends on the arrangement of the verifiers and (honest) provers. In the case of classical PMIPs, we show a dichotomy: if the arrangement satisfies what we call the "min-ball" condition, then the class is NEXP, otherwise it is PSPACE. We then exhibit an arrangement that does not satisfy the min-ball condition for which there is a quantum PMIP for EXP in the no pre-shared entanglement model. Our construction is based on positional cryptography and MIPs with no-signaling soundness. We introduce a new positional primitive, the positional hardcore bit, which allows a pair of spatially separated players to transmit a random bit to a particular location while guaranteeing that it remains strongly unguessable elsewhere.
We study multiprover interactive proof systems. The power of classical multiprover interactive proof systems, in which the provers do not share entanglement, was characterized in a famous work by Babai, Fortnow, and Lund (Computational Complexity 1991), whose main result was the equality MIP = NEXP. The power of quantum multiprover interactive proof systems, in which the provers are allowed to share entanglement, has proven to be much more difficult to characterize. The best known lower-bound on MIP* is NEXP ⊆ MIP* due to Ito and Vidick (FOCS 2012). As for upper bounds, MIP* could be as large as RE, the class of recursively enumerable languages. The main result of this work is the inclusion NEEXP = NTIME[22poly(n)] ⊆ MIP*. This is an exponential improvement over the prior lower bound and shows that proof systems with entangled provers are at least exponentially more powerful than classical provers. In our protocol the verifier delegates a classical, exponentially large MIP protocol for NEEXP to two entangled provers: the provers obtain their exponentially large questions by measuring their shared state, and use a classical PCP to certify the correctness of their exponentially-long answers. For the soundness of our protocol, it is crucial that each player should not only sample its own question correctly but also avoid performing measurements that would reveal the other player's sampled question. We ensure this by commanding the players to perform a complementary measurement, relying on the Heisenberg uncertainty principle to prevent the forbidden measurements from being performed.
We study multiprover interactive proof systems. The power of classical multiprover interactive proof systems, in which the provers do not share entanglement, was characterized in a famous work by Babai, Fortnow, and Lund (Computational Complexity 1991), whose main result was the equality MIP = NEXP. The power of quantum multiprover interactive proof systems, in which the provers are allowed to share entanglement, has proven to be much more difficult to characterize. The best known lower-bound on MIP* is NEXP, due to Ito and Vidick (FOCS 2012). As for upper bounds, MIP* could be as large as RE, the class of recursively enumerable languages. The main result of this work is the inclusion of NEEXP (nondeterministic doubly exponential time) in MIP*. This is an exponential improvement over the prior lower bound and shows that proof systems with entangled provers are at least exponentially more powerful than classical provers. In our protocol the verifier delegates a classical, exponentially large MIP protocol for NEEXP to two entangled provers: the provers obtain their exponentially large questions by measuring their shared state, and use a classical PCP to certify the correctness of their exponentially-long answers. For the soundness of our protocol, it is crucial that each player should not only sample its own question correctly but also avoid performing measurements that would reveal the other player's sampled question. We ensure this by commanding the players to perform a complementary measurement, relying on the Heisenberg uncertainty principle to prevent the forbidden measurements from being performed.