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arXiv:quant-ph/0504075

Quantum Information and the PCP Theorem

Ran Raz

Published Apr 10, 2005• 0 citations
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Abstract

We show how to encode $2^n$ (classical) bits $a_1,...,a_{2^n}$ by a single quantum state $|Ψ>$ of size O(n) qubits, such that: for any constant $k$ and any $i_1,...,i_k \in \{1,...,2^n\}$, the values of the bits $a_{i_1},...,a_{i_k}$ can be retrieved from $|Ψ>$ by a one-round Arthur-Merlin interactive protocol of size polynomial in $n$. This shows how to go around Holevo-Nayak's Theorem, using Arthur-Merlin proofs. We use the new representation to prove the following results: 1) Interactive proofs with quantum advice: We show that the class $QIP/qpoly$ contains ALL languages. That is, for any language $L$ (even non-recursive), the membership $x \in L$ (for $x$ of length $n$) can be proved by a polynomial-size quantum interactive proof, where the verifier is a polynomial-size quantum circuit with working space initiated with some quantum state $|Ψ_{L,n} >$ (depending only on $L$ and $n$). Moreover, the interactive proof that we give is of only one round, and the messages communicated are classical. 2) PCP with only one query: We show that the membership $x \in SAT$ (for $x$ of length $n$) can be proved by a logarithmic-size quantum state $|Ψ>$, together with a polynomial-size classical proof consisting of blocks of length $polylog(n)$ bits each, such that after measuring the state $|Ψ>$ the verifier only needs to read {\bf one} block of the classical proof. While the first result is a straight forward consequence of the new representation, the second requires an additional machinery of quantum low-degree-test that may be interesting in its own right.

DOI
10.48550/arxiv.quant-ph/0504075

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Our main result is that the membership x /spl epsi/ SAT (for x of length n) can be proved by a logarithmic-size quantum state |/spl Psi/>, together with a polynomial-size classical proof consisting of blocks of length polylog(n) bits each, such that after measuring the state |/spl Psi/> the verifier only needs to read one block of the classical proof. This shows that if a short quantum witness is available then a (classical) PCP with only one query is possible. Our second result is that the class QIP/qpoly contains all languages. That is, for any language L (even non-recursive), the membership x /spl epsi/ L (for x of length n) can be proved by a polynomial-size quantum interactive proof, where the verifier is a polynomial-size quantum circuit with working space initiated with some quantum state |/spl Psi//sub L,n/> (depending only on L and n). Moreover, the interactive proof that we give is of only one round, and the messages communicated are classical. The advice |/spl Psi//sub L,n/> given to the verifier can also be replaced by a classical probabilistic advice, as long as this advice is kept as a secret from the proven Our result can hence be interpreted as: the class IP/rpoly contains all languages. For the proof of the second result, we introduce the quantum low-degree-extension of a string of bits. The main result requires an additional machinery of quantum low-degree-test.

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We show how to encode 2n2^n2n (classical) bits a1,...,a2na_1,...,a_{2^n}a1​,...,a2n​ by a single quantum state ∣Ψ>|Ψ>∣Ψ> of size O(n) qubits, such that: for any constant kkk and any i1,...,ik∈{1,...,2n}i_1,...,i_k \in \{1,...,2^n\}i1​,...,ik​∈{1,...,2n}, the values of the bits ai1,...,aika_{i_1},...,a_{i_k}ai1​​,...,aik​​ can be retrieved from ∣Ψ>|Ψ>∣Ψ> by a one-round Arthur-Merlin interactive protocol of size polynomial in nnn. This shows how to go around Holevo-Nayak's Theorem, using Arthur-Merlin proofs. We use the new representation to prove the following results: 1) Interactive proofs with quantum advice: We show that the class QIP/qpolyQIP/qpolyQIP/qpoly contains ALL languages. That is, for any language LLL (even non-recursive), the membership x∈Lx \in Lx∈L (for xxx of length nnn) can be proved by a polynomial-size quantum interactive proof, where the verifier is a polynomial-size quantum circuit with working space initiated with some quantum state ∣ΨL,n>|Ψ_{L,n} >∣ΨL,n​> (depending only on LLL and nnn). Moreover, the interactive proof that we give is of only one round, and the messages communicated are classical. 2) PCP with only one query: We show that the membership x∈SATx \in SATx∈SAT (for xxx of length nnn) can be proved by a logarithmic-size quantum state ∣Ψ>|Ψ>∣Ψ>, together with a polynomial-size classical proof consisting of blocks of length polylog(n)polylog(n)polylog(n) bits each, such that after measuring the state ∣Ψ>|Ψ>∣Ψ> the verifier only needs to read {\bf one} block of the classical proof. While the first result is a straight forward consequence of the new representation, the second requires an additional machinery of quantum low-degree-test that may be interesting in its own right.
Our main result is that the membership x ∈ SAT (for x of length n) can be proved by a logarithmic-size quantum state |Ψ〉, together with a polynomial-size classical proof consisting of blocks of length polylog(n) bits each, such that after measuring the state |Ψ 〉 the verifier only needs to read one block of the classical proof. This shows that if a short quantum witness is available then a (classical) PCP with only one query is possible. Our second result is that the class QIP/qpoly contains all languages. That is, for any language L (even non-recursive), the membership x ∈ L (for x of length n) can be proved by a polynomial-size quantum interactive proof, where the verifier is a polynomial-size quantum circuit with working space initiated with some quantum state |ΨL,n 〉 (depending only on L and n). Moreover, the interactive proof that we give is of only one round, and the messages communicated are classical. The advice |ΨL,n 〉 given to the verifier can also be replaced by a classical probabilistic advice, as long as this advice is kept as a secret from the prover. Our result can hence be interpreted as: the class IP/rpoly contains all languages. For the proof of the second result, we introduce the quantum low-degree-extension of a string of bits. The main result requires an additional machinery of quantum lowdegree-test.
Our main result is that the membership x /spl epsi/ SAT (for x of length n) can be proved by a logarithmic-size quantum state |/spl Psi/>, together with a polynomial-size classical proof consisting of blocks of length polylog(n) bits each, such that after measuring the state |/spl Psi/> the verifier only needs to read one block of the classical proof. This shows that if a short quantum witness is available then a (classical) PCP with only one query is possible. Our second result is that the class QIP/qpoly contains all languages. That is, for any language L (even non-recursive), the membership x /spl epsi/ L (for x of length n) can be proved by a polynomial-size quantum interactive proof, where the verifier is a polynomial-size quantum circuit with working space initiated with some quantum state |/spl Psi//sub L,n/> (depending only on L and n). Moreover, the interactive proof that we give is of only one round, and the messages communicated are classical. The advice |/spl Psi//sub L,n/> given to the verifier can also be replaced by a classical probabilistic advice, as long as this advice is kept as a secret from the proven Our result can hence be interpreted as: the class IP/rpoly contains all languages. For the proof of the second result, we introduce the quantum low-degree-extension of a string of bits. The main result requires an additional machinery of quantum low-degree-test.
We initiate the study of quantum Interactive Oracle Proofs (qIOPs), a generalization of both quantum Probabilistically Checkable Proofs and quantum Interactive Proofs, as well as a quantum analogue of classical Interactive Oracle Proofs. In the model of quantum Interactive Oracle Proofs, we allow multiple rounds of quantum interaction between the quantum prover and the quantum verifier, but the verifier has limited access to quantum resources. This includes both queries to the prover's messages and the complexity of the quantum circuits applied by the verifier. The question of whether QMA admits a quantum interactive oracle proof system is a relaxation of the quantum PCP Conjecture. We show the following two main constructions of qIOPs, both of which are unconditional: - We construct a qIOP for QMA in which the verifier shares polynomially many EPR pairs with the prover at the start of the protocol and reads only a constant number of qubits from the prover's messages. - We provide a stronger construction of qIOP for QMA in which the verifier not only reads a constant number of qubits but also operates on a constant number of qubits overall, including those in their private registers. However, in this stronger setting, the communication complexity becomes exponential. This leaves open the question of whether strong qIOPs for QMA, with polynomial communication complexity, exist. As a key component of our construction, we introduce a novel single prover many-qubits test, which may be of independent interest.
For nearly three decades, the model of interactive proof systems and its variants have been central to many important and exciting developments in computational complexity theory such as exact characterization of some well known complexity classes, development of probabilistically checkable proof systems and theory of hardness of approximation, and formalization of fundamental cryptographic primitives. \n \nOn the other hand, the theory of quantum information, which is primarily concerned with harnessing quantum mechanical features for algorithmic, cryptographic, and information processing tasks has found many applications. In the past three decades, quantum information has been used to develop unconditionally secure quantum cryptography protocols, efficient quantum algorithms for certain problems that are believed to be intractable in classical world, and communication efficient protocols. \n \nIn this thesis, we study the impact of quantum information on the models of interactive proof systems and their multi-prover variants. We study various quantum models and explore two questions. The first question we address pertains to the expressive power of such models with or without resource constraints. The second question is related to error reduction technique of such proof systems via parallel repetition. \n \nThe question related to the expressive power of models of quantum interactive proof systems and their variants lead us to the following results. \n \n(1) We show that the expressive power of quantum interactive proof systems is exactly PSPACE, the class of problems that can be solved by a polynomial-space deterministic Turing machines and that also admit a classical interactive proof systems. This result shows that in terms of complexity-theoretic characterization, both the models are equivalent. The result is obtained using an algorithmic technique known as the matrix multiplicative weights update method to solve a semidefinite program that characterizes the success probability of the quantum prover. \n\t \n(2) We show that polynomially many logarithmic-size unentangled quantum proofs are no more powerful than a classical proof if the verifier has the ability to process quantum information. This result follows from an observation that logarithmic-size quantum states can be efficiently represented classically and such classical representation can be used to efficiently generate the quantum state. \n\t \n(3) We also establish that the model of multi-prover quantum Merlin Arthur proof system, where the verifier is only allowed to apply nonadaptive unentangled measurement on each proof and then a quantum circuit on the classical outcomes, is no more powerful than QMA under the restriction that there are only polynomial number of outcomes per proof. This result follows from showing that such proof systems also admit a QMA verification procedure. \n \nThe question related to error reduction via parallel repetition lead us to following results on a class of two-prover one-round games with quantum provers and a class of multi-prover QMA proof systems. \n \n(1) We establish that for a certain class of two-prover one-round games known as XOR games, admit a perfect parallel repetition theorem in the following sense. When the provers play a collection of XOR games, an optimal strategy of the provers is to play each instance of the collection independently and optimally. In particular, the success probability of the quantum provers in the n-fold repetition of an XOR game G with quantum value w(G) is exactly (w(G))^n. \n\t \n(2) We show a parallel repetition theorem for two-prover one-round unique games. More specifically, we prove that if the quantum value of a unique game is 1-e, then the quantum value of n-fold repetition of the game is at most (1-e^2/49)^n. We also establish that for certain class of unique games, the quantum value of the n-fold repetition of the game is at most (1-e/4)^n. For the special case of XOR games, our proof technique gives an alternate proof of result mentioned above. \n\t \n3. Our final result on parallel repetition is concerned with SepQMA(m) proof systems, where the verifier receives m unentangled quantum proofs and the measurement operator corresponding to outcome "accept" is a fully separable operator. We give an alternate proof of a result of Harrow and Montanaro [HM10] that states that perfect parallel repetition theorem holds for such proof systems. \n \nThe first two results follow from the duality of semidefinite programs and the final result follows from cone programming duality.
We generalize quantum-classical PCPs, first introduced by Weggemans, Folkertsma and Cade (TQC 2024), to allow for qqq quantum queries to a polynomially-sized classical proof (QCPCPQ,c,s[q]\mathsf{QCPCP}_{Q,c,s}[q]QCPCPQ,c,s​[q]). Exploiting a connection with the polynomial method, we prove that for any constant qqq, promise gap c−s=Ω(1/poly(n))c-s = Ω(1/\text{poly}(n))c−s=Ω(1/poly(n)) and δ>0δ>0δ>0, we have QCPCPQ,c,s[q]⊆QCPCP1−δ,1/2+δ[3]⊆BQ⋅NP\mathsf{QCPCP}_{Q,c,s}[q] \subseteq \mathsf{QCPCP}_{1-δ,1/2+δ}[3] \subseteq \mathsf{BQ} \cdot \mathsf{NP}QCPCPQ,c,s​[q]⊆QCPCP1−δ,1/2+δ​[3]⊆BQ⋅NP, where BQ⋅NP\mathsf{BQ} \cdot \mathsf{NP}BQ⋅NP is the class of promise problems with quantum reductions to an NP\mathsf{NP}NP-complete problem. Surprisingly, this shows that we can amplify the promise gap from inverse polynomial to constant for constant query quantum-classical PCPs, and that any quantum-classical PCP making any constant number of quantum queries can be simulated by one that makes only three classical queries. Nevertheless, even though we can achieve promise gap amplification, our result also gives strong evidence that there exists no constant query quantum-classical PCP for QCMA\mathsf{QCMA}QCMA, as it is unlikely that QCMA⊆BQ⋅NP\mathsf{QCMA} \subseteq \mathsf{BQ} \cdot \mathsf{NP}QCMA⊆BQ⋅NP, which we support by giving oracular evidence. In the (poly-)logarithmic query regime, we show for any positive integer ccc, there exists an oracle relative to which QCPCP[O(log⁡cn)]⊊QCPCPQ[O(log⁡cn)]\mathsf{QCPCP}[\mathcal{O}(\log^c n)] \subsetneq \mathsf{QCPCP}_Q[\mathcal{O}(\log^c n)]QCPCP[O(logcn)]⊊QCPCPQ​[O(logcn)], contrasting the constant query case where the equivalence of both query models holds relative to any oracle. Finally, we connect our results to more general quantum-classical interactive proof systems.
This paper studies whether quantum proofs are more powerful than classical proofs, or in complexity terms, whether QMA=QCMA. We prove three results about this question. First, we give a "quantum oracle separation" between QMA and QCMA. More concretely, we show that any quantum algorithm needs Ω(2n/(m+1))Ω(\sqrt{2^n/(m+1)})Ω(2n/(m+1)​) queries to find an nnn-qubit "marked state" ∣ψ⟩\lvertψ\rangle∣ψ⟩, even if given an mmm-bit classical description of ∣ψ⟩\lvertψ\rangle∣ψ⟩ together with a quantum black box that recognizes ∣ψ⟩\lvertψ\rangle∣ψ⟩. Second, we give an explicit QCMA protocol that nearly achieves this lower bound. Third, we show that, in the one previously-known case where quantum proofs seemed to provide an exponential advantage, classical proofs are basically just as powerful. In particular, Watrous gave a QMA protocol for verifying non-membership in finite groups. Under plausible group-theoretic assumptions, we give a QCMA protocol for the same problem. Even with no assumptions, our protocol makes only polynomially many queries to the group oracle. We end with some conjectures about quantum versus classical oracles, and about the possibility of a classical oracle separation between QMA and QCMA.