Paper 2020/1391
Interactive Proofs for Quantum BlackBox Computations
Jiang Zhang, Yu Yu, Dengguo Feng, Shuqin Fan, Zhenfeng Zhang, and Kang Yang
Abstract
In this paper, we initiate the study of interactive proofs for the promise problem $\mathsf{QBBC}$ (i.e., quantum blackbox computations), which consists of a quantum device $\mathcal{D}(x\rangle y\rangle) =x\rangle D_x(y\rangle)$ acting on $(n+m)$ qubits for some selfjoint unitary $D_x$ (i.e., $D_x(D_x(y\rangle)) = y\rangle$), a classical device $\mathcal{R}_F$ deciding the inputoutput relation of some unknown function $F:\{0,1\}^n \rightarrow \{0,1\}^m$, and two reals $0< b < a \leq 1$. Let $p(\mathcal{D},x) = \ x,F(x)\rangle \langle x,F(x) \mathcal{D}(x\rangle 0^m\rangle)\^2$ be the probability of obtaining $(x,F(x))$ as a result of a standard measurement of the $(n+m)$qubit state returned by $\mathcal{D}$ on input $x\rangle 0^m\rangle$. The task of the problem $\mathsf{QBBC}(\mathcal{D},\mathcal{R}_F,a,b)$ is to distinguish between two cases for all $x\in \{0,1\}^n$: \\ $\bullet$ YES Instance: $p(\mathcal{D},x) \geq a$; $\bullet$ NO Instance: $p(\mathcal{D},x) \leq b$. First, we show that for any constant $15/16< a \leq 1$, the problem $\mathsf{QBBC}(\mathcal{D},\mathcal{R}_F,a,b)$ has an efficient tworound interactive proof $(\mathcal{P}^{\mathcal{D}},\mathcal{V}^{\mathcal{R}_F})$ which basically allows a verifier $\mathcal{V}$, given a classical blackbox device $\mathcal{R}_F$, to efficiently verify if the prover $\mathcal{P}$ has a quantum blackbox device $\mathcal{D}$ (correctly) computing $F$. This proof system achieves completeness $\frac{1 + a}{2}$ and soundness error $\frac{31}{32} + \frac{\epsilon}{2} + \mathsf{negl}(n)$ for any constant $\max(0,b\frac{15}{16})<\epsilon<a  \frac{15}{16}$, given that the verifier $\mathcal{V}$ has some (limited) quantum capabilities. In terms of query complexities, the prover $\mathcal{P}^\mathcal{D}$ will make at most two quantum queries to $\mathcal{D}$, while the verifier $\mathcal{V}^{\mathcal{R}_F}$ only makes a single classical query to $\mathcal{R}_F$. This result is based on an information versus disturbance lemma, which may be of independent interest. Second, under the learning with errors (LWE) assumption in (Regev 2005), we show that the problem $\mathsf{QBBC}(\mathcal{D},\mathcal{R}_F,a,b)$ can even have an efficient interactive proof $(\mathcal{P}^{\mathcal{D}},\mathcal{V}^{\mathcal{R}_F})$ with a fully classical verifier $\mathcal{V}$ that does not have any quantum capability. This proof system achieves completeness $\frac{1 + a}{2}\mathsf{negl}(n)$ and soundness error $\frac{1+b}{2} + \mathsf{negl}(n)$, and thus applies to any $\mathsf{QBBC}(\mathcal{D},\mathcal{R}_F,a,b)$ with constants $0< b<a \leq 1$. Moreover, this proof system has the same query complexities as above. This result is based on the techniques introduced in (Brakerski et al. 2018) and (Mahadev 2018). As an application, we show that the problem of distinguishing the random oracle model (ROM) and the quantum random oracle model (QROM) in cryptography can be naturally seen as a $\mathsf{QBBC}$ problem. By applying the above result, we immediately obtain a separation between ROM and QROM under the standard LWE assumption.
Note: This is a major update of https://eprint.iacr.org/2019/1101 with new results.
Metadata
 Available format(s)
 Category
 Foundations
 Publication info
 Preprint. MINOR revision.
 Keywords
 Quantum ComputationInteractive ProofsROMQROMSeparations
 Contact author(s)

jiangzhang09 @ gmail com
yuyu @ yuyu hk
feng @ tca iscas ac cn
shuqinfan78 @ 163 com
zfzhang @ tca iscas ac cn
yangk @ sklc org  History
 20201119: last of 3 revisions
 20201110: received
 See all versions
 Short URL
 https://ia.cr/2020/1391
 License

CC BY
BibTeX
@misc{cryptoeprint:2020/1391, author = {Jiang Zhang and Yu Yu and Dengguo Feng and Shuqin Fan and Zhenfeng Zhang and Kang Yang}, title = {Interactive Proofs for Quantum BlackBox Computations}, howpublished = {Cryptology ePrint Archive, Paper 2020/1391}, year = {2020}, note = {\url{https://eprint.iacr.org/2020/1391}}, url = {https://eprint.iacr.org/2020/1391} }