Classical Verification of Quantum Computations in Linear Time

In the quantum computation verification problem, a quantum server wants to convince a client that the output of evaluating a quantum circuit C is some result that it claims. This problem is considered very important both theoretically and practically in quantum computation [1], [2], [3]. The client is considered to be limited in computational power, and one desirable property is that the client can be completely classical, which leads to the classical verification of quantum computation (CVQC) problem. In terms of the time complexity of server-side quantum computations (which typically dominate the total time complexity of both the client and the server), the fastest single-server CVQC protocol so far has complexity $O(\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}(\kappa)|C|^{3})$ where $|C|$ is the size of the circuit to be verified and $\kappa$ is the security parameter, given by Mahadev [4]. This leads to a similar cubic time blowup in many existing protocols including multiparty quantum computation, zero knowledge and obfuscation [5], [6], [7], [8], [9], [10]. Considering the preciousness of quantum computation resources, this cubic complexity barrier could be a big obstacle for theoretical and practical development of protocols for these problems.In this work, by developing new techniques, we give a new CVQC protocol with complexity $O(\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}(\kappa)|C|)$ (in terms of the total time complexity of both the client and the server), which is significantly faster than existing protocols. Our protocol is secure in the quantum random oracle model [11] assuming the existence of noisy trapdoor claw-free functions [12], which are both extensively used assumptions in quantum cryptography. Along the way, we also give a new classical channel remote state preparation protocol for states in $\displaystyle \left\{|+\theta\rangle=\frac{1}{\sqrt{2}}(|0\rangle+e^{\mathrm{i}\theta\pi/4}|1\rangle)\ :\ \theta\in\left\{0,1\cdots 7\right\}\right\}$, another basic primitive in quantum cryptography. Our protocol allows for parallel verifiable preparation of L independently random states in this form (up to a constant overall error and a possibly unbounded server-side simulator), and runs in only O(poly($\kappa$)L) time and constant rounds; for comparison, existing works (even for possibly simpler state families) all require very large or unestimated time and round complexities [13], [14], [15], [16].