Classical Verification of Quantum Computations

We present the first protocol allowing a classical computer to interactively verify the result of an efficient quantum computation. We achieve this by constructing a measurement protocol, which enables a classical verifier to use a quantum prover as a trusted measurement device. The protocol forces the prover to behave as follows: the prover must construct an n qubit state of his choice, measure each qubit in the Hadamard or standard basis as directed by the verifier, and report the measurement results to the verifier. The soundness of this protocol is enforced based on the assumption that the learning with errors problem is computationally intractable for efficient quantum machines.

[1]  W. Banaszczyk New bounds in some transference theorems in the geometry of numbers , 1993 .

[2]  Umesh V. Vazirani,et al.  A classical leash for a quantum system: command of quantum systems via rigidity of CHSH games , 2012, ITCS '13.

[3]  Chris Peikert,et al.  Trapdoors for Lattices: Simpler, Tighter, Faster, Smaller , 2012, IACR Cryptol. ePrint Arch..

[4]  Elham Kashefi,et al.  Universal Blind Quantum Computation , 2008, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[5]  Tomoyuki Morimae,et al.  Quantum proofs can be verified using only single qubit measurements , 2015, ArXiv.

[6]  Tomoyuki Morimae,et al.  Post hoc verification with a single prover , 2016 .

[7]  Elham Kashefi,et al.  Verification of Quantum Computation: An Overview of Existing Approaches , 2017, Theory of Computing Systems.

[8]  Yael Tauman Kalai,et al.  Robustness of the Learning with Errors Assumption , 2010, ICS.

[9]  Stephan Krenn,et al.  Learning with Rounding, Revisited: New Reduction, Properties and Applications , 2013, IACR Cryptol. ePrint Arch..

[10]  E. Kashefi,et al.  Unconditionally verifiable blind quantum computation , 2012, 1203.5217.

[11]  Brent Waters,et al.  Homomorphic Encryption from Learning with Errors: Conceptually-Simpler, Asymptotically-Faster, Attribute-Based , 2013, CRYPTO.

[12]  Chris Peikert,et al.  Pseudorandomness of ring-LWE for any ring and modulus , 2017, STOC.

[13]  J. Biamonte,et al.  Realizable Hamiltonians for Universal Adiabatic Quantum Computers , 2007, 0704.1287.

[14]  Elad Eban,et al.  Interactive Proofs For Quantum Computations , 2017, 1704.04487.

[15]  Chris Peikert,et al.  Public-key cryptosystems from the worst-case shortest vector problem: extended abstract , 2009, STOC '09.

[16]  Joseph Fitzsimons,et al.  Post hoc verification of quantum computation , 2015, Physical review letters.

[17]  Zvika Brakerski,et al.  Certifiable Randomness from a Single Quantum Device , 2018, ArXiv.

[18]  Oded Regev,et al.  On lattices, learning with errors, random linear codes, and cryptography , 2005, STOC '05.

[19]  Damien Stehlé,et al.  Classical hardness of learning with errors , 2013, STOC '13.

[20]  Urmila Mahadev,et al.  Classical Homomorphic Encryption for Quantum Circuits , 2017, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS).

[21]  Yaoyun Shi Both Toffoli and controlled-NOT need little help to do universal quantum computing , 2003, Quantum Inf. Comput..