# (Pseudo) Random Quantum States with Binary Phase

## Abstract

We prove a quantum information-theoretic conjecture due to Ji, Liu and Song (CRYPTO 2018) which suggested that a uniform superposition with random *binary* phase is statistically indistinguishable from a Haar random state. That is, any polynomial number of copies of the aforementioned state is within exponentially small trace distance from the same number of copies of a Haar random state.

As a consequence, we get a provable elementary construction of *pseudorandom* quantum states from post-quantum pseudorandom functions. Generating pseudorandom quantum states is desirable for physical applications as well as for computational tasks such as quantum money. We observe that replacing the pseudorandom function with a (2*t*)-wise independent function (either in our construction or in previous work), results in an explicit construction for *quantum state t-designs* for all *t*. In fact, we show that the circuit complexity (in terms of both circuit size and depth) of constructing *t*-designs is bounded by that of (2*t*)-wise independent functions. Explicitly, while in prior literature *t*-designs required linear depth (for \(t > 2\)), this observation shows that polylogarithmic depth suffices for all *t*.

We note that our constructions yield pseudorandom states and state designs with only real-valued amplitudes, which was not previously known. Furthermore, generating these states require quantum circuit of restricted form: applying one layer of Hadamard gates, followed by a sequence of Toffoli gates. This structure may be useful for efficiency and simplicity of implementation.

## Notes

### Acknowledgments

We thank Henry Yuen and Vinod Vaikuntanathan for insightful discussions. In particular thanks to Henry for pointing us to the [6] result. We thank the anonymous reviewers for their useful comments. We also thank Aram Harrow for providing advice regarding the state of the art.

## References

- 1.Akavia, A., Bogdanov, A., Guo, S., Kamath, A., Rosen, A.: Candidate weak pseudorandom functions in ac0 mod2. In: Electronic Colloquium on Computational Complexity (ECCC), vol. 21, p. 33 (2014)Google Scholar
- 2.Ambainis, A., Emerson, J.: Quantum t-designs: t-wise independence in the quantum world. In: Twenty-Second Annual IEEE Conference on Computational Complexity (CCC 2007), pp. 129–140. IEEE (2007)Google Scholar
- 3.Brakerski, Z., Shmueli, O.: (pseudo) random quantum states with binary phase. CoRR, abs/1906.10611 (2019)Google Scholar
- 4.Dankert, C., Cleve, R., Emerson, J., Livine, E.: Exact and approximate unitary 2-designs and their application to fidelity estimation. Phys. Rev. A
**80**(1), 012304 (2009)CrossRefGoogle Scholar - 5.Harrow, A.W., Low, R.A.: Random quantum circuits are approximate 2-designs. Commun. Math. Phys.
**291**(1), 257–302 (2009)MathSciNetCrossRefGoogle Scholar - 6.Ji, Z., Liu, Y.-K., Song, F.: Pseudorandom quantum states. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10993, pp. 126–152. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96878-0_5CrossRefGoogle Scholar
- 7.Kueng, R., Gross, D.: Qubit stabilizer states are complex projective 3-designs. arXiv preprint arXiv:1510.02767 (2015)
- 8.Lloyd, S.: Capacity of the noisy quantum channel. Phys. Rev. A
**55**(3), 1613 (1997)MathSciNetCrossRefGoogle Scholar - 9.Nakata, Y., Koashi, M., Murao, M.: Generating a state t-design by diagonal quantum circuits. New J. Phys.
**16**(5), 053043 (2014)CrossRefGoogle Scholar - 10.Nakata, Y., Murao, M.: Diagonal-unitary 2-design and their implementations by quantum circuits. Int. J. Quantum Inf.
**11**(07), 1350062 (2013)MathSciNetCrossRefGoogle Scholar - 11.Naor, M., Reingold, O.: Synthesizers and their application to the parallel construction of pseudo-random functions. J. Comput. Syst. Sci.
**58**(2), 336–375 (1999)MathSciNetCrossRefGoogle Scholar - 12.Nest, M.: Classical simulation of quantum computation, the gottesman-knill theorem, and slightly beyond. arXiv preprint arXiv:0811.0898 (2008)
- 13.Popescu, S., Short, A.J., Winter, A.: Entanglement and the foundations of statistical mechanics. Nat. Phys.
**2**(11), 754 (2006)CrossRefGoogle Scholar - 14.Renes, J.M., Blume-Kohout, R., Scott, A.J., Caves, C.M.: Symmetric informationally complete quantum measurements. J. Math. Phys.
**45**(6), 2171–2180 (2004)MathSciNetCrossRefGoogle Scholar - 15.Zhandry, M.: How to construct quantum random functions. In: 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, pp. 679–687. IEEE (2012)Google Scholar