International Association for Cryptologic Research

International Association
for Cryptologic Research


Paper: (Pseudo) Random Quantum States with Binary Phase

Zvika Brakerski
Omri Shmueli
DOI: 10.1007/978-3-030-36030-6_10
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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 (2t)-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 (2t)-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.
  title={(Pseudo) Random Quantum States with Binary Phase},
  booktitle={Theory of Cryptography},
  series={Lecture Notes in Computer Science},
  author={Zvika Brakerski and Omri Shmueli},